diff --git a/.gitattributes b/.gitattributes index d0915896a23098a7f6421e1f6c7e65c7f8abb036..76dd128b27a638c52c9c3bdeb3a5d885d201afaa 100644 --- a/.gitattributes +++ b/.gitattributes @@ -704,3 +704,5 @@ deepseek/lib/python3.10/site-packages/numpy/core/__pycache__/_add_newdocs.cpytho deepseekvl2/lib/python3.10/site-packages/torch/_inductor/__pycache__/ir.cpython-310.pyc filter=lfs diff=lfs merge=lfs -text infer_4_33_0/lib/python3.10/site-packages/aiohttp/_websocket/reader_c.cpython-310-x86_64-linux-gnu.so filter=lfs diff=lfs merge=lfs -text deepseekvl2/lib/python3.10/site-packages/regex/__pycache__/test_regex.cpython-310.pyc filter=lfs diff=lfs merge=lfs -text +falcon/lib/python3.10/site-packages/nvidia/cusolver/lib/libcusolver.so.11 filter=lfs diff=lfs merge=lfs -text +deepseekvl2/lib/python3.10/site-packages/numpy/lib/tests/__pycache__/test_function_base.cpython-310.pyc filter=lfs diff=lfs merge=lfs -text diff --git a/deepseekvl2/lib/python3.10/site-packages/numpy/lib/tests/__pycache__/test_function_base.cpython-310.pyc b/deepseekvl2/lib/python3.10/site-packages/numpy/lib/tests/__pycache__/test_function_base.cpython-310.pyc new file mode 100644 index 0000000000000000000000000000000000000000..39f44d3657d257afe40162ab52128461622ee952 --- /dev/null +++ b/deepseekvl2/lib/python3.10/site-packages/numpy/lib/tests/__pycache__/test_function_base.cpython-310.pyc @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:aceb3f531fe0d2d5845bffb8cd81a939a3b88f59b7a5ef8d08622a4e5e2181d3 +size 143252 diff --git a/deepseekvl2/lib/python3.10/site-packages/numpy/lib/tests/data/python3.npy b/deepseekvl2/lib/python3.10/site-packages/numpy/lib/tests/data/python3.npy new file mode 100644 index 0000000000000000000000000000000000000000..1de3124ee4eaf42cfc3625cdfa88767de07df9c4 --- /dev/null +++ b/deepseekvl2/lib/python3.10/site-packages/numpy/lib/tests/data/python3.npy @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:5f469dde101a2c65e283d2ed487028f8180ebcb9457cf60c9d9b952314668fed +size 96 diff --git a/deepseekvl2/lib/python3.10/site-packages/numpy/polynomial/__init__.pyi b/deepseekvl2/lib/python3.10/site-packages/numpy/polynomial/__init__.pyi new file mode 100644 index 0000000000000000000000000000000000000000..c9d1c27a96c2d8ccfeb9e378a2599c2e70003ee4 --- /dev/null +++ b/deepseekvl2/lib/python3.10/site-packages/numpy/polynomial/__init__.pyi @@ -0,0 +1,22 @@ +from numpy._pytesttester import PytestTester + +from numpy.polynomial import ( + chebyshev as chebyshev, + hermite as hermite, + hermite_e as hermite_e, + laguerre as laguerre, + legendre as legendre, + polynomial as polynomial, +) +from numpy.polynomial.chebyshev import Chebyshev as Chebyshev +from numpy.polynomial.hermite import Hermite as Hermite +from numpy.polynomial.hermite_e import HermiteE as HermiteE +from numpy.polynomial.laguerre import Laguerre as Laguerre +from numpy.polynomial.legendre import Legendre as Legendre +from numpy.polynomial.polynomial import Polynomial as Polynomial + +__all__: list[str] +__path__: list[str] +test: PytestTester + +def set_default_printstyle(style): ... diff --git a/deepseekvl2/lib/python3.10/site-packages/numpy/polynomial/__pycache__/_polybase.cpython-310.pyc b/deepseekvl2/lib/python3.10/site-packages/numpy/polynomial/__pycache__/_polybase.cpython-310.pyc new file mode 100644 index 0000000000000000000000000000000000000000..dcabdc692c21a60bdaba017900ae71c3d097b8e7 Binary files /dev/null and b/deepseekvl2/lib/python3.10/site-packages/numpy/polynomial/__pycache__/_polybase.cpython-310.pyc differ diff --git a/deepseekvl2/lib/python3.10/site-packages/numpy/polynomial/__pycache__/hermite.cpython-310.pyc b/deepseekvl2/lib/python3.10/site-packages/numpy/polynomial/__pycache__/hermite.cpython-310.pyc new file mode 100644 index 0000000000000000000000000000000000000000..9a25067e57b8e461517a78fe6e4374cbdf160da2 Binary files /dev/null and b/deepseekvl2/lib/python3.10/site-packages/numpy/polynomial/__pycache__/hermite.cpython-310.pyc differ diff --git a/deepseekvl2/lib/python3.10/site-packages/numpy/polynomial/__pycache__/laguerre.cpython-310.pyc b/deepseekvl2/lib/python3.10/site-packages/numpy/polynomial/__pycache__/laguerre.cpython-310.pyc new file mode 100644 index 0000000000000000000000000000000000000000..d16e24abc23c97f26a809ca0c5a33ba11d57dd24 Binary files /dev/null and b/deepseekvl2/lib/python3.10/site-packages/numpy/polynomial/__pycache__/laguerre.cpython-310.pyc differ diff --git a/deepseekvl2/lib/python3.10/site-packages/numpy/polynomial/__pycache__/polynomial.cpython-310.pyc b/deepseekvl2/lib/python3.10/site-packages/numpy/polynomial/__pycache__/polynomial.cpython-310.pyc new file mode 100644 index 0000000000000000000000000000000000000000..43222f85cf86733d7f2d30d46a46e0df858e16ae Binary files /dev/null and b/deepseekvl2/lib/python3.10/site-packages/numpy/polynomial/__pycache__/polynomial.cpython-310.pyc differ diff --git a/deepseekvl2/lib/python3.10/site-packages/numpy/polynomial/__pycache__/polyutils.cpython-310.pyc b/deepseekvl2/lib/python3.10/site-packages/numpy/polynomial/__pycache__/polyutils.cpython-310.pyc new file mode 100644 index 0000000000000000000000000000000000000000..a66a07b7283aa913a9b3bf5304f99e5e49968b66 Binary files /dev/null and b/deepseekvl2/lib/python3.10/site-packages/numpy/polynomial/__pycache__/polyutils.cpython-310.pyc differ diff --git a/deepseekvl2/lib/python3.10/site-packages/numpy/polynomial/_polybase.py b/deepseekvl2/lib/python3.10/site-packages/numpy/polynomial/_polybase.py new file mode 100644 index 0000000000000000000000000000000000000000..9730574cf22e22823aaa0c77be9e630425cb2f79 --- /dev/null +++ b/deepseekvl2/lib/python3.10/site-packages/numpy/polynomial/_polybase.py @@ -0,0 +1,1206 @@ +""" +Abstract base class for the various polynomial Classes. + +The ABCPolyBase class provides the methods needed to implement the common API +for the various polynomial classes. It operates as a mixin, but uses the +abc module from the stdlib, hence it is only available for Python >= 2.6. + +""" +import os +import abc +import numbers + +import numpy as np +from . import polyutils as pu + +__all__ = ['ABCPolyBase'] + +class ABCPolyBase(abc.ABC): + """An abstract base class for immutable series classes. + + ABCPolyBase provides the standard Python numerical methods + '+', '-', '*', '//', '%', 'divmod', '**', and '()' along with the + methods listed below. + + .. versionadded:: 1.9.0 + + Parameters + ---------- + coef : array_like + Series coefficients in order of increasing degree, i.e., + ``(1, 2, 3)`` gives ``1*P_0(x) + 2*P_1(x) + 3*P_2(x)``, where + ``P_i`` is the basis polynomials of degree ``i``. + domain : (2,) array_like, optional + Domain to use. The interval ``[domain[0], domain[1]]`` is mapped + to the interval ``[window[0], window[1]]`` by shifting and scaling. + The default value is the derived class domain. + window : (2,) array_like, optional + Window, see domain for its use. The default value is the + derived class window. + symbol : str, optional + Symbol used to represent the independent variable in string + representations of the polynomial expression, e.g. for printing. + The symbol must be a valid Python identifier. Default value is 'x'. + + .. versionadded:: 1.24 + + Attributes + ---------- + coef : (N,) ndarray + Series coefficients in order of increasing degree. + domain : (2,) ndarray + Domain that is mapped to window. + window : (2,) ndarray + Window that domain is mapped to. + symbol : str + Symbol representing the independent variable. + + Class Attributes + ---------------- + maxpower : int + Maximum power allowed, i.e., the largest number ``n`` such that + ``p(x)**n`` is allowed. This is to limit runaway polynomial size. + domain : (2,) ndarray + Default domain of the class. + window : (2,) ndarray + Default window of the class. + + """ + + # Not hashable + __hash__ = None + + # Opt out of numpy ufuncs and Python ops with ndarray subclasses. + __array_ufunc__ = None + + # Limit runaway size. T_n^m has degree n*m + maxpower = 100 + + # Unicode character mappings for improved __str__ + _superscript_mapping = str.maketrans({ + "0": "⁰", + "1": "¹", + "2": "²", + "3": "³", + "4": "⁴", + "5": "⁵", + "6": "⁶", + "7": "⁷", + "8": "⁸", + "9": "⁹" + }) + _subscript_mapping = str.maketrans({ + "0": "₀", + "1": "₁", + "2": "₂", + "3": "₃", + "4": "₄", + "5": "₅", + "6": "₆", + "7": "₇", + "8": "₈", + "9": "₉" + }) + # Some fonts don't support full unicode character ranges necessary for + # the full set of superscripts and subscripts, including common/default + # fonts in Windows shells/terminals. Therefore, default to ascii-only + # printing on windows. + _use_unicode = not os.name == 'nt' + + @property + def symbol(self): + return self._symbol + + @property + @abc.abstractmethod + def domain(self): + pass + + @property + @abc.abstractmethod + def window(self): + pass + + @property + @abc.abstractmethod + def basis_name(self): + pass + + @staticmethod + @abc.abstractmethod + def _add(c1, c2): + pass + + @staticmethod + @abc.abstractmethod + def _sub(c1, c2): + pass + + @staticmethod + @abc.abstractmethod + def _mul(c1, c2): + pass + + @staticmethod + @abc.abstractmethod + def _div(c1, c2): + pass + + @staticmethod + @abc.abstractmethod + def _pow(c, pow, maxpower=None): + pass + + @staticmethod + @abc.abstractmethod + def _val(x, c): + pass + + @staticmethod + @abc.abstractmethod + def _int(c, m, k, lbnd, scl): + pass + + @staticmethod + @abc.abstractmethod + def _der(c, m, scl): + pass + + @staticmethod + @abc.abstractmethod + def _fit(x, y, deg, rcond, full): + pass + + @staticmethod + @abc.abstractmethod + def _line(off, scl): + pass + + @staticmethod + @abc.abstractmethod + def _roots(c): + pass + + @staticmethod + @abc.abstractmethod + def _fromroots(r): + pass + + def has_samecoef(self, other): + """Check if coefficients match. + + .. versionadded:: 1.6.0 + + Parameters + ---------- + other : class instance + The other class must have the ``coef`` attribute. + + Returns + ------- + bool : boolean + True if the coefficients are the same, False otherwise. + + """ + if len(self.coef) != len(other.coef): + return False + elif not np.all(self.coef == other.coef): + return False + else: + return True + + def has_samedomain(self, other): + """Check if domains match. + + .. versionadded:: 1.6.0 + + Parameters + ---------- + other : class instance + The other class must have the ``domain`` attribute. + + Returns + ------- + bool : boolean + True if the domains are the same, False otherwise. + + """ + return np.all(self.domain == other.domain) + + def has_samewindow(self, other): + """Check if windows match. + + .. versionadded:: 1.6.0 + + Parameters + ---------- + other : class instance + The other class must have the ``window`` attribute. + + Returns + ------- + bool : boolean + True if the windows are the same, False otherwise. + + """ + return np.all(self.window == other.window) + + def has_sametype(self, other): + """Check if types match. + + .. versionadded:: 1.7.0 + + Parameters + ---------- + other : object + Class instance. + + Returns + ------- + bool : boolean + True if other is same class as self + + """ + return isinstance(other, self.__class__) + + def _get_coefficients(self, other): + """Interpret other as polynomial coefficients. + + The `other` argument is checked to see if it is of the same + class as self with identical domain and window. If so, + return its coefficients, otherwise return `other`. + + .. versionadded:: 1.9.0 + + Parameters + ---------- + other : anything + Object to be checked. + + Returns + ------- + coef + The coefficients of`other` if it is a compatible instance, + of ABCPolyBase, otherwise `other`. + + Raises + ------ + TypeError + When `other` is an incompatible instance of ABCPolyBase. + + """ + if isinstance(other, ABCPolyBase): + if not isinstance(other, self.__class__): + raise TypeError("Polynomial types differ") + elif not np.all(self.domain == other.domain): + raise TypeError("Domains differ") + elif not np.all(self.window == other.window): + raise TypeError("Windows differ") + elif self.symbol != other.symbol: + raise ValueError("Polynomial symbols differ") + return other.coef + return other + + def __init__(self, coef, domain=None, window=None, symbol='x'): + [coef] = pu.as_series([coef], trim=False) + self.coef = coef + + if domain is not None: + [domain] = pu.as_series([domain], trim=False) + if len(domain) != 2: + raise ValueError("Domain has wrong number of elements.") + self.domain = domain + + if window is not None: + [window] = pu.as_series([window], trim=False) + if len(window) != 2: + raise ValueError("Window has wrong number of elements.") + self.window = window + + # Validation for symbol + try: + if not symbol.isidentifier(): + raise ValueError( + "Symbol string must be a valid Python identifier" + ) + # If a user passes in something other than a string, the above + # results in an AttributeError. Catch this and raise a more + # informative exception + except AttributeError: + raise TypeError("Symbol must be a non-empty string") + + self._symbol = symbol + + def __repr__(self): + coef = repr(self.coef)[6:-1] + domain = repr(self.domain)[6:-1] + window = repr(self.window)[6:-1] + name = self.__class__.__name__ + return (f"{name}({coef}, domain={domain}, window={window}, " + f"symbol='{self.symbol}')") + + def __format__(self, fmt_str): + if fmt_str == '': + return self.__str__() + if fmt_str not in ('ascii', 'unicode'): + raise ValueError( + f"Unsupported format string '{fmt_str}' passed to " + f"{self.__class__}.__format__. Valid options are " + f"'ascii' and 'unicode'" + ) + if fmt_str == 'ascii': + return self._generate_string(self._str_term_ascii) + return self._generate_string(self._str_term_unicode) + + def __str__(self): + if self._use_unicode: + return self._generate_string(self._str_term_unicode) + return self._generate_string(self._str_term_ascii) + + def _generate_string(self, term_method): + """ + Generate the full string representation of the polynomial, using + ``term_method`` to generate each polynomial term. + """ + # Get configuration for line breaks + linewidth = np.get_printoptions().get('linewidth', 75) + if linewidth < 1: + linewidth = 1 + out = pu.format_float(self.coef[0]) + for i, coef in enumerate(self.coef[1:]): + out += " " + power = str(i + 1) + # Polynomial coefficient + # The coefficient array can be an object array with elements that + # will raise a TypeError with >= 0 (e.g. strings or Python + # complex). In this case, represent the coefficient as-is. + try: + if coef >= 0: + next_term = f"+ " + pu.format_float(coef, parens=True) + else: + next_term = f"- " + pu.format_float(-coef, parens=True) + except TypeError: + next_term = f"+ {coef}" + # Polynomial term + next_term += term_method(power, self.symbol) + # Length of the current line with next term added + line_len = len(out.split('\n')[-1]) + len(next_term) + # If not the last term in the polynomial, it will be two + # characters longer due to the +/- with the next term + if i < len(self.coef[1:]) - 1: + line_len += 2 + # Handle linebreaking + if line_len >= linewidth: + next_term = next_term.replace(" ", "\n", 1) + out += next_term + return out + + @classmethod + def _str_term_unicode(cls, i, arg_str): + """ + String representation of single polynomial term using unicode + characters for superscripts and subscripts. + """ + if cls.basis_name is None: + raise NotImplementedError( + "Subclasses must define either a basis_name, or override " + "_str_term_unicode(cls, i, arg_str)" + ) + return (f"·{cls.basis_name}{i.translate(cls._subscript_mapping)}" + f"({arg_str})") + + @classmethod + def _str_term_ascii(cls, i, arg_str): + """ + String representation of a single polynomial term using ** and _ to + represent superscripts and subscripts, respectively. + """ + if cls.basis_name is None: + raise NotImplementedError( + "Subclasses must define either a basis_name, or override " + "_str_term_ascii(cls, i, arg_str)" + ) + return f" {cls.basis_name}_{i}({arg_str})" + + @classmethod + def _repr_latex_term(cls, i, arg_str, needs_parens): + if cls.basis_name is None: + raise NotImplementedError( + "Subclasses must define either a basis name, or override " + "_repr_latex_term(i, arg_str, needs_parens)") + # since we always add parens, we don't care if the expression needs them + return f"{{{cls.basis_name}}}_{{{i}}}({arg_str})" + + @staticmethod + def _repr_latex_scalar(x, parens=False): + # TODO: we're stuck with disabling math formatting until we handle + # exponents in this function + return r'\text{{{}}}'.format(pu.format_float(x, parens=parens)) + + def _repr_latex_(self): + # get the scaled argument string to the basis functions + off, scale = self.mapparms() + if off == 0 and scale == 1: + term = self.symbol + needs_parens = False + elif scale == 1: + term = f"{self._repr_latex_scalar(off)} + {self.symbol}" + needs_parens = True + elif off == 0: + term = f"{self._repr_latex_scalar(scale)}{self.symbol}" + needs_parens = True + else: + term = ( + f"{self._repr_latex_scalar(off)} + " + f"{self._repr_latex_scalar(scale)}{self.symbol}" + ) + needs_parens = True + + mute = r"\color{{LightGray}}{{{}}}".format + + parts = [] + for i, c in enumerate(self.coef): + # prevent duplication of + and - signs + if i == 0: + coef_str = f"{self._repr_latex_scalar(c)}" + elif not isinstance(c, numbers.Real): + coef_str = f" + ({self._repr_latex_scalar(c)})" + elif not np.signbit(c): + coef_str = f" + {self._repr_latex_scalar(c, parens=True)}" + else: + coef_str = f" - {self._repr_latex_scalar(-c, parens=True)}" + + # produce the string for the term + term_str = self._repr_latex_term(i, term, needs_parens) + if term_str == '1': + part = coef_str + else: + part = rf"{coef_str}\,{term_str}" + + if c == 0: + part = mute(part) + + parts.append(part) + + if parts: + body = ''.join(parts) + else: + # in case somehow there are no coefficients at all + body = '0' + + return rf"${self.symbol} \mapsto {body}$" + + + + # Pickle and copy + + def __getstate__(self): + ret = self.__dict__.copy() + ret['coef'] = self.coef.copy() + ret['domain'] = self.domain.copy() + ret['window'] = self.window.copy() + ret['symbol'] = self.symbol + return ret + + def __setstate__(self, dict): + self.__dict__ = dict + + # Call + + def __call__(self, arg): + off, scl = pu.mapparms(self.domain, self.window) + arg = off + scl*arg + return self._val(arg, self.coef) + + def __iter__(self): + return iter(self.coef) + + def __len__(self): + return len(self.coef) + + # Numeric properties. + + def __neg__(self): + return self.__class__( + -self.coef, self.domain, self.window, self.symbol + ) + + def __pos__(self): + return self + + def __add__(self, other): + othercoef = self._get_coefficients(other) + try: + coef = self._add(self.coef, othercoef) + except Exception: + return NotImplemented + return self.__class__(coef, self.domain, self.window, self.symbol) + + def __sub__(self, other): + othercoef = self._get_coefficients(other) + try: + coef = self._sub(self.coef, othercoef) + except Exception: + return NotImplemented + return self.__class__(coef, self.domain, self.window, self.symbol) + + def __mul__(self, other): + othercoef = self._get_coefficients(other) + try: + coef = self._mul(self.coef, othercoef) + except Exception: + return NotImplemented + return self.__class__(coef, self.domain, self.window, self.symbol) + + def __truediv__(self, other): + # there is no true divide if the rhs is not a Number, although it + # could return the first n elements of an infinite series. + # It is hard to see where n would come from, though. + if not isinstance(other, numbers.Number) or isinstance(other, bool): + raise TypeError( + f"unsupported types for true division: " + f"'{type(self)}', '{type(other)}'" + ) + return self.__floordiv__(other) + + def __floordiv__(self, other): + res = self.__divmod__(other) + if res is NotImplemented: + return res + return res[0] + + def __mod__(self, other): + res = self.__divmod__(other) + if res is NotImplemented: + return res + return res[1] + + def __divmod__(self, other): + othercoef = self._get_coefficients(other) + try: + quo, rem = self._div(self.coef, othercoef) + except ZeroDivisionError: + raise + except Exception: + return NotImplemented + quo = self.__class__(quo, self.domain, self.window, self.symbol) + rem = self.__class__(rem, self.domain, self.window, self.symbol) + return quo, rem + + def __pow__(self, other): + coef = self._pow(self.coef, other, maxpower=self.maxpower) + res = self.__class__(coef, self.domain, self.window, self.symbol) + return res + + def __radd__(self, other): + try: + coef = self._add(other, self.coef) + except Exception: + return NotImplemented + return self.__class__(coef, self.domain, self.window, self.symbol) + + def __rsub__(self, other): + try: + coef = self._sub(other, self.coef) + except Exception: + return NotImplemented + return self.__class__(coef, self.domain, self.window, self.symbol) + + def __rmul__(self, other): + try: + coef = self._mul(other, self.coef) + except Exception: + return NotImplemented + return self.__class__(coef, self.domain, self.window, self.symbol) + + def __rdiv__(self, other): + # set to __floordiv__ /. + return self.__rfloordiv__(other) + + def __rtruediv__(self, other): + # An instance of ABCPolyBase is not considered a + # Number. + return NotImplemented + + def __rfloordiv__(self, other): + res = self.__rdivmod__(other) + if res is NotImplemented: + return res + return res[0] + + def __rmod__(self, other): + res = self.__rdivmod__(other) + if res is NotImplemented: + return res + return res[1] + + def __rdivmod__(self, other): + try: + quo, rem = self._div(other, self.coef) + except ZeroDivisionError: + raise + except Exception: + return NotImplemented + quo = self.__class__(quo, self.domain, self.window, self.symbol) + rem = self.__class__(rem, self.domain, self.window, self.symbol) + return quo, rem + + def __eq__(self, other): + res = (isinstance(other, self.__class__) and + np.all(self.domain == other.domain) and + np.all(self.window == other.window) and + (self.coef.shape == other.coef.shape) and + np.all(self.coef == other.coef) and + (self.symbol == other.symbol)) + return res + + def __ne__(self, other): + return not self.__eq__(other) + + # + # Extra methods. + # + + def copy(self): + """Return a copy. + + Returns + ------- + new_series : series + Copy of self. + + """ + return self.__class__(self.coef, self.domain, self.window, self.symbol) + + def degree(self): + """The degree of the series. + + .. versionadded:: 1.5.0 + + Returns + ------- + degree : int + Degree of the series, one less than the number of coefficients. + + Examples + -------- + + Create a polynomial object for ``1 + 7*x + 4*x**2``: + + >>> poly = np.polynomial.Polynomial([1, 7, 4]) + >>> print(poly) + 1.0 + 7.0·x + 4.0·x² + >>> poly.degree() + 2 + + Note that this method does not check for non-zero coefficients. + You must trim the polynomial to remove any trailing zeroes: + + >>> poly = np.polynomial.Polynomial([1, 7, 0]) + >>> print(poly) + 1.0 + 7.0·x + 0.0·x² + >>> poly.degree() + 2 + >>> poly.trim().degree() + 1 + + """ + return len(self) - 1 + + def cutdeg(self, deg): + """Truncate series to the given degree. + + Reduce the degree of the series to `deg` by discarding the + high order terms. If `deg` is greater than the current degree a + copy of the current series is returned. This can be useful in least + squares where the coefficients of the high degree terms may be very + small. + + .. versionadded:: 1.5.0 + + Parameters + ---------- + deg : non-negative int + The series is reduced to degree `deg` by discarding the high + order terms. The value of `deg` must be a non-negative integer. + + Returns + ------- + new_series : series + New instance of series with reduced degree. + + """ + return self.truncate(deg + 1) + + def trim(self, tol=0): + """Remove trailing coefficients + + Remove trailing coefficients until a coefficient is reached whose + absolute value greater than `tol` or the beginning of the series is + reached. If all the coefficients would be removed the series is set + to ``[0]``. A new series instance is returned with the new + coefficients. The current instance remains unchanged. + + Parameters + ---------- + tol : non-negative number. + All trailing coefficients less than `tol` will be removed. + + Returns + ------- + new_series : series + New instance of series with trimmed coefficients. + + """ + coef = pu.trimcoef(self.coef, tol) + return self.__class__(coef, self.domain, self.window, self.symbol) + + def truncate(self, size): + """Truncate series to length `size`. + + Reduce the series to length `size` by discarding the high + degree terms. The value of `size` must be a positive integer. This + can be useful in least squares where the coefficients of the + high degree terms may be very small. + + Parameters + ---------- + size : positive int + The series is reduced to length `size` by discarding the high + degree terms. The value of `size` must be a positive integer. + + Returns + ------- + new_series : series + New instance of series with truncated coefficients. + + """ + isize = int(size) + if isize != size or isize < 1: + raise ValueError("size must be a positive integer") + if isize >= len(self.coef): + coef = self.coef + else: + coef = self.coef[:isize] + return self.__class__(coef, self.domain, self.window, self.symbol) + + def convert(self, domain=None, kind=None, window=None): + """Convert series to a different kind and/or domain and/or window. + + Parameters + ---------- + domain : array_like, optional + The domain of the converted series. If the value is None, + the default domain of `kind` is used. + kind : class, optional + The polynomial series type class to which the current instance + should be converted. If kind is None, then the class of the + current instance is used. + window : array_like, optional + The window of the converted series. If the value is None, + the default window of `kind` is used. + + Returns + ------- + new_series : series + The returned class can be of different type than the current + instance and/or have a different domain and/or different + window. + + Notes + ----- + Conversion between domains and class types can result in + numerically ill defined series. + + """ + if kind is None: + kind = self.__class__ + if domain is None: + domain = kind.domain + if window is None: + window = kind.window + return self(kind.identity(domain, window=window, symbol=self.symbol)) + + def mapparms(self): + """Return the mapping parameters. + + The returned values define a linear map ``off + scl*x`` that is + applied to the input arguments before the series is evaluated. The + map depends on the ``domain`` and ``window``; if the current + ``domain`` is equal to the ``window`` the resulting map is the + identity. If the coefficients of the series instance are to be + used by themselves outside this class, then the linear function + must be substituted for the ``x`` in the standard representation of + the base polynomials. + + Returns + ------- + off, scl : float or complex + The mapping function is defined by ``off + scl*x``. + + Notes + ----- + If the current domain is the interval ``[l1, r1]`` and the window + is ``[l2, r2]``, then the linear mapping function ``L`` is + defined by the equations:: + + L(l1) = l2 + L(r1) = r2 + + """ + return pu.mapparms(self.domain, self.window) + + def integ(self, m=1, k=[], lbnd=None): + """Integrate. + + Return a series instance that is the definite integral of the + current series. + + Parameters + ---------- + m : non-negative int + The number of integrations to perform. + k : array_like + Integration constants. The first constant is applied to the + first integration, the second to the second, and so on. The + list of values must less than or equal to `m` in length and any + missing values are set to zero. + lbnd : Scalar + The lower bound of the definite integral. + + Returns + ------- + new_series : series + A new series representing the integral. The domain is the same + as the domain of the integrated series. + + """ + off, scl = self.mapparms() + if lbnd is None: + lbnd = 0 + else: + lbnd = off + scl*lbnd + coef = self._int(self.coef, m, k, lbnd, 1./scl) + return self.__class__(coef, self.domain, self.window, self.symbol) + + def deriv(self, m=1): + """Differentiate. + + Return a series instance of that is the derivative of the current + series. + + Parameters + ---------- + m : non-negative int + Find the derivative of order `m`. + + Returns + ------- + new_series : series + A new series representing the derivative. The domain is the same + as the domain of the differentiated series. + + """ + off, scl = self.mapparms() + coef = self._der(self.coef, m, scl) + return self.__class__(coef, self.domain, self.window, self.symbol) + + def roots(self): + """Return the roots of the series polynomial. + + Compute the roots for the series. Note that the accuracy of the + roots decreases the further outside the `domain` they lie. + + Returns + ------- + roots : ndarray + Array containing the roots of the series. + + """ + roots = self._roots(self.coef) + return pu.mapdomain(roots, self.window, self.domain) + + def linspace(self, n=100, domain=None): + """Return x, y values at equally spaced points in domain. + + Returns the x, y values at `n` linearly spaced points across the + domain. Here y is the value of the polynomial at the points x. By + default the domain is the same as that of the series instance. + This method is intended mostly as a plotting aid. + + .. versionadded:: 1.5.0 + + Parameters + ---------- + n : int, optional + Number of point pairs to return. The default value is 100. + domain : {None, array_like}, optional + If not None, the specified domain is used instead of that of + the calling instance. It should be of the form ``[beg,end]``. + The default is None which case the class domain is used. + + Returns + ------- + x, y : ndarray + x is equal to linspace(self.domain[0], self.domain[1], n) and + y is the series evaluated at element of x. + + """ + if domain is None: + domain = self.domain + x = np.linspace(domain[0], domain[1], n) + y = self(x) + return x, y + + @classmethod + def fit(cls, x, y, deg, domain=None, rcond=None, full=False, w=None, + window=None, symbol='x'): + """Least squares fit to data. + + Return a series instance that is the least squares fit to the data + `y` sampled at `x`. The domain of the returned instance can be + specified and this will often result in a superior fit with less + chance of ill conditioning. + + Parameters + ---------- + x : array_like, shape (M,) + x-coordinates of the M sample points ``(x[i], y[i])``. + y : array_like, shape (M,) + y-coordinates of the M sample points ``(x[i], y[i])``. + deg : int or 1-D array_like + Degree(s) of the fitting polynomials. If `deg` is a single integer + all terms up to and including the `deg`'th term are included in the + fit. For NumPy versions >= 1.11.0 a list of integers specifying the + degrees of the terms to include may be used instead. + domain : {None, [beg, end], []}, optional + Domain to use for the returned series. If ``None``, + then a minimal domain that covers the points `x` is chosen. If + ``[]`` the class domain is used. The default value was the + class domain in NumPy 1.4 and ``None`` in later versions. + The ``[]`` option was added in numpy 1.5.0. + rcond : float, optional + Relative condition number of the fit. Singular values smaller + than this relative to the largest singular value will be + ignored. The default value is len(x)*eps, where eps is the + relative precision of the float type, about 2e-16 in most + cases. + full : bool, optional + Switch determining nature of return value. When it is False + (the default) just the coefficients are returned, when True + diagnostic information from the singular value decomposition is + also returned. + w : array_like, shape (M,), optional + Weights. If not None, the weight ``w[i]`` applies to the unsquared + residual ``y[i] - y_hat[i]`` at ``x[i]``. Ideally the weights are + chosen so that the errors of the products ``w[i]*y[i]`` all have + the same variance. When using inverse-variance weighting, use + ``w[i] = 1/sigma(y[i])``. The default value is None. + + .. versionadded:: 1.5.0 + window : {[beg, end]}, optional + Window to use for the returned series. The default + value is the default class domain + + .. versionadded:: 1.6.0 + symbol : str, optional + Symbol representing the independent variable. Default is 'x'. + + Returns + ------- + new_series : series + A series that represents the least squares fit to the data and + has the domain and window specified in the call. If the + coefficients for the unscaled and unshifted basis polynomials are + of interest, do ``new_series.convert().coef``. + + [resid, rank, sv, rcond] : list + These values are only returned if ``full == True`` + + - resid -- sum of squared residuals of the least squares fit + - rank -- the numerical rank of the scaled Vandermonde matrix + - sv -- singular values of the scaled Vandermonde matrix + - rcond -- value of `rcond`. + + For more details, see `linalg.lstsq`. + + """ + if domain is None: + domain = pu.getdomain(x) + elif type(domain) is list and len(domain) == 0: + domain = cls.domain + + if window is None: + window = cls.window + + xnew = pu.mapdomain(x, domain, window) + res = cls._fit(xnew, y, deg, w=w, rcond=rcond, full=full) + if full: + [coef, status] = res + return ( + cls(coef, domain=domain, window=window, symbol=symbol), status + ) + else: + coef = res + return cls(coef, domain=domain, window=window, symbol=symbol) + + @classmethod + def fromroots(cls, roots, domain=[], window=None, symbol='x'): + """Return series instance that has the specified roots. + + Returns a series representing the product + ``(x - r[0])*(x - r[1])*...*(x - r[n-1])``, where ``r`` is a + list of roots. + + Parameters + ---------- + roots : array_like + List of roots. + domain : {[], None, array_like}, optional + Domain for the resulting series. If None the domain is the + interval from the smallest root to the largest. If [] the + domain is the class domain. The default is []. + window : {None, array_like}, optional + Window for the returned series. If None the class window is + used. The default is None. + symbol : str, optional + Symbol representing the independent variable. Default is 'x'. + + Returns + ------- + new_series : series + Series with the specified roots. + + """ + [roots] = pu.as_series([roots], trim=False) + if domain is None: + domain = pu.getdomain(roots) + elif type(domain) is list and len(domain) == 0: + domain = cls.domain + + if window is None: + window = cls.window + + deg = len(roots) + off, scl = pu.mapparms(domain, window) + rnew = off + scl*roots + coef = cls._fromroots(rnew) / scl**deg + return cls(coef, domain=domain, window=window, symbol=symbol) + + @classmethod + def identity(cls, domain=None, window=None, symbol='x'): + """Identity function. + + If ``p`` is the returned series, then ``p(x) == x`` for all + values of x. + + Parameters + ---------- + domain : {None, array_like}, optional + If given, the array must be of the form ``[beg, end]``, where + ``beg`` and ``end`` are the endpoints of the domain. If None is + given then the class domain is used. The default is None. + window : {None, array_like}, optional + If given, the resulting array must be if the form + ``[beg, end]``, where ``beg`` and ``end`` are the endpoints of + the window. If None is given then the class window is used. The + default is None. + symbol : str, optional + Symbol representing the independent variable. Default is 'x'. + + Returns + ------- + new_series : series + Series of representing the identity. + + """ + if domain is None: + domain = cls.domain + if window is None: + window = cls.window + off, scl = pu.mapparms(window, domain) + coef = cls._line(off, scl) + return cls(coef, domain, window, symbol) + + @classmethod + def basis(cls, deg, domain=None, window=None, symbol='x'): + """Series basis polynomial of degree `deg`. + + Returns the series representing the basis polynomial of degree `deg`. + + .. versionadded:: 1.7.0 + + Parameters + ---------- + deg : int + Degree of the basis polynomial for the series. Must be >= 0. + domain : {None, array_like}, optional + If given, the array must be of the form ``[beg, end]``, where + ``beg`` and ``end`` are the endpoints of the domain. If None is + given then the class domain is used. The default is None. + window : {None, array_like}, optional + If given, the resulting array must be if the form + ``[beg, end]``, where ``beg`` and ``end`` are the endpoints of + the window. If None is given then the class window is used. The + default is None. + symbol : str, optional + Symbol representing the independent variable. Default is 'x'. + + Returns + ------- + new_series : series + A series with the coefficient of the `deg` term set to one and + all others zero. + + """ + if domain is None: + domain = cls.domain + if window is None: + window = cls.window + ideg = int(deg) + + if ideg != deg or ideg < 0: + raise ValueError("deg must be non-negative integer") + return cls([0]*ideg + [1], domain, window, symbol) + + @classmethod + def cast(cls, series, domain=None, window=None): + """Convert series to series of this class. + + The `series` is expected to be an instance of some polynomial + series of one of the types supported by by the numpy.polynomial + module, but could be some other class that supports the convert + method. + + .. versionadded:: 1.7.0 + + Parameters + ---------- + series : series + The series instance to be converted. + domain : {None, array_like}, optional + If given, the array must be of the form ``[beg, end]``, where + ``beg`` and ``end`` are the endpoints of the domain. If None is + given then the class domain is used. The default is None. + window : {None, array_like}, optional + If given, the resulting array must be if the form + ``[beg, end]``, where ``beg`` and ``end`` are the endpoints of + the window. If None is given then the class window is used. The + default is None. + + Returns + ------- + new_series : series + A series of the same kind as the calling class and equal to + `series` when evaluated. + + See Also + -------- + convert : similar instance method + + """ + if domain is None: + domain = cls.domain + if window is None: + window = cls.window + return series.convert(domain, cls, window) diff --git a/deepseekvl2/lib/python3.10/site-packages/numpy/polynomial/chebyshev.py b/deepseekvl2/lib/python3.10/site-packages/numpy/polynomial/chebyshev.py new file mode 100644 index 0000000000000000000000000000000000000000..efbe13e0cadb27e29bea430a858dea5110621a0c --- /dev/null +++ b/deepseekvl2/lib/python3.10/site-packages/numpy/polynomial/chebyshev.py @@ -0,0 +1,2082 @@ +""" +==================================================== +Chebyshev Series (:mod:`numpy.polynomial.chebyshev`) +==================================================== + +This module provides a number of objects (mostly functions) useful for +dealing with Chebyshev series, including a `Chebyshev` class that +encapsulates the usual arithmetic operations. (General information +on how this module represents and works with such polynomials is in the +docstring for its "parent" sub-package, `numpy.polynomial`). + +Classes +------- + +.. autosummary:: + :toctree: generated/ + + Chebyshev + + +Constants +--------- + +.. autosummary:: + :toctree: generated/ + + chebdomain + chebzero + chebone + chebx + +Arithmetic +---------- + +.. autosummary:: + :toctree: generated/ + + chebadd + chebsub + chebmulx + chebmul + chebdiv + chebpow + chebval + chebval2d + chebval3d + chebgrid2d + chebgrid3d + +Calculus +-------- + +.. autosummary:: + :toctree: generated/ + + chebder + chebint + +Misc Functions +-------------- + +.. autosummary:: + :toctree: generated/ + + chebfromroots + chebroots + chebvander + chebvander2d + chebvander3d + chebgauss + chebweight + chebcompanion + chebfit + chebpts1 + chebpts2 + chebtrim + chebline + cheb2poly + poly2cheb + chebinterpolate + +See also +-------- +`numpy.polynomial` + +Notes +----- +The implementations of multiplication, division, integration, and +differentiation use the algebraic identities [1]_: + +.. math:: + T_n(x) = \\frac{z^n + z^{-n}}{2} \\\\ + z\\frac{dx}{dz} = \\frac{z - z^{-1}}{2}. + +where + +.. math:: x = \\frac{z + z^{-1}}{2}. + +These identities allow a Chebyshev series to be expressed as a finite, +symmetric Laurent series. In this module, this sort of Laurent series +is referred to as a "z-series." + +References +---------- +.. [1] A. T. Benjamin, et al., "Combinatorial Trigonometry with Chebyshev + Polynomials," *Journal of Statistical Planning and Inference 14*, 2008 + (https://web.archive.org/web/20080221202153/https://www.math.hmc.edu/~benjamin/papers/CombTrig.pdf, pg. 4) + +""" +import numpy as np +import numpy.linalg as la +from numpy.core.multiarray import normalize_axis_index + +from . import polyutils as pu +from ._polybase import ABCPolyBase + +__all__ = [ + 'chebzero', 'chebone', 'chebx', 'chebdomain', 'chebline', 'chebadd', + 'chebsub', 'chebmulx', 'chebmul', 'chebdiv', 'chebpow', 'chebval', + 'chebder', 'chebint', 'cheb2poly', 'poly2cheb', 'chebfromroots', + 'chebvander', 'chebfit', 'chebtrim', 'chebroots', 'chebpts1', + 'chebpts2', 'Chebyshev', 'chebval2d', 'chebval3d', 'chebgrid2d', + 'chebgrid3d', 'chebvander2d', 'chebvander3d', 'chebcompanion', + 'chebgauss', 'chebweight', 'chebinterpolate'] + +chebtrim = pu.trimcoef + +# +# A collection of functions for manipulating z-series. These are private +# functions and do minimal error checking. +# + +def _cseries_to_zseries(c): + """Convert Chebyshev series to z-series. + + Convert a Chebyshev series to the equivalent z-series. The result is + never an empty array. The dtype of the return is the same as that of + the input. No checks are run on the arguments as this routine is for + internal use. + + Parameters + ---------- + c : 1-D ndarray + Chebyshev coefficients, ordered from low to high + + Returns + ------- + zs : 1-D ndarray + Odd length symmetric z-series, ordered from low to high. + + """ + n = c.size + zs = np.zeros(2*n-1, dtype=c.dtype) + zs[n-1:] = c/2 + return zs + zs[::-1] + + +def _zseries_to_cseries(zs): + """Convert z-series to a Chebyshev series. + + Convert a z series to the equivalent Chebyshev series. The result is + never an empty array. The dtype of the return is the same as that of + the input. No checks are run on the arguments as this routine is for + internal use. + + Parameters + ---------- + zs : 1-D ndarray + Odd length symmetric z-series, ordered from low to high. + + Returns + ------- + c : 1-D ndarray + Chebyshev coefficients, ordered from low to high. + + """ + n = (zs.size + 1)//2 + c = zs[n-1:].copy() + c[1:n] *= 2 + return c + + +def _zseries_mul(z1, z2): + """Multiply two z-series. + + Multiply two z-series to produce a z-series. + + Parameters + ---------- + z1, z2 : 1-D ndarray + The arrays must be 1-D but this is not checked. + + Returns + ------- + product : 1-D ndarray + The product z-series. + + Notes + ----- + This is simply convolution. If symmetric/anti-symmetric z-series are + denoted by S/A then the following rules apply: + + S*S, A*A -> S + S*A, A*S -> A + + """ + return np.convolve(z1, z2) + + +def _zseries_div(z1, z2): + """Divide the first z-series by the second. + + Divide `z1` by `z2` and return the quotient and remainder as z-series. + Warning: this implementation only applies when both z1 and z2 have the + same symmetry, which is sufficient for present purposes. + + Parameters + ---------- + z1, z2 : 1-D ndarray + The arrays must be 1-D and have the same symmetry, but this is not + checked. + + Returns + ------- + + (quotient, remainder) : 1-D ndarrays + Quotient and remainder as z-series. + + Notes + ----- + This is not the same as polynomial division on account of the desired form + of the remainder. If symmetric/anti-symmetric z-series are denoted by S/A + then the following rules apply: + + S/S -> S,S + A/A -> S,A + + The restriction to types of the same symmetry could be fixed but seems like + unneeded generality. There is no natural form for the remainder in the case + where there is no symmetry. + + """ + z1 = z1.copy() + z2 = z2.copy() + lc1 = len(z1) + lc2 = len(z2) + if lc2 == 1: + z1 /= z2 + return z1, z1[:1]*0 + elif lc1 < lc2: + return z1[:1]*0, z1 + else: + dlen = lc1 - lc2 + scl = z2[0] + z2 /= scl + quo = np.empty(dlen + 1, dtype=z1.dtype) + i = 0 + j = dlen + while i < j: + r = z1[i] + quo[i] = z1[i] + quo[dlen - i] = r + tmp = r*z2 + z1[i:i+lc2] -= tmp + z1[j:j+lc2] -= tmp + i += 1 + j -= 1 + r = z1[i] + quo[i] = r + tmp = r*z2 + z1[i:i+lc2] -= tmp + quo /= scl + rem = z1[i+1:i-1+lc2].copy() + return quo, rem + + +def _zseries_der(zs): + """Differentiate a z-series. + + The derivative is with respect to x, not z. This is achieved using the + chain rule and the value of dx/dz given in the module notes. + + Parameters + ---------- + zs : z-series + The z-series to differentiate. + + Returns + ------- + derivative : z-series + The derivative + + Notes + ----- + The zseries for x (ns) has been multiplied by two in order to avoid + using floats that are incompatible with Decimal and likely other + specialized scalar types. This scaling has been compensated by + multiplying the value of zs by two also so that the two cancels in the + division. + + """ + n = len(zs)//2 + ns = np.array([-1, 0, 1], dtype=zs.dtype) + zs *= np.arange(-n, n+1)*2 + d, r = _zseries_div(zs, ns) + return d + + +def _zseries_int(zs): + """Integrate a z-series. + + The integral is with respect to x, not z. This is achieved by a change + of variable using dx/dz given in the module notes. + + Parameters + ---------- + zs : z-series + The z-series to integrate + + Returns + ------- + integral : z-series + The indefinite integral + + Notes + ----- + The zseries for x (ns) has been multiplied by two in order to avoid + using floats that are incompatible with Decimal and likely other + specialized scalar types. This scaling has been compensated by + dividing the resulting zs by two. + + """ + n = 1 + len(zs)//2 + ns = np.array([-1, 0, 1], dtype=zs.dtype) + zs = _zseries_mul(zs, ns) + div = np.arange(-n, n+1)*2 + zs[:n] /= div[:n] + zs[n+1:] /= div[n+1:] + zs[n] = 0 + return zs + +# +# Chebyshev series functions +# + + +def poly2cheb(pol): + """ + Convert a polynomial to a Chebyshev series. + + Convert an array representing the coefficients of a polynomial (relative + to the "standard" basis) ordered from lowest degree to highest, to an + array of the coefficients of the equivalent Chebyshev series, ordered + from lowest to highest degree. + + Parameters + ---------- + pol : array_like + 1-D array containing the polynomial coefficients + + Returns + ------- + c : ndarray + 1-D array containing the coefficients of the equivalent Chebyshev + series. + + See Also + -------- + cheb2poly + + Notes + ----- + The easy way to do conversions between polynomial basis sets + is to use the convert method of a class instance. + + Examples + -------- + >>> from numpy import polynomial as P + >>> p = P.Polynomial(range(4)) + >>> p + Polynomial([0., 1., 2., 3.], domain=[-1, 1], window=[-1, 1]) + >>> c = p.convert(kind=P.Chebyshev) + >>> c + Chebyshev([1. , 3.25, 1. , 0.75], domain=[-1., 1.], window=[-1., 1.]) + >>> P.chebyshev.poly2cheb(range(4)) + array([1. , 3.25, 1. , 0.75]) + + """ + [pol] = pu.as_series([pol]) + deg = len(pol) - 1 + res = 0 + for i in range(deg, -1, -1): + res = chebadd(chebmulx(res), pol[i]) + return res + + +def cheb2poly(c): + """ + Convert a Chebyshev series to a polynomial. + + Convert an array representing the coefficients of a Chebyshev series, + ordered from lowest degree to highest, to an array of the coefficients + of the equivalent polynomial (relative to the "standard" basis) ordered + from lowest to highest degree. + + Parameters + ---------- + c : array_like + 1-D array containing the Chebyshev series coefficients, ordered + from lowest order term to highest. + + Returns + ------- + pol : ndarray + 1-D array containing the coefficients of the equivalent polynomial + (relative to the "standard" basis) ordered from lowest order term + to highest. + + See Also + -------- + poly2cheb + + Notes + ----- + The easy way to do conversions between polynomial basis sets + is to use the convert method of a class instance. + + Examples + -------- + >>> from numpy import polynomial as P + >>> c = P.Chebyshev(range(4)) + >>> c + Chebyshev([0., 1., 2., 3.], domain=[-1, 1], window=[-1, 1]) + >>> p = c.convert(kind=P.Polynomial) + >>> p + Polynomial([-2., -8., 4., 12.], domain=[-1., 1.], window=[-1., 1.]) + >>> P.chebyshev.cheb2poly(range(4)) + array([-2., -8., 4., 12.]) + + """ + from .polynomial import polyadd, polysub, polymulx + + [c] = pu.as_series([c]) + n = len(c) + if n < 3: + return c + else: + c0 = c[-2] + c1 = c[-1] + # i is the current degree of c1 + for i in range(n - 1, 1, -1): + tmp = c0 + c0 = polysub(c[i - 2], c1) + c1 = polyadd(tmp, polymulx(c1)*2) + return polyadd(c0, polymulx(c1)) + + +# +# These are constant arrays are of integer type so as to be compatible +# with the widest range of other types, such as Decimal. +# + +# Chebyshev default domain. +chebdomain = np.array([-1, 1]) + +# Chebyshev coefficients representing zero. +chebzero = np.array([0]) + +# Chebyshev coefficients representing one. +chebone = np.array([1]) + +# Chebyshev coefficients representing the identity x. +chebx = np.array([0, 1]) + + +def chebline(off, scl): + """ + Chebyshev series whose graph is a straight line. + + Parameters + ---------- + off, scl : scalars + The specified line is given by ``off + scl*x``. + + Returns + ------- + y : ndarray + This module's representation of the Chebyshev series for + ``off + scl*x``. + + See Also + -------- + numpy.polynomial.polynomial.polyline + numpy.polynomial.legendre.legline + numpy.polynomial.laguerre.lagline + numpy.polynomial.hermite.hermline + numpy.polynomial.hermite_e.hermeline + + Examples + -------- + >>> import numpy.polynomial.chebyshev as C + >>> C.chebline(3,2) + array([3, 2]) + >>> C.chebval(-3, C.chebline(3,2)) # should be -3 + -3.0 + + """ + if scl != 0: + return np.array([off, scl]) + else: + return np.array([off]) + + +def chebfromroots(roots): + """ + Generate a Chebyshev series with given roots. + + The function returns the coefficients of the polynomial + + .. math:: p(x) = (x - r_0) * (x - r_1) * ... * (x - r_n), + + in Chebyshev form, where the `r_n` are the roots specified in `roots`. + If a zero has multiplicity n, then it must appear in `roots` n times. + For instance, if 2 is a root of multiplicity three and 3 is a root of + multiplicity 2, then `roots` looks something like [2, 2, 2, 3, 3]. The + roots can appear in any order. + + If the returned coefficients are `c`, then + + .. math:: p(x) = c_0 + c_1 * T_1(x) + ... + c_n * T_n(x) + + The coefficient of the last term is not generally 1 for monic + polynomials in Chebyshev form. + + Parameters + ---------- + roots : array_like + Sequence containing the roots. + + Returns + ------- + out : ndarray + 1-D array of coefficients. If all roots are real then `out` is a + real array, if some of the roots are complex, then `out` is complex + even if all the coefficients in the result are real (see Examples + below). + + See Also + -------- + numpy.polynomial.polynomial.polyfromroots + numpy.polynomial.legendre.legfromroots + numpy.polynomial.laguerre.lagfromroots + numpy.polynomial.hermite.hermfromroots + numpy.polynomial.hermite_e.hermefromroots + + Examples + -------- + >>> import numpy.polynomial.chebyshev as C + >>> C.chebfromroots((-1,0,1)) # x^3 - x relative to the standard basis + array([ 0. , -0.25, 0. , 0.25]) + >>> j = complex(0,1) + >>> C.chebfromroots((-j,j)) # x^2 + 1 relative to the standard basis + array([1.5+0.j, 0. +0.j, 0.5+0.j]) + + """ + return pu._fromroots(chebline, chebmul, roots) + + +def chebadd(c1, c2): + """ + Add one Chebyshev series to another. + + Returns the sum of two Chebyshev series `c1` + `c2`. The arguments + are sequences of coefficients ordered from lowest order term to + highest, i.e., [1,2,3] represents the series ``T_0 + 2*T_1 + 3*T_2``. + + Parameters + ---------- + c1, c2 : array_like + 1-D arrays of Chebyshev series coefficients ordered from low to + high. + + Returns + ------- + out : ndarray + Array representing the Chebyshev series of their sum. + + See Also + -------- + chebsub, chebmulx, chebmul, chebdiv, chebpow + + Notes + ----- + Unlike multiplication, division, etc., the sum of two Chebyshev series + is a Chebyshev series (without having to "reproject" the result onto + the basis set) so addition, just like that of "standard" polynomials, + is simply "component-wise." + + Examples + -------- + >>> from numpy.polynomial import chebyshev as C + >>> c1 = (1,2,3) + >>> c2 = (3,2,1) + >>> C.chebadd(c1,c2) + array([4., 4., 4.]) + + """ + return pu._add(c1, c2) + + +def chebsub(c1, c2): + """ + Subtract one Chebyshev series from another. + + Returns the difference of two Chebyshev series `c1` - `c2`. The + sequences of coefficients are from lowest order term to highest, i.e., + [1,2,3] represents the series ``T_0 + 2*T_1 + 3*T_2``. + + Parameters + ---------- + c1, c2 : array_like + 1-D arrays of Chebyshev series coefficients ordered from low to + high. + + Returns + ------- + out : ndarray + Of Chebyshev series coefficients representing their difference. + + See Also + -------- + chebadd, chebmulx, chebmul, chebdiv, chebpow + + Notes + ----- + Unlike multiplication, division, etc., the difference of two Chebyshev + series is a Chebyshev series (without having to "reproject" the result + onto the basis set) so subtraction, just like that of "standard" + polynomials, is simply "component-wise." + + Examples + -------- + >>> from numpy.polynomial import chebyshev as C + >>> c1 = (1,2,3) + >>> c2 = (3,2,1) + >>> C.chebsub(c1,c2) + array([-2., 0., 2.]) + >>> C.chebsub(c2,c1) # -C.chebsub(c1,c2) + array([ 2., 0., -2.]) + + """ + return pu._sub(c1, c2) + + +def chebmulx(c): + """Multiply a Chebyshev series by x. + + Multiply the polynomial `c` by x, where x is the independent + variable. + + + Parameters + ---------- + c : array_like + 1-D array of Chebyshev series coefficients ordered from low to + high. + + Returns + ------- + out : ndarray + Array representing the result of the multiplication. + + Notes + ----- + + .. versionadded:: 1.5.0 + + Examples + -------- + >>> from numpy.polynomial import chebyshev as C + >>> C.chebmulx([1,2,3]) + array([1. , 2.5, 1. , 1.5]) + + """ + # c is a trimmed copy + [c] = pu.as_series([c]) + # The zero series needs special treatment + if len(c) == 1 and c[0] == 0: + return c + + prd = np.empty(len(c) + 1, dtype=c.dtype) + prd[0] = c[0]*0 + prd[1] = c[0] + if len(c) > 1: + tmp = c[1:]/2 + prd[2:] = tmp + prd[0:-2] += tmp + return prd + + +def chebmul(c1, c2): + """ + Multiply one Chebyshev series by another. + + Returns the product of two Chebyshev series `c1` * `c2`. The arguments + are sequences of coefficients, from lowest order "term" to highest, + e.g., [1,2,3] represents the series ``T_0 + 2*T_1 + 3*T_2``. + + Parameters + ---------- + c1, c2 : array_like + 1-D arrays of Chebyshev series coefficients ordered from low to + high. + + Returns + ------- + out : ndarray + Of Chebyshev series coefficients representing their product. + + See Also + -------- + chebadd, chebsub, chebmulx, chebdiv, chebpow + + Notes + ----- + In general, the (polynomial) product of two C-series results in terms + that are not in the Chebyshev polynomial basis set. Thus, to express + the product as a C-series, it is typically necessary to "reproject" + the product onto said basis set, which typically produces + "unintuitive live" (but correct) results; see Examples section below. + + Examples + -------- + >>> from numpy.polynomial import chebyshev as C + >>> c1 = (1,2,3) + >>> c2 = (3,2,1) + >>> C.chebmul(c1,c2) # multiplication requires "reprojection" + array([ 6.5, 12. , 12. , 4. , 1.5]) + + """ + # c1, c2 are trimmed copies + [c1, c2] = pu.as_series([c1, c2]) + z1 = _cseries_to_zseries(c1) + z2 = _cseries_to_zseries(c2) + prd = _zseries_mul(z1, z2) + ret = _zseries_to_cseries(prd) + return pu.trimseq(ret) + + +def chebdiv(c1, c2): + """ + Divide one Chebyshev series by another. + + Returns the quotient-with-remainder of two Chebyshev series + `c1` / `c2`. The arguments are sequences of coefficients from lowest + order "term" to highest, e.g., [1,2,3] represents the series + ``T_0 + 2*T_1 + 3*T_2``. + + Parameters + ---------- + c1, c2 : array_like + 1-D arrays of Chebyshev series coefficients ordered from low to + high. + + Returns + ------- + [quo, rem] : ndarrays + Of Chebyshev series coefficients representing the quotient and + remainder. + + See Also + -------- + chebadd, chebsub, chebmulx, chebmul, chebpow + + Notes + ----- + In general, the (polynomial) division of one C-series by another + results in quotient and remainder terms that are not in the Chebyshev + polynomial basis set. Thus, to express these results as C-series, it + is typically necessary to "reproject" the results onto said basis + set, which typically produces "unintuitive" (but correct) results; + see Examples section below. + + Examples + -------- + >>> from numpy.polynomial import chebyshev as C + >>> c1 = (1,2,3) + >>> c2 = (3,2,1) + >>> C.chebdiv(c1,c2) # quotient "intuitive," remainder not + (array([3.]), array([-8., -4.])) + >>> c2 = (0,1,2,3) + >>> C.chebdiv(c2,c1) # neither "intuitive" + (array([0., 2.]), array([-2., -4.])) + + """ + # c1, c2 are trimmed copies + [c1, c2] = pu.as_series([c1, c2]) + if c2[-1] == 0: + raise ZeroDivisionError() + + # note: this is more efficient than `pu._div(chebmul, c1, c2)` + lc1 = len(c1) + lc2 = len(c2) + if lc1 < lc2: + return c1[:1]*0, c1 + elif lc2 == 1: + return c1/c2[-1], c1[:1]*0 + else: + z1 = _cseries_to_zseries(c1) + z2 = _cseries_to_zseries(c2) + quo, rem = _zseries_div(z1, z2) + quo = pu.trimseq(_zseries_to_cseries(quo)) + rem = pu.trimseq(_zseries_to_cseries(rem)) + return quo, rem + + +def chebpow(c, pow, maxpower=16): + """Raise a Chebyshev series to a power. + + Returns the Chebyshev series `c` raised to the power `pow`. The + argument `c` is a sequence of coefficients ordered from low to high. + i.e., [1,2,3] is the series ``T_0 + 2*T_1 + 3*T_2.`` + + Parameters + ---------- + c : array_like + 1-D array of Chebyshev series coefficients ordered from low to + high. + pow : integer + Power to which the series will be raised + maxpower : integer, optional + Maximum power allowed. This is mainly to limit growth of the series + to unmanageable size. Default is 16 + + Returns + ------- + coef : ndarray + Chebyshev series of power. + + See Also + -------- + chebadd, chebsub, chebmulx, chebmul, chebdiv + + Examples + -------- + >>> from numpy.polynomial import chebyshev as C + >>> C.chebpow([1, 2, 3, 4], 2) + array([15.5, 22. , 16. , ..., 12.5, 12. , 8. ]) + + """ + # note: this is more efficient than `pu._pow(chebmul, c1, c2)`, as it + # avoids converting between z and c series repeatedly + + # c is a trimmed copy + [c] = pu.as_series([c]) + power = int(pow) + if power != pow or power < 0: + raise ValueError("Power must be a non-negative integer.") + elif maxpower is not None and power > maxpower: + raise ValueError("Power is too large") + elif power == 0: + return np.array([1], dtype=c.dtype) + elif power == 1: + return c + else: + # This can be made more efficient by using powers of two + # in the usual way. + zs = _cseries_to_zseries(c) + prd = zs + for i in range(2, power + 1): + prd = np.convolve(prd, zs) + return _zseries_to_cseries(prd) + + +def chebder(c, m=1, scl=1, axis=0): + """ + Differentiate a Chebyshev series. + + Returns the Chebyshev series coefficients `c` differentiated `m` times + along `axis`. At each iteration the result is multiplied by `scl` (the + scaling factor is for use in a linear change of variable). The argument + `c` is an array of coefficients from low to high degree along each + axis, e.g., [1,2,3] represents the series ``1*T_0 + 2*T_1 + 3*T_2`` + while [[1,2],[1,2]] represents ``1*T_0(x)*T_0(y) + 1*T_1(x)*T_0(y) + + 2*T_0(x)*T_1(y) + 2*T_1(x)*T_1(y)`` if axis=0 is ``x`` and axis=1 is + ``y``. + + Parameters + ---------- + c : array_like + Array of Chebyshev series coefficients. If c is multidimensional + the different axis correspond to different variables with the + degree in each axis given by the corresponding index. + m : int, optional + Number of derivatives taken, must be non-negative. (Default: 1) + scl : scalar, optional + Each differentiation is multiplied by `scl`. The end result is + multiplication by ``scl**m``. This is for use in a linear change of + variable. (Default: 1) + axis : int, optional + Axis over which the derivative is taken. (Default: 0). + + .. versionadded:: 1.7.0 + + Returns + ------- + der : ndarray + Chebyshev series of the derivative. + + See Also + -------- + chebint + + Notes + ----- + In general, the result of differentiating a C-series needs to be + "reprojected" onto the C-series basis set. Thus, typically, the + result of this function is "unintuitive," albeit correct; see Examples + section below. + + Examples + -------- + >>> from numpy.polynomial import chebyshev as C + >>> c = (1,2,3,4) + >>> C.chebder(c) + array([14., 12., 24.]) + >>> C.chebder(c,3) + array([96.]) + >>> C.chebder(c,scl=-1) + array([-14., -12., -24.]) + >>> C.chebder(c,2,-1) + array([12., 96.]) + + """ + c = np.array(c, ndmin=1, copy=True) + if c.dtype.char in '?bBhHiIlLqQpP': + c = c.astype(np.double) + cnt = pu._deprecate_as_int(m, "the order of derivation") + iaxis = pu._deprecate_as_int(axis, "the axis") + if cnt < 0: + raise ValueError("The order of derivation must be non-negative") + iaxis = normalize_axis_index(iaxis, c.ndim) + + if cnt == 0: + return c + + c = np.moveaxis(c, iaxis, 0) + n = len(c) + if cnt >= n: + c = c[:1]*0 + else: + for i in range(cnt): + n = n - 1 + c *= scl + der = np.empty((n,) + c.shape[1:], dtype=c.dtype) + for j in range(n, 2, -1): + der[j - 1] = (2*j)*c[j] + c[j - 2] += (j*c[j])/(j - 2) + if n > 1: + der[1] = 4*c[2] + der[0] = c[1] + c = der + c = np.moveaxis(c, 0, iaxis) + return c + + +def chebint(c, m=1, k=[], lbnd=0, scl=1, axis=0): + """ + Integrate a Chebyshev series. + + Returns the Chebyshev series coefficients `c` integrated `m` times from + `lbnd` along `axis`. At each iteration the resulting series is + **multiplied** by `scl` and an integration constant, `k`, is added. + The scaling factor is for use in a linear change of variable. ("Buyer + beware": note that, depending on what one is doing, one may want `scl` + to be the reciprocal of what one might expect; for more information, + see the Notes section below.) The argument `c` is an array of + coefficients from low to high degree along each axis, e.g., [1,2,3] + represents the series ``T_0 + 2*T_1 + 3*T_2`` while [[1,2],[1,2]] + represents ``1*T_0(x)*T_0(y) + 1*T_1(x)*T_0(y) + 2*T_0(x)*T_1(y) + + 2*T_1(x)*T_1(y)`` if axis=0 is ``x`` and axis=1 is ``y``. + + Parameters + ---------- + c : array_like + Array of Chebyshev series coefficients. If c is multidimensional + the different axis correspond to different variables with the + degree in each axis given by the corresponding index. + m : int, optional + Order of integration, must be positive. (Default: 1) + k : {[], list, scalar}, optional + Integration constant(s). The value of the first integral at zero + is the first value in the list, the value of the second integral + at zero is the second value, etc. If ``k == []`` (the default), + all constants are set to zero. If ``m == 1``, a single scalar can + be given instead of a list. + lbnd : scalar, optional + The lower bound of the integral. (Default: 0) + scl : scalar, optional + Following each integration the result is *multiplied* by `scl` + before the integration constant is added. (Default: 1) + axis : int, optional + Axis over which the integral is taken. (Default: 0). + + .. versionadded:: 1.7.0 + + Returns + ------- + S : ndarray + C-series coefficients of the integral. + + Raises + ------ + ValueError + If ``m < 1``, ``len(k) > m``, ``np.ndim(lbnd) != 0``, or + ``np.ndim(scl) != 0``. + + See Also + -------- + chebder + + Notes + ----- + Note that the result of each integration is *multiplied* by `scl`. + Why is this important to note? Say one is making a linear change of + variable :math:`u = ax + b` in an integral relative to `x`. Then + :math:`dx = du/a`, so one will need to set `scl` equal to + :math:`1/a`- perhaps not what one would have first thought. + + Also note that, in general, the result of integrating a C-series needs + to be "reprojected" onto the C-series basis set. Thus, typically, + the result of this function is "unintuitive," albeit correct; see + Examples section below. + + Examples + -------- + >>> from numpy.polynomial import chebyshev as C + >>> c = (1,2,3) + >>> C.chebint(c) + array([ 0.5, -0.5, 0.5, 0.5]) + >>> C.chebint(c,3) + array([ 0.03125 , -0.1875 , 0.04166667, -0.05208333, 0.01041667, # may vary + 0.00625 ]) + >>> C.chebint(c, k=3) + array([ 3.5, -0.5, 0.5, 0.5]) + >>> C.chebint(c,lbnd=-2) + array([ 8.5, -0.5, 0.5, 0.5]) + >>> C.chebint(c,scl=-2) + array([-1., 1., -1., -1.]) + + """ + c = np.array(c, ndmin=1, copy=True) + if c.dtype.char in '?bBhHiIlLqQpP': + c = c.astype(np.double) + if not np.iterable(k): + k = [k] + cnt = pu._deprecate_as_int(m, "the order of integration") + iaxis = pu._deprecate_as_int(axis, "the axis") + if cnt < 0: + raise ValueError("The order of integration must be non-negative") + if len(k) > cnt: + raise ValueError("Too many integration constants") + if np.ndim(lbnd) != 0: + raise ValueError("lbnd must be a scalar.") + if np.ndim(scl) != 0: + raise ValueError("scl must be a scalar.") + iaxis = normalize_axis_index(iaxis, c.ndim) + + if cnt == 0: + return c + + c = np.moveaxis(c, iaxis, 0) + k = list(k) + [0]*(cnt - len(k)) + for i in range(cnt): + n = len(c) + c *= scl + if n == 1 and np.all(c[0] == 0): + c[0] += k[i] + else: + tmp = np.empty((n + 1,) + c.shape[1:], dtype=c.dtype) + tmp[0] = c[0]*0 + tmp[1] = c[0] + if n > 1: + tmp[2] = c[1]/4 + for j in range(2, n): + tmp[j + 1] = c[j]/(2*(j + 1)) + tmp[j - 1] -= c[j]/(2*(j - 1)) + tmp[0] += k[i] - chebval(lbnd, tmp) + c = tmp + c = np.moveaxis(c, 0, iaxis) + return c + + +def chebval(x, c, tensor=True): + """ + Evaluate a Chebyshev series at points x. + + If `c` is of length `n + 1`, this function returns the value: + + .. math:: p(x) = c_0 * T_0(x) + c_1 * T_1(x) + ... + c_n * T_n(x) + + The parameter `x` is converted to an array only if it is a tuple or a + list, otherwise it is treated as a scalar. In either case, either `x` + or its elements must support multiplication and addition both with + themselves and with the elements of `c`. + + If `c` is a 1-D array, then `p(x)` will have the same shape as `x`. If + `c` is multidimensional, then the shape of the result depends on the + value of `tensor`. If `tensor` is true the shape will be c.shape[1:] + + x.shape. If `tensor` is false the shape will be c.shape[1:]. Note that + scalars have shape (,). + + Trailing zeros in the coefficients will be used in the evaluation, so + they should be avoided if efficiency is a concern. + + Parameters + ---------- + x : array_like, compatible object + If `x` is a list or tuple, it is converted to an ndarray, otherwise + it is left unchanged and treated as a scalar. In either case, `x` + or its elements must support addition and multiplication with + themselves and with the elements of `c`. + c : array_like + Array of coefficients ordered so that the coefficients for terms of + degree n are contained in c[n]. If `c` is multidimensional the + remaining indices enumerate multiple polynomials. In the two + dimensional case the coefficients may be thought of as stored in + the columns of `c`. + tensor : boolean, optional + If True, the shape of the coefficient array is extended with ones + on the right, one for each dimension of `x`. Scalars have dimension 0 + for this action. The result is that every column of coefficients in + `c` is evaluated for every element of `x`. If False, `x` is broadcast + over the columns of `c` for the evaluation. This keyword is useful + when `c` is multidimensional. The default value is True. + + .. versionadded:: 1.7.0 + + Returns + ------- + values : ndarray, algebra_like + The shape of the return value is described above. + + See Also + -------- + chebval2d, chebgrid2d, chebval3d, chebgrid3d + + Notes + ----- + The evaluation uses Clenshaw recursion, aka synthetic division. + + """ + c = np.array(c, ndmin=1, copy=True) + if c.dtype.char in '?bBhHiIlLqQpP': + c = c.astype(np.double) + if isinstance(x, (tuple, list)): + x = np.asarray(x) + if isinstance(x, np.ndarray) and tensor: + c = c.reshape(c.shape + (1,)*x.ndim) + + if len(c) == 1: + c0 = c[0] + c1 = 0 + elif len(c) == 2: + c0 = c[0] + c1 = c[1] + else: + x2 = 2*x + c0 = c[-2] + c1 = c[-1] + for i in range(3, len(c) + 1): + tmp = c0 + c0 = c[-i] - c1 + c1 = tmp + c1*x2 + return c0 + c1*x + + +def chebval2d(x, y, c): + """ + Evaluate a 2-D Chebyshev series at points (x, y). + + This function returns the values: + + .. math:: p(x,y) = \\sum_{i,j} c_{i,j} * T_i(x) * T_j(y) + + The parameters `x` and `y` are converted to arrays only if they are + tuples or a lists, otherwise they are treated as a scalars and they + must have the same shape after conversion. In either case, either `x` + and `y` or their elements must support multiplication and addition both + with themselves and with the elements of `c`. + + If `c` is a 1-D array a one is implicitly appended to its shape to make + it 2-D. The shape of the result will be c.shape[2:] + x.shape. + + Parameters + ---------- + x, y : array_like, compatible objects + The two dimensional series is evaluated at the points `(x, y)`, + where `x` and `y` must have the same shape. If `x` or `y` is a list + or tuple, it is first converted to an ndarray, otherwise it is left + unchanged and if it isn't an ndarray it is treated as a scalar. + c : array_like + Array of coefficients ordered so that the coefficient of the term + of multi-degree i,j is contained in ``c[i,j]``. If `c` has + dimension greater than 2 the remaining indices enumerate multiple + sets of coefficients. + + Returns + ------- + values : ndarray, compatible object + The values of the two dimensional Chebyshev series at points formed + from pairs of corresponding values from `x` and `y`. + + See Also + -------- + chebval, chebgrid2d, chebval3d, chebgrid3d + + Notes + ----- + + .. versionadded:: 1.7.0 + + """ + return pu._valnd(chebval, c, x, y) + + +def chebgrid2d(x, y, c): + """ + Evaluate a 2-D Chebyshev series on the Cartesian product of x and y. + + This function returns the values: + + .. math:: p(a,b) = \\sum_{i,j} c_{i,j} * T_i(a) * T_j(b), + + where the points `(a, b)` consist of all pairs formed by taking + `a` from `x` and `b` from `y`. The resulting points form a grid with + `x` in the first dimension and `y` in the second. + + The parameters `x` and `y` are converted to arrays only if they are + tuples or a lists, otherwise they are treated as a scalars. In either + case, either `x` and `y` or their elements must support multiplication + and addition both with themselves and with the elements of `c`. + + If `c` has fewer than two dimensions, ones are implicitly appended to + its shape to make it 2-D. The shape of the result will be c.shape[2:] + + x.shape + y.shape. + + Parameters + ---------- + x, y : array_like, compatible objects + The two dimensional series is evaluated at the points in the + Cartesian product of `x` and `y`. If `x` or `y` is a list or + tuple, it is first converted to an ndarray, otherwise it is left + unchanged and, if it isn't an ndarray, it is treated as a scalar. + c : array_like + Array of coefficients ordered so that the coefficient of the term of + multi-degree i,j is contained in `c[i,j]`. If `c` has dimension + greater than two the remaining indices enumerate multiple sets of + coefficients. + + Returns + ------- + values : ndarray, compatible object + The values of the two dimensional Chebyshev series at points in the + Cartesian product of `x` and `y`. + + See Also + -------- + chebval, chebval2d, chebval3d, chebgrid3d + + Notes + ----- + + .. versionadded:: 1.7.0 + + """ + return pu._gridnd(chebval, c, x, y) + + +def chebval3d(x, y, z, c): + """ + Evaluate a 3-D Chebyshev series at points (x, y, z). + + This function returns the values: + + .. math:: p(x,y,z) = \\sum_{i,j,k} c_{i,j,k} * T_i(x) * T_j(y) * T_k(z) + + The parameters `x`, `y`, and `z` are converted to arrays only if + they are tuples or a lists, otherwise they are treated as a scalars and + they must have the same shape after conversion. In either case, either + `x`, `y`, and `z` or their elements must support multiplication and + addition both with themselves and with the elements of `c`. + + If `c` has fewer than 3 dimensions, ones are implicitly appended to its + shape to make it 3-D. The shape of the result will be c.shape[3:] + + x.shape. + + Parameters + ---------- + x, y, z : array_like, compatible object + The three dimensional series is evaluated at the points + `(x, y, z)`, where `x`, `y`, and `z` must have the same shape. If + any of `x`, `y`, or `z` is a list or tuple, it is first converted + to an ndarray, otherwise it is left unchanged and if it isn't an + ndarray it is treated as a scalar. + c : array_like + Array of coefficients ordered so that the coefficient of the term of + multi-degree i,j,k is contained in ``c[i,j,k]``. If `c` has dimension + greater than 3 the remaining indices enumerate multiple sets of + coefficients. + + Returns + ------- + values : ndarray, compatible object + The values of the multidimensional polynomial on points formed with + triples of corresponding values from `x`, `y`, and `z`. + + See Also + -------- + chebval, chebval2d, chebgrid2d, chebgrid3d + + Notes + ----- + + .. versionadded:: 1.7.0 + + """ + return pu._valnd(chebval, c, x, y, z) + + +def chebgrid3d(x, y, z, c): + """ + Evaluate a 3-D Chebyshev series on the Cartesian product of x, y, and z. + + This function returns the values: + + .. math:: p(a,b,c) = \\sum_{i,j,k} c_{i,j,k} * T_i(a) * T_j(b) * T_k(c) + + where the points `(a, b, c)` consist of all triples formed by taking + `a` from `x`, `b` from `y`, and `c` from `z`. The resulting points form + a grid with `x` in the first dimension, `y` in the second, and `z` in + the third. + + The parameters `x`, `y`, and `z` are converted to arrays only if they + are tuples or a lists, otherwise they are treated as a scalars. In + either case, either `x`, `y`, and `z` or their elements must support + multiplication and addition both with themselves and with the elements + of `c`. + + If `c` has fewer than three dimensions, ones are implicitly appended to + its shape to make it 3-D. The shape of the result will be c.shape[3:] + + x.shape + y.shape + z.shape. + + Parameters + ---------- + x, y, z : array_like, compatible objects + The three dimensional series is evaluated at the points in the + Cartesian product of `x`, `y`, and `z`. If `x`,`y`, or `z` is a + list or tuple, it is first converted to an ndarray, otherwise it is + left unchanged and, if it isn't an ndarray, it is treated as a + scalar. + c : array_like + Array of coefficients ordered so that the coefficients for terms of + degree i,j are contained in ``c[i,j]``. If `c` has dimension + greater than two the remaining indices enumerate multiple sets of + coefficients. + + Returns + ------- + values : ndarray, compatible object + The values of the two dimensional polynomial at points in the Cartesian + product of `x` and `y`. + + See Also + -------- + chebval, chebval2d, chebgrid2d, chebval3d + + Notes + ----- + + .. versionadded:: 1.7.0 + + """ + return pu._gridnd(chebval, c, x, y, z) + + +def chebvander(x, deg): + """Pseudo-Vandermonde matrix of given degree. + + Returns the pseudo-Vandermonde matrix of degree `deg` and sample points + `x`. The pseudo-Vandermonde matrix is defined by + + .. math:: V[..., i] = T_i(x), + + where `0 <= i <= deg`. The leading indices of `V` index the elements of + `x` and the last index is the degree of the Chebyshev polynomial. + + If `c` is a 1-D array of coefficients of length `n + 1` and `V` is the + matrix ``V = chebvander(x, n)``, then ``np.dot(V, c)`` and + ``chebval(x, c)`` are the same up to roundoff. This equivalence is + useful both for least squares fitting and for the evaluation of a large + number of Chebyshev series of the same degree and sample points. + + Parameters + ---------- + x : array_like + Array of points. The dtype is converted to float64 or complex128 + depending on whether any of the elements are complex. If `x` is + scalar it is converted to a 1-D array. + deg : int + Degree of the resulting matrix. + + Returns + ------- + vander : ndarray + The pseudo Vandermonde matrix. The shape of the returned matrix is + ``x.shape + (deg + 1,)``, where The last index is the degree of the + corresponding Chebyshev polynomial. The dtype will be the same as + the converted `x`. + + """ + ideg = pu._deprecate_as_int(deg, "deg") + if ideg < 0: + raise ValueError("deg must be non-negative") + + x = np.array(x, copy=False, ndmin=1) + 0.0 + dims = (ideg + 1,) + x.shape + dtyp = x.dtype + v = np.empty(dims, dtype=dtyp) + # Use forward recursion to generate the entries. + v[0] = x*0 + 1 + if ideg > 0: + x2 = 2*x + v[1] = x + for i in range(2, ideg + 1): + v[i] = v[i-1]*x2 - v[i-2] + return np.moveaxis(v, 0, -1) + + +def chebvander2d(x, y, deg): + """Pseudo-Vandermonde matrix of given degrees. + + Returns the pseudo-Vandermonde matrix of degrees `deg` and sample + points `(x, y)`. The pseudo-Vandermonde matrix is defined by + + .. math:: V[..., (deg[1] + 1)*i + j] = T_i(x) * T_j(y), + + where `0 <= i <= deg[0]` and `0 <= j <= deg[1]`. The leading indices of + `V` index the points `(x, y)` and the last index encodes the degrees of + the Chebyshev polynomials. + + If ``V = chebvander2d(x, y, [xdeg, ydeg])``, then the columns of `V` + correspond to the elements of a 2-D coefficient array `c` of shape + (xdeg + 1, ydeg + 1) in the order + + .. math:: c_{00}, c_{01}, c_{02} ... , c_{10}, c_{11}, c_{12} ... + + and ``np.dot(V, c.flat)`` and ``chebval2d(x, y, c)`` will be the same + up to roundoff. This equivalence is useful both for least squares + fitting and for the evaluation of a large number of 2-D Chebyshev + series of the same degrees and sample points. + + Parameters + ---------- + x, y : array_like + Arrays of point coordinates, all of the same shape. The dtypes + will be converted to either float64 or complex128 depending on + whether any of the elements are complex. Scalars are converted to + 1-D arrays. + deg : list of ints + List of maximum degrees of the form [x_deg, y_deg]. + + Returns + ------- + vander2d : ndarray + The shape of the returned matrix is ``x.shape + (order,)``, where + :math:`order = (deg[0]+1)*(deg[1]+1)`. The dtype will be the same + as the converted `x` and `y`. + + See Also + -------- + chebvander, chebvander3d, chebval2d, chebval3d + + Notes + ----- + + .. versionadded:: 1.7.0 + + """ + return pu._vander_nd_flat((chebvander, chebvander), (x, y), deg) + + +def chebvander3d(x, y, z, deg): + """Pseudo-Vandermonde matrix of given degrees. + + Returns the pseudo-Vandermonde matrix of degrees `deg` and sample + points `(x, y, z)`. If `l, m, n` are the given degrees in `x, y, z`, + then The pseudo-Vandermonde matrix is defined by + + .. math:: V[..., (m+1)(n+1)i + (n+1)j + k] = T_i(x)*T_j(y)*T_k(z), + + where `0 <= i <= l`, `0 <= j <= m`, and `0 <= j <= n`. The leading + indices of `V` index the points `(x, y, z)` and the last index encodes + the degrees of the Chebyshev polynomials. + + If ``V = chebvander3d(x, y, z, [xdeg, ydeg, zdeg])``, then the columns + of `V` correspond to the elements of a 3-D coefficient array `c` of + shape (xdeg + 1, ydeg + 1, zdeg + 1) in the order + + .. math:: c_{000}, c_{001}, c_{002},... , c_{010}, c_{011}, c_{012},... + + and ``np.dot(V, c.flat)`` and ``chebval3d(x, y, z, c)`` will be the + same up to roundoff. This equivalence is useful both for least squares + fitting and for the evaluation of a large number of 3-D Chebyshev + series of the same degrees and sample points. + + Parameters + ---------- + x, y, z : array_like + Arrays of point coordinates, all of the same shape. The dtypes will + be converted to either float64 or complex128 depending on whether + any of the elements are complex. Scalars are converted to 1-D + arrays. + deg : list of ints + List of maximum degrees of the form [x_deg, y_deg, z_deg]. + + Returns + ------- + vander3d : ndarray + The shape of the returned matrix is ``x.shape + (order,)``, where + :math:`order = (deg[0]+1)*(deg[1]+1)*(deg[2]+1)`. The dtype will + be the same as the converted `x`, `y`, and `z`. + + See Also + -------- + chebvander, chebvander3d, chebval2d, chebval3d + + Notes + ----- + + .. versionadded:: 1.7.0 + + """ + return pu._vander_nd_flat((chebvander, chebvander, chebvander), (x, y, z), deg) + + +def chebfit(x, y, deg, rcond=None, full=False, w=None): + """ + Least squares fit of Chebyshev series to data. + + Return the coefficients of a Chebyshev series of degree `deg` that is the + least squares fit to the data values `y` given at points `x`. If `y` is + 1-D the returned coefficients will also be 1-D. If `y` is 2-D multiple + fits are done, one for each column of `y`, and the resulting + coefficients are stored in the corresponding columns of a 2-D return. + The fitted polynomial(s) are in the form + + .. math:: p(x) = c_0 + c_1 * T_1(x) + ... + c_n * T_n(x), + + where `n` is `deg`. + + Parameters + ---------- + x : array_like, shape (M,) + x-coordinates of the M sample points ``(x[i], y[i])``. + y : array_like, shape (M,) or (M, K) + y-coordinates of the sample points. Several data sets of sample + points sharing the same x-coordinates can be fitted at once by + passing in a 2D-array that contains one dataset per column. + deg : int or 1-D array_like + Degree(s) of the fitting polynomials. If `deg` is a single integer, + all terms up to and including the `deg`'th term are included in the + fit. For NumPy versions >= 1.11.0 a list of integers specifying the + degrees of the terms to include may be used instead. + rcond : float, optional + Relative condition number of the fit. Singular values smaller than + this relative to the largest singular value will be ignored. The + default value is len(x)*eps, where eps is the relative precision of + the float type, about 2e-16 in most cases. + full : bool, optional + Switch determining nature of return value. When it is False (the + default) just the coefficients are returned, when True diagnostic + information from the singular value decomposition is also returned. + w : array_like, shape (`M`,), optional + Weights. If not None, the weight ``w[i]`` applies to the unsquared + residual ``y[i] - y_hat[i]`` at ``x[i]``. Ideally the weights are + chosen so that the errors of the products ``w[i]*y[i]`` all have the + same variance. When using inverse-variance weighting, use + ``w[i] = 1/sigma(y[i])``. The default value is None. + + .. versionadded:: 1.5.0 + + Returns + ------- + coef : ndarray, shape (M,) or (M, K) + Chebyshev coefficients ordered from low to high. If `y` was 2-D, + the coefficients for the data in column k of `y` are in column + `k`. + + [residuals, rank, singular_values, rcond] : list + These values are only returned if ``full == True`` + + - residuals -- sum of squared residuals of the least squares fit + - rank -- the numerical rank of the scaled Vandermonde matrix + - singular_values -- singular values of the scaled Vandermonde matrix + - rcond -- value of `rcond`. + + For more details, see `numpy.linalg.lstsq`. + + Warns + ----- + RankWarning + The rank of the coefficient matrix in the least-squares fit is + deficient. The warning is only raised if ``full == False``. The + warnings can be turned off by + + >>> import warnings + >>> warnings.simplefilter('ignore', np.RankWarning) + + See Also + -------- + numpy.polynomial.polynomial.polyfit + numpy.polynomial.legendre.legfit + numpy.polynomial.laguerre.lagfit + numpy.polynomial.hermite.hermfit + numpy.polynomial.hermite_e.hermefit + chebval : Evaluates a Chebyshev series. + chebvander : Vandermonde matrix of Chebyshev series. + chebweight : Chebyshev weight function. + numpy.linalg.lstsq : Computes a least-squares fit from the matrix. + scipy.interpolate.UnivariateSpline : Computes spline fits. + + Notes + ----- + The solution is the coefficients of the Chebyshev series `p` that + minimizes the sum of the weighted squared errors + + .. math:: E = \\sum_j w_j^2 * |y_j - p(x_j)|^2, + + where :math:`w_j` are the weights. This problem is solved by setting up + as the (typically) overdetermined matrix equation + + .. math:: V(x) * c = w * y, + + where `V` is the weighted pseudo Vandermonde matrix of `x`, `c` are the + coefficients to be solved for, `w` are the weights, and `y` are the + observed values. This equation is then solved using the singular value + decomposition of `V`. + + If some of the singular values of `V` are so small that they are + neglected, then a `RankWarning` will be issued. This means that the + coefficient values may be poorly determined. Using a lower order fit + will usually get rid of the warning. The `rcond` parameter can also be + set to a value smaller than its default, but the resulting fit may be + spurious and have large contributions from roundoff error. + + Fits using Chebyshev series are usually better conditioned than fits + using power series, but much can depend on the distribution of the + sample points and the smoothness of the data. If the quality of the fit + is inadequate splines may be a good alternative. + + References + ---------- + .. [1] Wikipedia, "Curve fitting", + https://en.wikipedia.org/wiki/Curve_fitting + + Examples + -------- + + """ + return pu._fit(chebvander, x, y, deg, rcond, full, w) + + +def chebcompanion(c): + """Return the scaled companion matrix of c. + + The basis polynomials are scaled so that the companion matrix is + symmetric when `c` is a Chebyshev basis polynomial. This provides + better eigenvalue estimates than the unscaled case and for basis + polynomials the eigenvalues are guaranteed to be real if + `numpy.linalg.eigvalsh` is used to obtain them. + + Parameters + ---------- + c : array_like + 1-D array of Chebyshev series coefficients ordered from low to high + degree. + + Returns + ------- + mat : ndarray + Scaled companion matrix of dimensions (deg, deg). + + Notes + ----- + + .. versionadded:: 1.7.0 + + """ + # c is a trimmed copy + [c] = pu.as_series([c]) + if len(c) < 2: + raise ValueError('Series must have maximum degree of at least 1.') + if len(c) == 2: + return np.array([[-c[0]/c[1]]]) + + n = len(c) - 1 + mat = np.zeros((n, n), dtype=c.dtype) + scl = np.array([1.] + [np.sqrt(.5)]*(n-1)) + top = mat.reshape(-1)[1::n+1] + bot = mat.reshape(-1)[n::n+1] + top[0] = np.sqrt(.5) + top[1:] = 1/2 + bot[...] = top + mat[:, -1] -= (c[:-1]/c[-1])*(scl/scl[-1])*.5 + return mat + + +def chebroots(c): + """ + Compute the roots of a Chebyshev series. + + Return the roots (a.k.a. "zeros") of the polynomial + + .. math:: p(x) = \\sum_i c[i] * T_i(x). + + Parameters + ---------- + c : 1-D array_like + 1-D array of coefficients. + + Returns + ------- + out : ndarray + Array of the roots of the series. If all the roots are real, + then `out` is also real, otherwise it is complex. + + See Also + -------- + numpy.polynomial.polynomial.polyroots + numpy.polynomial.legendre.legroots + numpy.polynomial.laguerre.lagroots + numpy.polynomial.hermite.hermroots + numpy.polynomial.hermite_e.hermeroots + + Notes + ----- + The root estimates are obtained as the eigenvalues of the companion + matrix, Roots far from the origin of the complex plane may have large + errors due to the numerical instability of the series for such + values. Roots with multiplicity greater than 1 will also show larger + errors as the value of the series near such points is relatively + insensitive to errors in the roots. Isolated roots near the origin can + be improved by a few iterations of Newton's method. + + The Chebyshev series basis polynomials aren't powers of `x` so the + results of this function may seem unintuitive. + + Examples + -------- + >>> import numpy.polynomial.chebyshev as cheb + >>> cheb.chebroots((-1, 1,-1, 1)) # T3 - T2 + T1 - T0 has real roots + array([ -5.00000000e-01, 2.60860684e-17, 1.00000000e+00]) # may vary + + """ + # c is a trimmed copy + [c] = pu.as_series([c]) + if len(c) < 2: + return np.array([], dtype=c.dtype) + if len(c) == 2: + return np.array([-c[0]/c[1]]) + + # rotated companion matrix reduces error + m = chebcompanion(c)[::-1,::-1] + r = la.eigvals(m) + r.sort() + return r + + +def chebinterpolate(func, deg, args=()): + """Interpolate a function at the Chebyshev points of the first kind. + + Returns the Chebyshev series that interpolates `func` at the Chebyshev + points of the first kind in the interval [-1, 1]. The interpolating + series tends to a minmax approximation to `func` with increasing `deg` + if the function is continuous in the interval. + + .. versionadded:: 1.14.0 + + Parameters + ---------- + func : function + The function to be approximated. It must be a function of a single + variable of the form ``f(x, a, b, c...)``, where ``a, b, c...`` are + extra arguments passed in the `args` parameter. + deg : int + Degree of the interpolating polynomial + args : tuple, optional + Extra arguments to be used in the function call. Default is no extra + arguments. + + Returns + ------- + coef : ndarray, shape (deg + 1,) + Chebyshev coefficients of the interpolating series ordered from low to + high. + + Examples + -------- + >>> import numpy.polynomial.chebyshev as C + >>> C.chebfromfunction(lambda x: np.tanh(x) + 0.5, 8) + array([ 5.00000000e-01, 8.11675684e-01, -9.86864911e-17, + -5.42457905e-02, -2.71387850e-16, 4.51658839e-03, + 2.46716228e-17, -3.79694221e-04, -3.26899002e-16]) + + Notes + ----- + + The Chebyshev polynomials used in the interpolation are orthogonal when + sampled at the Chebyshev points of the first kind. If it is desired to + constrain some of the coefficients they can simply be set to the desired + value after the interpolation, no new interpolation or fit is needed. This + is especially useful if it is known apriori that some of coefficients are + zero. For instance, if the function is even then the coefficients of the + terms of odd degree in the result can be set to zero. + + """ + deg = np.asarray(deg) + + # check arguments. + if deg.ndim > 0 or deg.dtype.kind not in 'iu' or deg.size == 0: + raise TypeError("deg must be an int") + if deg < 0: + raise ValueError("expected deg >= 0") + + order = deg + 1 + xcheb = chebpts1(order) + yfunc = func(xcheb, *args) + m = chebvander(xcheb, deg) + c = np.dot(m.T, yfunc) + c[0] /= order + c[1:] /= 0.5*order + + return c + + +def chebgauss(deg): + """ + Gauss-Chebyshev quadrature. + + Computes the sample points and weights for Gauss-Chebyshev quadrature. + These sample points and weights will correctly integrate polynomials of + degree :math:`2*deg - 1` or less over the interval :math:`[-1, 1]` with + the weight function :math:`f(x) = 1/\\sqrt{1 - x^2}`. + + Parameters + ---------- + deg : int + Number of sample points and weights. It must be >= 1. + + Returns + ------- + x : ndarray + 1-D ndarray containing the sample points. + y : ndarray + 1-D ndarray containing the weights. + + Notes + ----- + + .. versionadded:: 1.7.0 + + The results have only been tested up to degree 100, higher degrees may + be problematic. For Gauss-Chebyshev there are closed form solutions for + the sample points and weights. If n = `deg`, then + + .. math:: x_i = \\cos(\\pi (2 i - 1) / (2 n)) + + .. math:: w_i = \\pi / n + + """ + ideg = pu._deprecate_as_int(deg, "deg") + if ideg <= 0: + raise ValueError("deg must be a positive integer") + + x = np.cos(np.pi * np.arange(1, 2*ideg, 2) / (2.0*ideg)) + w = np.ones(ideg)*(np.pi/ideg) + + return x, w + + +def chebweight(x): + """ + The weight function of the Chebyshev polynomials. + + The weight function is :math:`1/\\sqrt{1 - x^2}` and the interval of + integration is :math:`[-1, 1]`. The Chebyshev polynomials are + orthogonal, but not normalized, with respect to this weight function. + + Parameters + ---------- + x : array_like + Values at which the weight function will be computed. + + Returns + ------- + w : ndarray + The weight function at `x`. + + Notes + ----- + + .. versionadded:: 1.7.0 + + """ + w = 1./(np.sqrt(1. + x) * np.sqrt(1. - x)) + return w + + +def chebpts1(npts): + """ + Chebyshev points of the first kind. + + The Chebyshev points of the first kind are the points ``cos(x)``, + where ``x = [pi*(k + .5)/npts for k in range(npts)]``. + + Parameters + ---------- + npts : int + Number of sample points desired. + + Returns + ------- + pts : ndarray + The Chebyshev points of the first kind. + + See Also + -------- + chebpts2 + + Notes + ----- + + .. versionadded:: 1.5.0 + + """ + _npts = int(npts) + if _npts != npts: + raise ValueError("npts must be integer") + if _npts < 1: + raise ValueError("npts must be >= 1") + + x = 0.5 * np.pi / _npts * np.arange(-_npts+1, _npts+1, 2) + return np.sin(x) + + +def chebpts2(npts): + """ + Chebyshev points of the second kind. + + The Chebyshev points of the second kind are the points ``cos(x)``, + where ``x = [pi*k/(npts - 1) for k in range(npts)]`` sorted in ascending + order. + + Parameters + ---------- + npts : int + Number of sample points desired. + + Returns + ------- + pts : ndarray + The Chebyshev points of the second kind. + + Notes + ----- + + .. versionadded:: 1.5.0 + + """ + _npts = int(npts) + if _npts != npts: + raise ValueError("npts must be integer") + if _npts < 2: + raise ValueError("npts must be >= 2") + + x = np.linspace(-np.pi, 0, _npts) + return np.cos(x) + + +# +# Chebyshev series class +# + +class Chebyshev(ABCPolyBase): + """A Chebyshev series class. + + The Chebyshev class provides the standard Python numerical methods + '+', '-', '*', '//', '%', 'divmod', '**', and '()' as well as the + methods listed below. + + Parameters + ---------- + coef : array_like + Chebyshev coefficients in order of increasing degree, i.e., + ``(1, 2, 3)`` gives ``1*T_0(x) + 2*T_1(x) + 3*T_2(x)``. + domain : (2,) array_like, optional + Domain to use. The interval ``[domain[0], domain[1]]`` is mapped + to the interval ``[window[0], window[1]]`` by shifting and scaling. + The default value is [-1, 1]. + window : (2,) array_like, optional + Window, see `domain` for its use. The default value is [-1, 1]. + + .. versionadded:: 1.6.0 + symbol : str, optional + Symbol used to represent the independent variable in string + representations of the polynomial expression, e.g. for printing. + The symbol must be a valid Python identifier. Default value is 'x'. + + .. versionadded:: 1.24 + + """ + # Virtual Functions + _add = staticmethod(chebadd) + _sub = staticmethod(chebsub) + _mul = staticmethod(chebmul) + _div = staticmethod(chebdiv) + _pow = staticmethod(chebpow) + _val = staticmethod(chebval) + _int = staticmethod(chebint) + _der = staticmethod(chebder) + _fit = staticmethod(chebfit) + _line = staticmethod(chebline) + _roots = staticmethod(chebroots) + _fromroots = staticmethod(chebfromroots) + + @classmethod + def interpolate(cls, func, deg, domain=None, args=()): + """Interpolate a function at the Chebyshev points of the first kind. + + Returns the series that interpolates `func` at the Chebyshev points of + the first kind scaled and shifted to the `domain`. The resulting series + tends to a minmax approximation of `func` when the function is + continuous in the domain. + + .. versionadded:: 1.14.0 + + Parameters + ---------- + func : function + The function to be interpolated. It must be a function of a single + variable of the form ``f(x, a, b, c...)``, where ``a, b, c...`` are + extra arguments passed in the `args` parameter. + deg : int + Degree of the interpolating polynomial. + domain : {None, [beg, end]}, optional + Domain over which `func` is interpolated. The default is None, in + which case the domain is [-1, 1]. + args : tuple, optional + Extra arguments to be used in the function call. Default is no + extra arguments. + + Returns + ------- + polynomial : Chebyshev instance + Interpolating Chebyshev instance. + + Notes + ----- + See `numpy.polynomial.chebfromfunction` for more details. + + """ + if domain is None: + domain = cls.domain + xfunc = lambda x: func(pu.mapdomain(x, cls.window, domain), *args) + coef = chebinterpolate(xfunc, deg) + return cls(coef, domain=domain) + + # Virtual properties + domain = np.array(chebdomain) + window = np.array(chebdomain) + basis_name = 'T' diff --git a/deepseekvl2/lib/python3.10/site-packages/numpy/polynomial/chebyshev.pyi b/deepseekvl2/lib/python3.10/site-packages/numpy/polynomial/chebyshev.pyi new file mode 100644 index 0000000000000000000000000000000000000000..e8113dbae780263de1bd99ae841df16a4646d761 --- /dev/null +++ b/deepseekvl2/lib/python3.10/site-packages/numpy/polynomial/chebyshev.pyi @@ -0,0 +1,51 @@ +from typing import Any + +from numpy import ndarray, dtype, int_ +from numpy.polynomial._polybase import ABCPolyBase +from numpy.polynomial.polyutils import trimcoef + +__all__: list[str] + +chebtrim = trimcoef + +def poly2cheb(pol): ... +def cheb2poly(c): ... + +chebdomain: ndarray[Any, dtype[int_]] +chebzero: ndarray[Any, dtype[int_]] +chebone: ndarray[Any, dtype[int_]] +chebx: ndarray[Any, dtype[int_]] + +def chebline(off, scl): ... +def chebfromroots(roots): ... +def chebadd(c1, c2): ... +def chebsub(c1, c2): ... +def chebmulx(c): ... +def chebmul(c1, c2): ... +def chebdiv(c1, c2): ... +def chebpow(c, pow, maxpower=...): ... +def chebder(c, m=..., scl=..., axis=...): ... +def chebint(c, m=..., k = ..., lbnd=..., scl=..., axis=...): ... +def chebval(x, c, tensor=...): ... +def chebval2d(x, y, c): ... +def chebgrid2d(x, y, c): ... +def chebval3d(x, y, z, c): ... +def chebgrid3d(x, y, z, c): ... +def chebvander(x, deg): ... +def chebvander2d(x, y, deg): ... +def chebvander3d(x, y, z, deg): ... +def chebfit(x, y, deg, rcond=..., full=..., w=...): ... +def chebcompanion(c): ... +def chebroots(c): ... +def chebinterpolate(func, deg, args = ...): ... +def chebgauss(deg): ... +def chebweight(x): ... +def chebpts1(npts): ... +def chebpts2(npts): ... + +class Chebyshev(ABCPolyBase): + @classmethod + def interpolate(cls, func, deg, domain=..., args = ...): ... + domain: Any + window: Any + basis_name: Any diff --git a/deepseekvl2/lib/python3.10/site-packages/numpy/polynomial/hermite.py b/deepseekvl2/lib/python3.10/site-packages/numpy/polynomial/hermite.py new file mode 100644 index 0000000000000000000000000000000000000000..210df25f5ca3ace7aaa8c7614936e305097a6195 --- /dev/null +++ b/deepseekvl2/lib/python3.10/site-packages/numpy/polynomial/hermite.py @@ -0,0 +1,1703 @@ +""" +============================================================== +Hermite Series, "Physicists" (:mod:`numpy.polynomial.hermite`) +============================================================== + +This module provides a number of objects (mostly functions) useful for +dealing with Hermite series, including a `Hermite` class that +encapsulates the usual arithmetic operations. (General information +on how this module represents and works with such polynomials is in the +docstring for its "parent" sub-package, `numpy.polynomial`). + +Classes +------- +.. autosummary:: + :toctree: generated/ + + Hermite + +Constants +--------- +.. autosummary:: + :toctree: generated/ + + hermdomain + hermzero + hermone + hermx + +Arithmetic +---------- +.. autosummary:: + :toctree: generated/ + + hermadd + hermsub + hermmulx + hermmul + hermdiv + hermpow + hermval + hermval2d + hermval3d + hermgrid2d + hermgrid3d + +Calculus +-------- +.. autosummary:: + :toctree: generated/ + + hermder + hermint + +Misc Functions +-------------- +.. autosummary:: + :toctree: generated/ + + hermfromroots + hermroots + hermvander + hermvander2d + hermvander3d + hermgauss + hermweight + hermcompanion + hermfit + hermtrim + hermline + herm2poly + poly2herm + +See also +-------- +`numpy.polynomial` + +""" +import numpy as np +import numpy.linalg as la +from numpy.core.multiarray import normalize_axis_index + +from . import polyutils as pu +from ._polybase import ABCPolyBase + +__all__ = [ + 'hermzero', 'hermone', 'hermx', 'hermdomain', 'hermline', 'hermadd', + 'hermsub', 'hermmulx', 'hermmul', 'hermdiv', 'hermpow', 'hermval', + 'hermder', 'hermint', 'herm2poly', 'poly2herm', 'hermfromroots', + 'hermvander', 'hermfit', 'hermtrim', 'hermroots', 'Hermite', + 'hermval2d', 'hermval3d', 'hermgrid2d', 'hermgrid3d', 'hermvander2d', + 'hermvander3d', 'hermcompanion', 'hermgauss', 'hermweight'] + +hermtrim = pu.trimcoef + + +def poly2herm(pol): + """ + poly2herm(pol) + + Convert a polynomial to a Hermite series. + + Convert an array representing the coefficients of a polynomial (relative + to the "standard" basis) ordered from lowest degree to highest, to an + array of the coefficients of the equivalent Hermite series, ordered + from lowest to highest degree. + + Parameters + ---------- + pol : array_like + 1-D array containing the polynomial coefficients + + Returns + ------- + c : ndarray + 1-D array containing the coefficients of the equivalent Hermite + series. + + See Also + -------- + herm2poly + + Notes + ----- + The easy way to do conversions between polynomial basis sets + is to use the convert method of a class instance. + + Examples + -------- + >>> from numpy.polynomial.hermite import poly2herm + >>> poly2herm(np.arange(4)) + array([1. , 2.75 , 0.5 , 0.375]) + + """ + [pol] = pu.as_series([pol]) + deg = len(pol) - 1 + res = 0 + for i in range(deg, -1, -1): + res = hermadd(hermmulx(res), pol[i]) + return res + + +def herm2poly(c): + """ + Convert a Hermite series to a polynomial. + + Convert an array representing the coefficients of a Hermite series, + ordered from lowest degree to highest, to an array of the coefficients + of the equivalent polynomial (relative to the "standard" basis) ordered + from lowest to highest degree. + + Parameters + ---------- + c : array_like + 1-D array containing the Hermite series coefficients, ordered + from lowest order term to highest. + + Returns + ------- + pol : ndarray + 1-D array containing the coefficients of the equivalent polynomial + (relative to the "standard" basis) ordered from lowest order term + to highest. + + See Also + -------- + poly2herm + + Notes + ----- + The easy way to do conversions between polynomial basis sets + is to use the convert method of a class instance. + + Examples + -------- + >>> from numpy.polynomial.hermite import herm2poly + >>> herm2poly([ 1. , 2.75 , 0.5 , 0.375]) + array([0., 1., 2., 3.]) + + """ + from .polynomial import polyadd, polysub, polymulx + + [c] = pu.as_series([c]) + n = len(c) + if n == 1: + return c + if n == 2: + c[1] *= 2 + return c + else: + c0 = c[-2] + c1 = c[-1] + # i is the current degree of c1 + for i in range(n - 1, 1, -1): + tmp = c0 + c0 = polysub(c[i - 2], c1*(2*(i - 1))) + c1 = polyadd(tmp, polymulx(c1)*2) + return polyadd(c0, polymulx(c1)*2) + +# +# These are constant arrays are of integer type so as to be compatible +# with the widest range of other types, such as Decimal. +# + +# Hermite +hermdomain = np.array([-1, 1]) + +# Hermite coefficients representing zero. +hermzero = np.array([0]) + +# Hermite coefficients representing one. +hermone = np.array([1]) + +# Hermite coefficients representing the identity x. +hermx = np.array([0, 1/2]) + + +def hermline(off, scl): + """ + Hermite series whose graph is a straight line. + + + + Parameters + ---------- + off, scl : scalars + The specified line is given by ``off + scl*x``. + + Returns + ------- + y : ndarray + This module's representation of the Hermite series for + ``off + scl*x``. + + See Also + -------- + numpy.polynomial.polynomial.polyline + numpy.polynomial.chebyshev.chebline + numpy.polynomial.legendre.legline + numpy.polynomial.laguerre.lagline + numpy.polynomial.hermite_e.hermeline + + Examples + -------- + >>> from numpy.polynomial.hermite import hermline, hermval + >>> hermval(0,hermline(3, 2)) + 3.0 + >>> hermval(1,hermline(3, 2)) + 5.0 + + """ + if scl != 0: + return np.array([off, scl/2]) + else: + return np.array([off]) + + +def hermfromroots(roots): + """ + Generate a Hermite series with given roots. + + The function returns the coefficients of the polynomial + + .. math:: p(x) = (x - r_0) * (x - r_1) * ... * (x - r_n), + + in Hermite form, where the `r_n` are the roots specified in `roots`. + If a zero has multiplicity n, then it must appear in `roots` n times. + For instance, if 2 is a root of multiplicity three and 3 is a root of + multiplicity 2, then `roots` looks something like [2, 2, 2, 3, 3]. The + roots can appear in any order. + + If the returned coefficients are `c`, then + + .. math:: p(x) = c_0 + c_1 * H_1(x) + ... + c_n * H_n(x) + + The coefficient of the last term is not generally 1 for monic + polynomials in Hermite form. + + Parameters + ---------- + roots : array_like + Sequence containing the roots. + + Returns + ------- + out : ndarray + 1-D array of coefficients. If all roots are real then `out` is a + real array, if some of the roots are complex, then `out` is complex + even if all the coefficients in the result are real (see Examples + below). + + See Also + -------- + numpy.polynomial.polynomial.polyfromroots + numpy.polynomial.legendre.legfromroots + numpy.polynomial.laguerre.lagfromroots + numpy.polynomial.chebyshev.chebfromroots + numpy.polynomial.hermite_e.hermefromroots + + Examples + -------- + >>> from numpy.polynomial.hermite import hermfromroots, hermval + >>> coef = hermfromroots((-1, 0, 1)) + >>> hermval((-1, 0, 1), coef) + array([0., 0., 0.]) + >>> coef = hermfromroots((-1j, 1j)) + >>> hermval((-1j, 1j), coef) + array([0.+0.j, 0.+0.j]) + + """ + return pu._fromroots(hermline, hermmul, roots) + + +def hermadd(c1, c2): + """ + Add one Hermite series to another. + + Returns the sum of two Hermite series `c1` + `c2`. The arguments + are sequences of coefficients ordered from lowest order term to + highest, i.e., [1,2,3] represents the series ``P_0 + 2*P_1 + 3*P_2``. + + Parameters + ---------- + c1, c2 : array_like + 1-D arrays of Hermite series coefficients ordered from low to + high. + + Returns + ------- + out : ndarray + Array representing the Hermite series of their sum. + + See Also + -------- + hermsub, hermmulx, hermmul, hermdiv, hermpow + + Notes + ----- + Unlike multiplication, division, etc., the sum of two Hermite series + is a Hermite series (without having to "reproject" the result onto + the basis set) so addition, just like that of "standard" polynomials, + is simply "component-wise." + + Examples + -------- + >>> from numpy.polynomial.hermite import hermadd + >>> hermadd([1, 2, 3], [1, 2, 3, 4]) + array([2., 4., 6., 4.]) + + """ + return pu._add(c1, c2) + + +def hermsub(c1, c2): + """ + Subtract one Hermite series from another. + + Returns the difference of two Hermite series `c1` - `c2`. The + sequences of coefficients are from lowest order term to highest, i.e., + [1,2,3] represents the series ``P_0 + 2*P_1 + 3*P_2``. + + Parameters + ---------- + c1, c2 : array_like + 1-D arrays of Hermite series coefficients ordered from low to + high. + + Returns + ------- + out : ndarray + Of Hermite series coefficients representing their difference. + + See Also + -------- + hermadd, hermmulx, hermmul, hermdiv, hermpow + + Notes + ----- + Unlike multiplication, division, etc., the difference of two Hermite + series is a Hermite series (without having to "reproject" the result + onto the basis set) so subtraction, just like that of "standard" + polynomials, is simply "component-wise." + + Examples + -------- + >>> from numpy.polynomial.hermite import hermsub + >>> hermsub([1, 2, 3, 4], [1, 2, 3]) + array([0., 0., 0., 4.]) + + """ + return pu._sub(c1, c2) + + +def hermmulx(c): + """Multiply a Hermite series by x. + + Multiply the Hermite series `c` by x, where x is the independent + variable. + + + Parameters + ---------- + c : array_like + 1-D array of Hermite series coefficients ordered from low to + high. + + Returns + ------- + out : ndarray + Array representing the result of the multiplication. + + See Also + -------- + hermadd, hermsub, hermmul, hermdiv, hermpow + + Notes + ----- + The multiplication uses the recursion relationship for Hermite + polynomials in the form + + .. math:: + + xP_i(x) = (P_{i + 1}(x)/2 + i*P_{i - 1}(x)) + + Examples + -------- + >>> from numpy.polynomial.hermite import hermmulx + >>> hermmulx([1, 2, 3]) + array([2. , 6.5, 1. , 1.5]) + + """ + # c is a trimmed copy + [c] = pu.as_series([c]) + # The zero series needs special treatment + if len(c) == 1 and c[0] == 0: + return c + + prd = np.empty(len(c) + 1, dtype=c.dtype) + prd[0] = c[0]*0 + prd[1] = c[0]/2 + for i in range(1, len(c)): + prd[i + 1] = c[i]/2 + prd[i - 1] += c[i]*i + return prd + + +def hermmul(c1, c2): + """ + Multiply one Hermite series by another. + + Returns the product of two Hermite series `c1` * `c2`. The arguments + are sequences of coefficients, from lowest order "term" to highest, + e.g., [1,2,3] represents the series ``P_0 + 2*P_1 + 3*P_2``. + + Parameters + ---------- + c1, c2 : array_like + 1-D arrays of Hermite series coefficients ordered from low to + high. + + Returns + ------- + out : ndarray + Of Hermite series coefficients representing their product. + + See Also + -------- + hermadd, hermsub, hermmulx, hermdiv, hermpow + + Notes + ----- + In general, the (polynomial) product of two C-series results in terms + that are not in the Hermite polynomial basis set. Thus, to express + the product as a Hermite series, it is necessary to "reproject" the + product onto said basis set, which may produce "unintuitive" (but + correct) results; see Examples section below. + + Examples + -------- + >>> from numpy.polynomial.hermite import hermmul + >>> hermmul([1, 2, 3], [0, 1, 2]) + array([52., 29., 52., 7., 6.]) + + """ + # s1, s2 are trimmed copies + [c1, c2] = pu.as_series([c1, c2]) + + if len(c1) > len(c2): + c = c2 + xs = c1 + else: + c = c1 + xs = c2 + + if len(c) == 1: + c0 = c[0]*xs + c1 = 0 + elif len(c) == 2: + c0 = c[0]*xs + c1 = c[1]*xs + else: + nd = len(c) + c0 = c[-2]*xs + c1 = c[-1]*xs + for i in range(3, len(c) + 1): + tmp = c0 + nd = nd - 1 + c0 = hermsub(c[-i]*xs, c1*(2*(nd - 1))) + c1 = hermadd(tmp, hermmulx(c1)*2) + return hermadd(c0, hermmulx(c1)*2) + + +def hermdiv(c1, c2): + """ + Divide one Hermite series by another. + + Returns the quotient-with-remainder of two Hermite series + `c1` / `c2`. The arguments are sequences of coefficients from lowest + order "term" to highest, e.g., [1,2,3] represents the series + ``P_0 + 2*P_1 + 3*P_2``. + + Parameters + ---------- + c1, c2 : array_like + 1-D arrays of Hermite series coefficients ordered from low to + high. + + Returns + ------- + [quo, rem] : ndarrays + Of Hermite series coefficients representing the quotient and + remainder. + + See Also + -------- + hermadd, hermsub, hermmulx, hermmul, hermpow + + Notes + ----- + In general, the (polynomial) division of one Hermite series by another + results in quotient and remainder terms that are not in the Hermite + polynomial basis set. Thus, to express these results as a Hermite + series, it is necessary to "reproject" the results onto the Hermite + basis set, which may produce "unintuitive" (but correct) results; see + Examples section below. + + Examples + -------- + >>> from numpy.polynomial.hermite import hermdiv + >>> hermdiv([ 52., 29., 52., 7., 6.], [0, 1, 2]) + (array([1., 2., 3.]), array([0.])) + >>> hermdiv([ 54., 31., 52., 7., 6.], [0, 1, 2]) + (array([1., 2., 3.]), array([2., 2.])) + >>> hermdiv([ 53., 30., 52., 7., 6.], [0, 1, 2]) + (array([1., 2., 3.]), array([1., 1.])) + + """ + return pu._div(hermmul, c1, c2) + + +def hermpow(c, pow, maxpower=16): + """Raise a Hermite series to a power. + + Returns the Hermite series `c` raised to the power `pow`. The + argument `c` is a sequence of coefficients ordered from low to high. + i.e., [1,2,3] is the series ``P_0 + 2*P_1 + 3*P_2.`` + + Parameters + ---------- + c : array_like + 1-D array of Hermite series coefficients ordered from low to + high. + pow : integer + Power to which the series will be raised + maxpower : integer, optional + Maximum power allowed. This is mainly to limit growth of the series + to unmanageable size. Default is 16 + + Returns + ------- + coef : ndarray + Hermite series of power. + + See Also + -------- + hermadd, hermsub, hermmulx, hermmul, hermdiv + + Examples + -------- + >>> from numpy.polynomial.hermite import hermpow + >>> hermpow([1, 2, 3], 2) + array([81., 52., 82., 12., 9.]) + + """ + return pu._pow(hermmul, c, pow, maxpower) + + +def hermder(c, m=1, scl=1, axis=0): + """ + Differentiate a Hermite series. + + Returns the Hermite series coefficients `c` differentiated `m` times + along `axis`. At each iteration the result is multiplied by `scl` (the + scaling factor is for use in a linear change of variable). The argument + `c` is an array of coefficients from low to high degree along each + axis, e.g., [1,2,3] represents the series ``1*H_0 + 2*H_1 + 3*H_2`` + while [[1,2],[1,2]] represents ``1*H_0(x)*H_0(y) + 1*H_1(x)*H_0(y) + + 2*H_0(x)*H_1(y) + 2*H_1(x)*H_1(y)`` if axis=0 is ``x`` and axis=1 is + ``y``. + + Parameters + ---------- + c : array_like + Array of Hermite series coefficients. If `c` is multidimensional the + different axis correspond to different variables with the degree in + each axis given by the corresponding index. + m : int, optional + Number of derivatives taken, must be non-negative. (Default: 1) + scl : scalar, optional + Each differentiation is multiplied by `scl`. The end result is + multiplication by ``scl**m``. This is for use in a linear change of + variable. (Default: 1) + axis : int, optional + Axis over which the derivative is taken. (Default: 0). + + .. versionadded:: 1.7.0 + + Returns + ------- + der : ndarray + Hermite series of the derivative. + + See Also + -------- + hermint + + Notes + ----- + In general, the result of differentiating a Hermite series does not + resemble the same operation on a power series. Thus the result of this + function may be "unintuitive," albeit correct; see Examples section + below. + + Examples + -------- + >>> from numpy.polynomial.hermite import hermder + >>> hermder([ 1. , 0.5, 0.5, 0.5]) + array([1., 2., 3.]) + >>> hermder([-0.5, 1./2., 1./8., 1./12., 1./16.], m=2) + array([1., 2., 3.]) + + """ + c = np.array(c, ndmin=1, copy=True) + if c.dtype.char in '?bBhHiIlLqQpP': + c = c.astype(np.double) + cnt = pu._deprecate_as_int(m, "the order of derivation") + iaxis = pu._deprecate_as_int(axis, "the axis") + if cnt < 0: + raise ValueError("The order of derivation must be non-negative") + iaxis = normalize_axis_index(iaxis, c.ndim) + + if cnt == 0: + return c + + c = np.moveaxis(c, iaxis, 0) + n = len(c) + if cnt >= n: + c = c[:1]*0 + else: + for i in range(cnt): + n = n - 1 + c *= scl + der = np.empty((n,) + c.shape[1:], dtype=c.dtype) + for j in range(n, 0, -1): + der[j - 1] = (2*j)*c[j] + c = der + c = np.moveaxis(c, 0, iaxis) + return c + + +def hermint(c, m=1, k=[], lbnd=0, scl=1, axis=0): + """ + Integrate a Hermite series. + + Returns the Hermite series coefficients `c` integrated `m` times from + `lbnd` along `axis`. At each iteration the resulting series is + **multiplied** by `scl` and an integration constant, `k`, is added. + The scaling factor is for use in a linear change of variable. ("Buyer + beware": note that, depending on what one is doing, one may want `scl` + to be the reciprocal of what one might expect; for more information, + see the Notes section below.) The argument `c` is an array of + coefficients from low to high degree along each axis, e.g., [1,2,3] + represents the series ``H_0 + 2*H_1 + 3*H_2`` while [[1,2],[1,2]] + represents ``1*H_0(x)*H_0(y) + 1*H_1(x)*H_0(y) + 2*H_0(x)*H_1(y) + + 2*H_1(x)*H_1(y)`` if axis=0 is ``x`` and axis=1 is ``y``. + + Parameters + ---------- + c : array_like + Array of Hermite series coefficients. If c is multidimensional the + different axis correspond to different variables with the degree in + each axis given by the corresponding index. + m : int, optional + Order of integration, must be positive. (Default: 1) + k : {[], list, scalar}, optional + Integration constant(s). The value of the first integral at + ``lbnd`` is the first value in the list, the value of the second + integral at ``lbnd`` is the second value, etc. If ``k == []`` (the + default), all constants are set to zero. If ``m == 1``, a single + scalar can be given instead of a list. + lbnd : scalar, optional + The lower bound of the integral. (Default: 0) + scl : scalar, optional + Following each integration the result is *multiplied* by `scl` + before the integration constant is added. (Default: 1) + axis : int, optional + Axis over which the integral is taken. (Default: 0). + + .. versionadded:: 1.7.0 + + Returns + ------- + S : ndarray + Hermite series coefficients of the integral. + + Raises + ------ + ValueError + If ``m < 0``, ``len(k) > m``, ``np.ndim(lbnd) != 0``, or + ``np.ndim(scl) != 0``. + + See Also + -------- + hermder + + Notes + ----- + Note that the result of each integration is *multiplied* by `scl`. + Why is this important to note? Say one is making a linear change of + variable :math:`u = ax + b` in an integral relative to `x`. Then + :math:`dx = du/a`, so one will need to set `scl` equal to + :math:`1/a` - perhaps not what one would have first thought. + + Also note that, in general, the result of integrating a C-series needs + to be "reprojected" onto the C-series basis set. Thus, typically, + the result of this function is "unintuitive," albeit correct; see + Examples section below. + + Examples + -------- + >>> from numpy.polynomial.hermite import hermint + >>> hermint([1,2,3]) # integrate once, value 0 at 0. + array([1. , 0.5, 0.5, 0.5]) + >>> hermint([1,2,3], m=2) # integrate twice, value & deriv 0 at 0 + array([-0.5 , 0.5 , 0.125 , 0.08333333, 0.0625 ]) # may vary + >>> hermint([1,2,3], k=1) # integrate once, value 1 at 0. + array([2. , 0.5, 0.5, 0.5]) + >>> hermint([1,2,3], lbnd=-1) # integrate once, value 0 at -1 + array([-2. , 0.5, 0.5, 0.5]) + >>> hermint([1,2,3], m=2, k=[1,2], lbnd=-1) + array([ 1.66666667, -0.5 , 0.125 , 0.08333333, 0.0625 ]) # may vary + + """ + c = np.array(c, ndmin=1, copy=True) + if c.dtype.char in '?bBhHiIlLqQpP': + c = c.astype(np.double) + if not np.iterable(k): + k = [k] + cnt = pu._deprecate_as_int(m, "the order of integration") + iaxis = pu._deprecate_as_int(axis, "the axis") + if cnt < 0: + raise ValueError("The order of integration must be non-negative") + if len(k) > cnt: + raise ValueError("Too many integration constants") + if np.ndim(lbnd) != 0: + raise ValueError("lbnd must be a scalar.") + if np.ndim(scl) != 0: + raise ValueError("scl must be a scalar.") + iaxis = normalize_axis_index(iaxis, c.ndim) + + if cnt == 0: + return c + + c = np.moveaxis(c, iaxis, 0) + k = list(k) + [0]*(cnt - len(k)) + for i in range(cnt): + n = len(c) + c *= scl + if n == 1 and np.all(c[0] == 0): + c[0] += k[i] + else: + tmp = np.empty((n + 1,) + c.shape[1:], dtype=c.dtype) + tmp[0] = c[0]*0 + tmp[1] = c[0]/2 + for j in range(1, n): + tmp[j + 1] = c[j]/(2*(j + 1)) + tmp[0] += k[i] - hermval(lbnd, tmp) + c = tmp + c = np.moveaxis(c, 0, iaxis) + return c + + +def hermval(x, c, tensor=True): + """ + Evaluate an Hermite series at points x. + + If `c` is of length `n + 1`, this function returns the value: + + .. math:: p(x) = c_0 * H_0(x) + c_1 * H_1(x) + ... + c_n * H_n(x) + + The parameter `x` is converted to an array only if it is a tuple or a + list, otherwise it is treated as a scalar. In either case, either `x` + or its elements must support multiplication and addition both with + themselves and with the elements of `c`. + + If `c` is a 1-D array, then `p(x)` will have the same shape as `x`. If + `c` is multidimensional, then the shape of the result depends on the + value of `tensor`. If `tensor` is true the shape will be c.shape[1:] + + x.shape. If `tensor` is false the shape will be c.shape[1:]. Note that + scalars have shape (,). + + Trailing zeros in the coefficients will be used in the evaluation, so + they should be avoided if efficiency is a concern. + + Parameters + ---------- + x : array_like, compatible object + If `x` is a list or tuple, it is converted to an ndarray, otherwise + it is left unchanged and treated as a scalar. In either case, `x` + or its elements must support addition and multiplication with + themselves and with the elements of `c`. + c : array_like + Array of coefficients ordered so that the coefficients for terms of + degree n are contained in c[n]. If `c` is multidimensional the + remaining indices enumerate multiple polynomials. In the two + dimensional case the coefficients may be thought of as stored in + the columns of `c`. + tensor : boolean, optional + If True, the shape of the coefficient array is extended with ones + on the right, one for each dimension of `x`. Scalars have dimension 0 + for this action. The result is that every column of coefficients in + `c` is evaluated for every element of `x`. If False, `x` is broadcast + over the columns of `c` for the evaluation. This keyword is useful + when `c` is multidimensional. The default value is True. + + .. versionadded:: 1.7.0 + + Returns + ------- + values : ndarray, algebra_like + The shape of the return value is described above. + + See Also + -------- + hermval2d, hermgrid2d, hermval3d, hermgrid3d + + Notes + ----- + The evaluation uses Clenshaw recursion, aka synthetic division. + + Examples + -------- + >>> from numpy.polynomial.hermite import hermval + >>> coef = [1,2,3] + >>> hermval(1, coef) + 11.0 + >>> hermval([[1,2],[3,4]], coef) + array([[ 11., 51.], + [115., 203.]]) + + """ + c = np.array(c, ndmin=1, copy=False) + if c.dtype.char in '?bBhHiIlLqQpP': + c = c.astype(np.double) + if isinstance(x, (tuple, list)): + x = np.asarray(x) + if isinstance(x, np.ndarray) and tensor: + c = c.reshape(c.shape + (1,)*x.ndim) + + x2 = x*2 + if len(c) == 1: + c0 = c[0] + c1 = 0 + elif len(c) == 2: + c0 = c[0] + c1 = c[1] + else: + nd = len(c) + c0 = c[-2] + c1 = c[-1] + for i in range(3, len(c) + 1): + tmp = c0 + nd = nd - 1 + c0 = c[-i] - c1*(2*(nd - 1)) + c1 = tmp + c1*x2 + return c0 + c1*x2 + + +def hermval2d(x, y, c): + """ + Evaluate a 2-D Hermite series at points (x, y). + + This function returns the values: + + .. math:: p(x,y) = \\sum_{i,j} c_{i,j} * H_i(x) * H_j(y) + + The parameters `x` and `y` are converted to arrays only if they are + tuples or a lists, otherwise they are treated as a scalars and they + must have the same shape after conversion. In either case, either `x` + and `y` or their elements must support multiplication and addition both + with themselves and with the elements of `c`. + + If `c` is a 1-D array a one is implicitly appended to its shape to make + it 2-D. The shape of the result will be c.shape[2:] + x.shape. + + Parameters + ---------- + x, y : array_like, compatible objects + The two dimensional series is evaluated at the points `(x, y)`, + where `x` and `y` must have the same shape. If `x` or `y` is a list + or tuple, it is first converted to an ndarray, otherwise it is left + unchanged and if it isn't an ndarray it is treated as a scalar. + c : array_like + Array of coefficients ordered so that the coefficient of the term + of multi-degree i,j is contained in ``c[i,j]``. If `c` has + dimension greater than two the remaining indices enumerate multiple + sets of coefficients. + + Returns + ------- + values : ndarray, compatible object + The values of the two dimensional polynomial at points formed with + pairs of corresponding values from `x` and `y`. + + See Also + -------- + hermval, hermgrid2d, hermval3d, hermgrid3d + + Notes + ----- + + .. versionadded:: 1.7.0 + + """ + return pu._valnd(hermval, c, x, y) + + +def hermgrid2d(x, y, c): + """ + Evaluate a 2-D Hermite series on the Cartesian product of x and y. + + This function returns the values: + + .. math:: p(a,b) = \\sum_{i,j} c_{i,j} * H_i(a) * H_j(b) + + where the points `(a, b)` consist of all pairs formed by taking + `a` from `x` and `b` from `y`. The resulting points form a grid with + `x` in the first dimension and `y` in the second. + + The parameters `x` and `y` are converted to arrays only if they are + tuples or a lists, otherwise they are treated as a scalars. In either + case, either `x` and `y` or their elements must support multiplication + and addition both with themselves and with the elements of `c`. + + If `c` has fewer than two dimensions, ones are implicitly appended to + its shape to make it 2-D. The shape of the result will be c.shape[2:] + + x.shape. + + Parameters + ---------- + x, y : array_like, compatible objects + The two dimensional series is evaluated at the points in the + Cartesian product of `x` and `y`. If `x` or `y` is a list or + tuple, it is first converted to an ndarray, otherwise it is left + unchanged and, if it isn't an ndarray, it is treated as a scalar. + c : array_like + Array of coefficients ordered so that the coefficients for terms of + degree i,j are contained in ``c[i,j]``. If `c` has dimension + greater than two the remaining indices enumerate multiple sets of + coefficients. + + Returns + ------- + values : ndarray, compatible object + The values of the two dimensional polynomial at points in the Cartesian + product of `x` and `y`. + + See Also + -------- + hermval, hermval2d, hermval3d, hermgrid3d + + Notes + ----- + + .. versionadded:: 1.7.0 + + """ + return pu._gridnd(hermval, c, x, y) + + +def hermval3d(x, y, z, c): + """ + Evaluate a 3-D Hermite series at points (x, y, z). + + This function returns the values: + + .. math:: p(x,y,z) = \\sum_{i,j,k} c_{i,j,k} * H_i(x) * H_j(y) * H_k(z) + + The parameters `x`, `y`, and `z` are converted to arrays only if + they are tuples or a lists, otherwise they are treated as a scalars and + they must have the same shape after conversion. In either case, either + `x`, `y`, and `z` or their elements must support multiplication and + addition both with themselves and with the elements of `c`. + + If `c` has fewer than 3 dimensions, ones are implicitly appended to its + shape to make it 3-D. The shape of the result will be c.shape[3:] + + x.shape. + + Parameters + ---------- + x, y, z : array_like, compatible object + The three dimensional series is evaluated at the points + `(x, y, z)`, where `x`, `y`, and `z` must have the same shape. If + any of `x`, `y`, or `z` is a list or tuple, it is first converted + to an ndarray, otherwise it is left unchanged and if it isn't an + ndarray it is treated as a scalar. + c : array_like + Array of coefficients ordered so that the coefficient of the term of + multi-degree i,j,k is contained in ``c[i,j,k]``. If `c` has dimension + greater than 3 the remaining indices enumerate multiple sets of + coefficients. + + Returns + ------- + values : ndarray, compatible object + The values of the multidimensional polynomial on points formed with + triples of corresponding values from `x`, `y`, and `z`. + + See Also + -------- + hermval, hermval2d, hermgrid2d, hermgrid3d + + Notes + ----- + + .. versionadded:: 1.7.0 + + """ + return pu._valnd(hermval, c, x, y, z) + + +def hermgrid3d(x, y, z, c): + """ + Evaluate a 3-D Hermite series on the Cartesian product of x, y, and z. + + This function returns the values: + + .. math:: p(a,b,c) = \\sum_{i,j,k} c_{i,j,k} * H_i(a) * H_j(b) * H_k(c) + + where the points `(a, b, c)` consist of all triples formed by taking + `a` from `x`, `b` from `y`, and `c` from `z`. The resulting points form + a grid with `x` in the first dimension, `y` in the second, and `z` in + the third. + + The parameters `x`, `y`, and `z` are converted to arrays only if they + are tuples or a lists, otherwise they are treated as a scalars. In + either case, either `x`, `y`, and `z` or their elements must support + multiplication and addition both with themselves and with the elements + of `c`. + + If `c` has fewer than three dimensions, ones are implicitly appended to + its shape to make it 3-D. The shape of the result will be c.shape[3:] + + x.shape + y.shape + z.shape. + + Parameters + ---------- + x, y, z : array_like, compatible objects + The three dimensional series is evaluated at the points in the + Cartesian product of `x`, `y`, and `z`. If `x`,`y`, or `z` is a + list or tuple, it is first converted to an ndarray, otherwise it is + left unchanged and, if it isn't an ndarray, it is treated as a + scalar. + c : array_like + Array of coefficients ordered so that the coefficients for terms of + degree i,j are contained in ``c[i,j]``. If `c` has dimension + greater than two the remaining indices enumerate multiple sets of + coefficients. + + Returns + ------- + values : ndarray, compatible object + The values of the two dimensional polynomial at points in the Cartesian + product of `x` and `y`. + + See Also + -------- + hermval, hermval2d, hermgrid2d, hermval3d + + Notes + ----- + + .. versionadded:: 1.7.0 + + """ + return pu._gridnd(hermval, c, x, y, z) + + +def hermvander(x, deg): + """Pseudo-Vandermonde matrix of given degree. + + Returns the pseudo-Vandermonde matrix of degree `deg` and sample points + `x`. The pseudo-Vandermonde matrix is defined by + + .. math:: V[..., i] = H_i(x), + + where `0 <= i <= deg`. The leading indices of `V` index the elements of + `x` and the last index is the degree of the Hermite polynomial. + + If `c` is a 1-D array of coefficients of length `n + 1` and `V` is the + array ``V = hermvander(x, n)``, then ``np.dot(V, c)`` and + ``hermval(x, c)`` are the same up to roundoff. This equivalence is + useful both for least squares fitting and for the evaluation of a large + number of Hermite series of the same degree and sample points. + + Parameters + ---------- + x : array_like + Array of points. The dtype is converted to float64 or complex128 + depending on whether any of the elements are complex. If `x` is + scalar it is converted to a 1-D array. + deg : int + Degree of the resulting matrix. + + Returns + ------- + vander : ndarray + The pseudo-Vandermonde matrix. The shape of the returned matrix is + ``x.shape + (deg + 1,)``, where The last index is the degree of the + corresponding Hermite polynomial. The dtype will be the same as + the converted `x`. + + Examples + -------- + >>> from numpy.polynomial.hermite import hermvander + >>> x = np.array([-1, 0, 1]) + >>> hermvander(x, 3) + array([[ 1., -2., 2., 4.], + [ 1., 0., -2., -0.], + [ 1., 2., 2., -4.]]) + + """ + ideg = pu._deprecate_as_int(deg, "deg") + if ideg < 0: + raise ValueError("deg must be non-negative") + + x = np.array(x, copy=False, ndmin=1) + 0.0 + dims = (ideg + 1,) + x.shape + dtyp = x.dtype + v = np.empty(dims, dtype=dtyp) + v[0] = x*0 + 1 + if ideg > 0: + x2 = x*2 + v[1] = x2 + for i in range(2, ideg + 1): + v[i] = (v[i-1]*x2 - v[i-2]*(2*(i - 1))) + return np.moveaxis(v, 0, -1) + + +def hermvander2d(x, y, deg): + """Pseudo-Vandermonde matrix of given degrees. + + Returns the pseudo-Vandermonde matrix of degrees `deg` and sample + points `(x, y)`. The pseudo-Vandermonde matrix is defined by + + .. math:: V[..., (deg[1] + 1)*i + j] = H_i(x) * H_j(y), + + where `0 <= i <= deg[0]` and `0 <= j <= deg[1]`. The leading indices of + `V` index the points `(x, y)` and the last index encodes the degrees of + the Hermite polynomials. + + If ``V = hermvander2d(x, y, [xdeg, ydeg])``, then the columns of `V` + correspond to the elements of a 2-D coefficient array `c` of shape + (xdeg + 1, ydeg + 1) in the order + + .. math:: c_{00}, c_{01}, c_{02} ... , c_{10}, c_{11}, c_{12} ... + + and ``np.dot(V, c.flat)`` and ``hermval2d(x, y, c)`` will be the same + up to roundoff. This equivalence is useful both for least squares + fitting and for the evaluation of a large number of 2-D Hermite + series of the same degrees and sample points. + + Parameters + ---------- + x, y : array_like + Arrays of point coordinates, all of the same shape. The dtypes + will be converted to either float64 or complex128 depending on + whether any of the elements are complex. Scalars are converted to 1-D + arrays. + deg : list of ints + List of maximum degrees of the form [x_deg, y_deg]. + + Returns + ------- + vander2d : ndarray + The shape of the returned matrix is ``x.shape + (order,)``, where + :math:`order = (deg[0]+1)*(deg[1]+1)`. The dtype will be the same + as the converted `x` and `y`. + + See Also + -------- + hermvander, hermvander3d, hermval2d, hermval3d + + Notes + ----- + + .. versionadded:: 1.7.0 + + """ + return pu._vander_nd_flat((hermvander, hermvander), (x, y), deg) + + +def hermvander3d(x, y, z, deg): + """Pseudo-Vandermonde matrix of given degrees. + + Returns the pseudo-Vandermonde matrix of degrees `deg` and sample + points `(x, y, z)`. If `l, m, n` are the given degrees in `x, y, z`, + then The pseudo-Vandermonde matrix is defined by + + .. math:: V[..., (m+1)(n+1)i + (n+1)j + k] = H_i(x)*H_j(y)*H_k(z), + + where `0 <= i <= l`, `0 <= j <= m`, and `0 <= j <= n`. The leading + indices of `V` index the points `(x, y, z)` and the last index encodes + the degrees of the Hermite polynomials. + + If ``V = hermvander3d(x, y, z, [xdeg, ydeg, zdeg])``, then the columns + of `V` correspond to the elements of a 3-D coefficient array `c` of + shape (xdeg + 1, ydeg + 1, zdeg + 1) in the order + + .. math:: c_{000}, c_{001}, c_{002},... , c_{010}, c_{011}, c_{012},... + + and ``np.dot(V, c.flat)`` and ``hermval3d(x, y, z, c)`` will be the + same up to roundoff. This equivalence is useful both for least squares + fitting and for the evaluation of a large number of 3-D Hermite + series of the same degrees and sample points. + + Parameters + ---------- + x, y, z : array_like + Arrays of point coordinates, all of the same shape. The dtypes will + be converted to either float64 or complex128 depending on whether + any of the elements are complex. Scalars are converted to 1-D + arrays. + deg : list of ints + List of maximum degrees of the form [x_deg, y_deg, z_deg]. + + Returns + ------- + vander3d : ndarray + The shape of the returned matrix is ``x.shape + (order,)``, where + :math:`order = (deg[0]+1)*(deg[1]+1)*(deg[2]+1)`. The dtype will + be the same as the converted `x`, `y`, and `z`. + + See Also + -------- + hermvander, hermvander3d, hermval2d, hermval3d + + Notes + ----- + + .. versionadded:: 1.7.0 + + """ + return pu._vander_nd_flat((hermvander, hermvander, hermvander), (x, y, z), deg) + + +def hermfit(x, y, deg, rcond=None, full=False, w=None): + """ + Least squares fit of Hermite series to data. + + Return the coefficients of a Hermite series of degree `deg` that is the + least squares fit to the data values `y` given at points `x`. If `y` is + 1-D the returned coefficients will also be 1-D. If `y` is 2-D multiple + fits are done, one for each column of `y`, and the resulting + coefficients are stored in the corresponding columns of a 2-D return. + The fitted polynomial(s) are in the form + + .. math:: p(x) = c_0 + c_1 * H_1(x) + ... + c_n * H_n(x), + + where `n` is `deg`. + + Parameters + ---------- + x : array_like, shape (M,) + x-coordinates of the M sample points ``(x[i], y[i])``. + y : array_like, shape (M,) or (M, K) + y-coordinates of the sample points. Several data sets of sample + points sharing the same x-coordinates can be fitted at once by + passing in a 2D-array that contains one dataset per column. + deg : int or 1-D array_like + Degree(s) of the fitting polynomials. If `deg` is a single integer + all terms up to and including the `deg`'th term are included in the + fit. For NumPy versions >= 1.11.0 a list of integers specifying the + degrees of the terms to include may be used instead. + rcond : float, optional + Relative condition number of the fit. Singular values smaller than + this relative to the largest singular value will be ignored. The + default value is len(x)*eps, where eps is the relative precision of + the float type, about 2e-16 in most cases. + full : bool, optional + Switch determining nature of return value. When it is False (the + default) just the coefficients are returned, when True diagnostic + information from the singular value decomposition is also returned. + w : array_like, shape (`M`,), optional + Weights. If not None, the weight ``w[i]`` applies to the unsquared + residual ``y[i] - y_hat[i]`` at ``x[i]``. Ideally the weights are + chosen so that the errors of the products ``w[i]*y[i]`` all have the + same variance. When using inverse-variance weighting, use + ``w[i] = 1/sigma(y[i])``. The default value is None. + + Returns + ------- + coef : ndarray, shape (M,) or (M, K) + Hermite coefficients ordered from low to high. If `y` was 2-D, + the coefficients for the data in column k of `y` are in column + `k`. + + [residuals, rank, singular_values, rcond] : list + These values are only returned if ``full == True`` + + - residuals -- sum of squared residuals of the least squares fit + - rank -- the numerical rank of the scaled Vandermonde matrix + - singular_values -- singular values of the scaled Vandermonde matrix + - rcond -- value of `rcond`. + + For more details, see `numpy.linalg.lstsq`. + + Warns + ----- + RankWarning + The rank of the coefficient matrix in the least-squares fit is + deficient. The warning is only raised if ``full == False``. The + warnings can be turned off by + + >>> import warnings + >>> warnings.simplefilter('ignore', np.RankWarning) + + See Also + -------- + numpy.polynomial.chebyshev.chebfit + numpy.polynomial.legendre.legfit + numpy.polynomial.laguerre.lagfit + numpy.polynomial.polynomial.polyfit + numpy.polynomial.hermite_e.hermefit + hermval : Evaluates a Hermite series. + hermvander : Vandermonde matrix of Hermite series. + hermweight : Hermite weight function + numpy.linalg.lstsq : Computes a least-squares fit from the matrix. + scipy.interpolate.UnivariateSpline : Computes spline fits. + + Notes + ----- + The solution is the coefficients of the Hermite series `p` that + minimizes the sum of the weighted squared errors + + .. math:: E = \\sum_j w_j^2 * |y_j - p(x_j)|^2, + + where the :math:`w_j` are the weights. This problem is solved by + setting up the (typically) overdetermined matrix equation + + .. math:: V(x) * c = w * y, + + where `V` is the weighted pseudo Vandermonde matrix of `x`, `c` are the + coefficients to be solved for, `w` are the weights, `y` are the + observed values. This equation is then solved using the singular value + decomposition of `V`. + + If some of the singular values of `V` are so small that they are + neglected, then a `RankWarning` will be issued. This means that the + coefficient values may be poorly determined. Using a lower order fit + will usually get rid of the warning. The `rcond` parameter can also be + set to a value smaller than its default, but the resulting fit may be + spurious and have large contributions from roundoff error. + + Fits using Hermite series are probably most useful when the data can be + approximated by ``sqrt(w(x)) * p(x)``, where `w(x)` is the Hermite + weight. In that case the weight ``sqrt(w(x[i]))`` should be used + together with data values ``y[i]/sqrt(w(x[i]))``. The weight function is + available as `hermweight`. + + References + ---------- + .. [1] Wikipedia, "Curve fitting", + https://en.wikipedia.org/wiki/Curve_fitting + + Examples + -------- + >>> from numpy.polynomial.hermite import hermfit, hermval + >>> x = np.linspace(-10, 10) + >>> err = np.random.randn(len(x))/10 + >>> y = hermval(x, [1, 2, 3]) + err + >>> hermfit(x, y, 2) + array([1.0218, 1.9986, 2.9999]) # may vary + + """ + return pu._fit(hermvander, x, y, deg, rcond, full, w) + + +def hermcompanion(c): + """Return the scaled companion matrix of c. + + The basis polynomials are scaled so that the companion matrix is + symmetric when `c` is an Hermite basis polynomial. This provides + better eigenvalue estimates than the unscaled case and for basis + polynomials the eigenvalues are guaranteed to be real if + `numpy.linalg.eigvalsh` is used to obtain them. + + Parameters + ---------- + c : array_like + 1-D array of Hermite series coefficients ordered from low to high + degree. + + Returns + ------- + mat : ndarray + Scaled companion matrix of dimensions (deg, deg). + + Notes + ----- + + .. versionadded:: 1.7.0 + + """ + # c is a trimmed copy + [c] = pu.as_series([c]) + if len(c) < 2: + raise ValueError('Series must have maximum degree of at least 1.') + if len(c) == 2: + return np.array([[-.5*c[0]/c[1]]]) + + n = len(c) - 1 + mat = np.zeros((n, n), dtype=c.dtype) + scl = np.hstack((1., 1./np.sqrt(2.*np.arange(n - 1, 0, -1)))) + scl = np.multiply.accumulate(scl)[::-1] + top = mat.reshape(-1)[1::n+1] + bot = mat.reshape(-1)[n::n+1] + top[...] = np.sqrt(.5*np.arange(1, n)) + bot[...] = top + mat[:, -1] -= scl*c[:-1]/(2.0*c[-1]) + return mat + + +def hermroots(c): + """ + Compute the roots of a Hermite series. + + Return the roots (a.k.a. "zeros") of the polynomial + + .. math:: p(x) = \\sum_i c[i] * H_i(x). + + Parameters + ---------- + c : 1-D array_like + 1-D array of coefficients. + + Returns + ------- + out : ndarray + Array of the roots of the series. If all the roots are real, + then `out` is also real, otherwise it is complex. + + See Also + -------- + numpy.polynomial.polynomial.polyroots + numpy.polynomial.legendre.legroots + numpy.polynomial.laguerre.lagroots + numpy.polynomial.chebyshev.chebroots + numpy.polynomial.hermite_e.hermeroots + + Notes + ----- + The root estimates are obtained as the eigenvalues of the companion + matrix, Roots far from the origin of the complex plane may have large + errors due to the numerical instability of the series for such + values. Roots with multiplicity greater than 1 will also show larger + errors as the value of the series near such points is relatively + insensitive to errors in the roots. Isolated roots near the origin can + be improved by a few iterations of Newton's method. + + The Hermite series basis polynomials aren't powers of `x` so the + results of this function may seem unintuitive. + + Examples + -------- + >>> from numpy.polynomial.hermite import hermroots, hermfromroots + >>> coef = hermfromroots([-1, 0, 1]) + >>> coef + array([0. , 0.25 , 0. , 0.125]) + >>> hermroots(coef) + array([-1.00000000e+00, -1.38777878e-17, 1.00000000e+00]) + + """ + # c is a trimmed copy + [c] = pu.as_series([c]) + if len(c) <= 1: + return np.array([], dtype=c.dtype) + if len(c) == 2: + return np.array([-.5*c[0]/c[1]]) + + # rotated companion matrix reduces error + m = hermcompanion(c)[::-1,::-1] + r = la.eigvals(m) + r.sort() + return r + + +def _normed_hermite_n(x, n): + """ + Evaluate a normalized Hermite polynomial. + + Compute the value of the normalized Hermite polynomial of degree ``n`` + at the points ``x``. + + + Parameters + ---------- + x : ndarray of double. + Points at which to evaluate the function + n : int + Degree of the normalized Hermite function to be evaluated. + + Returns + ------- + values : ndarray + The shape of the return value is described above. + + Notes + ----- + .. versionadded:: 1.10.0 + + This function is needed for finding the Gauss points and integration + weights for high degrees. The values of the standard Hermite functions + overflow when n >= 207. + + """ + if n == 0: + return np.full(x.shape, 1/np.sqrt(np.sqrt(np.pi))) + + c0 = 0. + c1 = 1./np.sqrt(np.sqrt(np.pi)) + nd = float(n) + for i in range(n - 1): + tmp = c0 + c0 = -c1*np.sqrt((nd - 1.)/nd) + c1 = tmp + c1*x*np.sqrt(2./nd) + nd = nd - 1.0 + return c0 + c1*x*np.sqrt(2) + + +def hermgauss(deg): + """ + Gauss-Hermite quadrature. + + Computes the sample points and weights for Gauss-Hermite quadrature. + These sample points and weights will correctly integrate polynomials of + degree :math:`2*deg - 1` or less over the interval :math:`[-\\inf, \\inf]` + with the weight function :math:`f(x) = \\exp(-x^2)`. + + Parameters + ---------- + deg : int + Number of sample points and weights. It must be >= 1. + + Returns + ------- + x : ndarray + 1-D ndarray containing the sample points. + y : ndarray + 1-D ndarray containing the weights. + + Notes + ----- + + .. versionadded:: 1.7.0 + + The results have only been tested up to degree 100, higher degrees may + be problematic. The weights are determined by using the fact that + + .. math:: w_k = c / (H'_n(x_k) * H_{n-1}(x_k)) + + where :math:`c` is a constant independent of :math:`k` and :math:`x_k` + is the k'th root of :math:`H_n`, and then scaling the results to get + the right value when integrating 1. + + """ + ideg = pu._deprecate_as_int(deg, "deg") + if ideg <= 0: + raise ValueError("deg must be a positive integer") + + # first approximation of roots. We use the fact that the companion + # matrix is symmetric in this case in order to obtain better zeros. + c = np.array([0]*deg + [1], dtype=np.float64) + m = hermcompanion(c) + x = la.eigvalsh(m) + + # improve roots by one application of Newton + dy = _normed_hermite_n(x, ideg) + df = _normed_hermite_n(x, ideg - 1) * np.sqrt(2*ideg) + x -= dy/df + + # compute the weights. We scale the factor to avoid possible numerical + # overflow. + fm = _normed_hermite_n(x, ideg - 1) + fm /= np.abs(fm).max() + w = 1/(fm * fm) + + # for Hermite we can also symmetrize + w = (w + w[::-1])/2 + x = (x - x[::-1])/2 + + # scale w to get the right value + w *= np.sqrt(np.pi) / w.sum() + + return x, w + + +def hermweight(x): + """ + Weight function of the Hermite polynomials. + + The weight function is :math:`\\exp(-x^2)` and the interval of + integration is :math:`[-\\inf, \\inf]`. the Hermite polynomials are + orthogonal, but not normalized, with respect to this weight function. + + Parameters + ---------- + x : array_like + Values at which the weight function will be computed. + + Returns + ------- + w : ndarray + The weight function at `x`. + + Notes + ----- + + .. versionadded:: 1.7.0 + + """ + w = np.exp(-x**2) + return w + + +# +# Hermite series class +# + +class Hermite(ABCPolyBase): + """An Hermite series class. + + The Hermite class provides the standard Python numerical methods + '+', '-', '*', '//', '%', 'divmod', '**', and '()' as well as the + attributes and methods listed in the `ABCPolyBase` documentation. + + Parameters + ---------- + coef : array_like + Hermite coefficients in order of increasing degree, i.e, + ``(1, 2, 3)`` gives ``1*H_0(x) + 2*H_1(X) + 3*H_2(x)``. + domain : (2,) array_like, optional + Domain to use. The interval ``[domain[0], domain[1]]`` is mapped + to the interval ``[window[0], window[1]]`` by shifting and scaling. + The default value is [-1, 1]. + window : (2,) array_like, optional + Window, see `domain` for its use. The default value is [-1, 1]. + + .. versionadded:: 1.6.0 + symbol : str, optional + Symbol used to represent the independent variable in string + representations of the polynomial expression, e.g. for printing. + The symbol must be a valid Python identifier. Default value is 'x'. + + .. versionadded:: 1.24 + + """ + # Virtual Functions + _add = staticmethod(hermadd) + _sub = staticmethod(hermsub) + _mul = staticmethod(hermmul) + _div = staticmethod(hermdiv) + _pow = staticmethod(hermpow) + _val = staticmethod(hermval) + _int = staticmethod(hermint) + _der = staticmethod(hermder) + _fit = staticmethod(hermfit) + _line = staticmethod(hermline) + _roots = staticmethod(hermroots) + _fromroots = staticmethod(hermfromroots) + + # Virtual properties + domain = np.array(hermdomain) + window = np.array(hermdomain) + basis_name = 'H' diff --git a/deepseekvl2/lib/python3.10/site-packages/numpy/polynomial/hermite_e.py b/deepseekvl2/lib/python3.10/site-packages/numpy/polynomial/hermite_e.py new file mode 100644 index 0000000000000000000000000000000000000000..bdf29405bee7788d5ca6a8677b8402b9a7af393e --- /dev/null +++ b/deepseekvl2/lib/python3.10/site-packages/numpy/polynomial/hermite_e.py @@ -0,0 +1,1695 @@ +""" +=================================================================== +HermiteE Series, "Probabilists" (:mod:`numpy.polynomial.hermite_e`) +=================================================================== + +This module provides a number of objects (mostly functions) useful for +dealing with Hermite_e series, including a `HermiteE` class that +encapsulates the usual arithmetic operations. (General information +on how this module represents and works with such polynomials is in the +docstring for its "parent" sub-package, `numpy.polynomial`). + +Classes +------- +.. autosummary:: + :toctree: generated/ + + HermiteE + +Constants +--------- +.. autosummary:: + :toctree: generated/ + + hermedomain + hermezero + hermeone + hermex + +Arithmetic +---------- +.. autosummary:: + :toctree: generated/ + + hermeadd + hermesub + hermemulx + hermemul + hermediv + hermepow + hermeval + hermeval2d + hermeval3d + hermegrid2d + hermegrid3d + +Calculus +-------- +.. autosummary:: + :toctree: generated/ + + hermeder + hermeint + +Misc Functions +-------------- +.. autosummary:: + :toctree: generated/ + + hermefromroots + hermeroots + hermevander + hermevander2d + hermevander3d + hermegauss + hermeweight + hermecompanion + hermefit + hermetrim + hermeline + herme2poly + poly2herme + +See also +-------- +`numpy.polynomial` + +""" +import numpy as np +import numpy.linalg as la +from numpy.core.multiarray import normalize_axis_index + +from . import polyutils as pu +from ._polybase import ABCPolyBase + +__all__ = [ + 'hermezero', 'hermeone', 'hermex', 'hermedomain', 'hermeline', + 'hermeadd', 'hermesub', 'hermemulx', 'hermemul', 'hermediv', + 'hermepow', 'hermeval', 'hermeder', 'hermeint', 'herme2poly', + 'poly2herme', 'hermefromroots', 'hermevander', 'hermefit', 'hermetrim', + 'hermeroots', 'HermiteE', 'hermeval2d', 'hermeval3d', 'hermegrid2d', + 'hermegrid3d', 'hermevander2d', 'hermevander3d', 'hermecompanion', + 'hermegauss', 'hermeweight'] + +hermetrim = pu.trimcoef + + +def poly2herme(pol): + """ + poly2herme(pol) + + Convert a polynomial to a Hermite series. + + Convert an array representing the coefficients of a polynomial (relative + to the "standard" basis) ordered from lowest degree to highest, to an + array of the coefficients of the equivalent Hermite series, ordered + from lowest to highest degree. + + Parameters + ---------- + pol : array_like + 1-D array containing the polynomial coefficients + + Returns + ------- + c : ndarray + 1-D array containing the coefficients of the equivalent Hermite + series. + + See Also + -------- + herme2poly + + Notes + ----- + The easy way to do conversions between polynomial basis sets + is to use the convert method of a class instance. + + Examples + -------- + >>> from numpy.polynomial.hermite_e import poly2herme + >>> poly2herme(np.arange(4)) + array([ 2., 10., 2., 3.]) + + """ + [pol] = pu.as_series([pol]) + deg = len(pol) - 1 + res = 0 + for i in range(deg, -1, -1): + res = hermeadd(hermemulx(res), pol[i]) + return res + + +def herme2poly(c): + """ + Convert a Hermite series to a polynomial. + + Convert an array representing the coefficients of a Hermite series, + ordered from lowest degree to highest, to an array of the coefficients + of the equivalent polynomial (relative to the "standard" basis) ordered + from lowest to highest degree. + + Parameters + ---------- + c : array_like + 1-D array containing the Hermite series coefficients, ordered + from lowest order term to highest. + + Returns + ------- + pol : ndarray + 1-D array containing the coefficients of the equivalent polynomial + (relative to the "standard" basis) ordered from lowest order term + to highest. + + See Also + -------- + poly2herme + + Notes + ----- + The easy way to do conversions between polynomial basis sets + is to use the convert method of a class instance. + + Examples + -------- + >>> from numpy.polynomial.hermite_e import herme2poly + >>> herme2poly([ 2., 10., 2., 3.]) + array([0., 1., 2., 3.]) + + """ + from .polynomial import polyadd, polysub, polymulx + + [c] = pu.as_series([c]) + n = len(c) + if n == 1: + return c + if n == 2: + return c + else: + c0 = c[-2] + c1 = c[-1] + # i is the current degree of c1 + for i in range(n - 1, 1, -1): + tmp = c0 + c0 = polysub(c[i - 2], c1*(i - 1)) + c1 = polyadd(tmp, polymulx(c1)) + return polyadd(c0, polymulx(c1)) + +# +# These are constant arrays are of integer type so as to be compatible +# with the widest range of other types, such as Decimal. +# + +# Hermite +hermedomain = np.array([-1, 1]) + +# Hermite coefficients representing zero. +hermezero = np.array([0]) + +# Hermite coefficients representing one. +hermeone = np.array([1]) + +# Hermite coefficients representing the identity x. +hermex = np.array([0, 1]) + + +def hermeline(off, scl): + """ + Hermite series whose graph is a straight line. + + Parameters + ---------- + off, scl : scalars + The specified line is given by ``off + scl*x``. + + Returns + ------- + y : ndarray + This module's representation of the Hermite series for + ``off + scl*x``. + + See Also + -------- + numpy.polynomial.polynomial.polyline + numpy.polynomial.chebyshev.chebline + numpy.polynomial.legendre.legline + numpy.polynomial.laguerre.lagline + numpy.polynomial.hermite.hermline + + Examples + -------- + >>> from numpy.polynomial.hermite_e import hermeline + >>> from numpy.polynomial.hermite_e import hermeline, hermeval + >>> hermeval(0,hermeline(3, 2)) + 3.0 + >>> hermeval(1,hermeline(3, 2)) + 5.0 + + """ + if scl != 0: + return np.array([off, scl]) + else: + return np.array([off]) + + +def hermefromroots(roots): + """ + Generate a HermiteE series with given roots. + + The function returns the coefficients of the polynomial + + .. math:: p(x) = (x - r_0) * (x - r_1) * ... * (x - r_n), + + in HermiteE form, where the `r_n` are the roots specified in `roots`. + If a zero has multiplicity n, then it must appear in `roots` n times. + For instance, if 2 is a root of multiplicity three and 3 is a root of + multiplicity 2, then `roots` looks something like [2, 2, 2, 3, 3]. The + roots can appear in any order. + + If the returned coefficients are `c`, then + + .. math:: p(x) = c_0 + c_1 * He_1(x) + ... + c_n * He_n(x) + + The coefficient of the last term is not generally 1 for monic + polynomials in HermiteE form. + + Parameters + ---------- + roots : array_like + Sequence containing the roots. + + Returns + ------- + out : ndarray + 1-D array of coefficients. If all roots are real then `out` is a + real array, if some of the roots are complex, then `out` is complex + even if all the coefficients in the result are real (see Examples + below). + + See Also + -------- + numpy.polynomial.polynomial.polyfromroots + numpy.polynomial.legendre.legfromroots + numpy.polynomial.laguerre.lagfromroots + numpy.polynomial.hermite.hermfromroots + numpy.polynomial.chebyshev.chebfromroots + + Examples + -------- + >>> from numpy.polynomial.hermite_e import hermefromroots, hermeval + >>> coef = hermefromroots((-1, 0, 1)) + >>> hermeval((-1, 0, 1), coef) + array([0., 0., 0.]) + >>> coef = hermefromroots((-1j, 1j)) + >>> hermeval((-1j, 1j), coef) + array([0.+0.j, 0.+0.j]) + + """ + return pu._fromroots(hermeline, hermemul, roots) + + +def hermeadd(c1, c2): + """ + Add one Hermite series to another. + + Returns the sum of two Hermite series `c1` + `c2`. The arguments + are sequences of coefficients ordered from lowest order term to + highest, i.e., [1,2,3] represents the series ``P_0 + 2*P_1 + 3*P_2``. + + Parameters + ---------- + c1, c2 : array_like + 1-D arrays of Hermite series coefficients ordered from low to + high. + + Returns + ------- + out : ndarray + Array representing the Hermite series of their sum. + + See Also + -------- + hermesub, hermemulx, hermemul, hermediv, hermepow + + Notes + ----- + Unlike multiplication, division, etc., the sum of two Hermite series + is a Hermite series (without having to "reproject" the result onto + the basis set) so addition, just like that of "standard" polynomials, + is simply "component-wise." + + Examples + -------- + >>> from numpy.polynomial.hermite_e import hermeadd + >>> hermeadd([1, 2, 3], [1, 2, 3, 4]) + array([2., 4., 6., 4.]) + + """ + return pu._add(c1, c2) + + +def hermesub(c1, c2): + """ + Subtract one Hermite series from another. + + Returns the difference of two Hermite series `c1` - `c2`. The + sequences of coefficients are from lowest order term to highest, i.e., + [1,2,3] represents the series ``P_0 + 2*P_1 + 3*P_2``. + + Parameters + ---------- + c1, c2 : array_like + 1-D arrays of Hermite series coefficients ordered from low to + high. + + Returns + ------- + out : ndarray + Of Hermite series coefficients representing their difference. + + See Also + -------- + hermeadd, hermemulx, hermemul, hermediv, hermepow + + Notes + ----- + Unlike multiplication, division, etc., the difference of two Hermite + series is a Hermite series (without having to "reproject" the result + onto the basis set) so subtraction, just like that of "standard" + polynomials, is simply "component-wise." + + Examples + -------- + >>> from numpy.polynomial.hermite_e import hermesub + >>> hermesub([1, 2, 3, 4], [1, 2, 3]) + array([0., 0., 0., 4.]) + + """ + return pu._sub(c1, c2) + + +def hermemulx(c): + """Multiply a Hermite series by x. + + Multiply the Hermite series `c` by x, where x is the independent + variable. + + + Parameters + ---------- + c : array_like + 1-D array of Hermite series coefficients ordered from low to + high. + + Returns + ------- + out : ndarray + Array representing the result of the multiplication. + + Notes + ----- + The multiplication uses the recursion relationship for Hermite + polynomials in the form + + .. math:: + + xP_i(x) = (P_{i + 1}(x) + iP_{i - 1}(x))) + + Examples + -------- + >>> from numpy.polynomial.hermite_e import hermemulx + >>> hermemulx([1, 2, 3]) + array([2., 7., 2., 3.]) + + """ + # c is a trimmed copy + [c] = pu.as_series([c]) + # The zero series needs special treatment + if len(c) == 1 and c[0] == 0: + return c + + prd = np.empty(len(c) + 1, dtype=c.dtype) + prd[0] = c[0]*0 + prd[1] = c[0] + for i in range(1, len(c)): + prd[i + 1] = c[i] + prd[i - 1] += c[i]*i + return prd + + +def hermemul(c1, c2): + """ + Multiply one Hermite series by another. + + Returns the product of two Hermite series `c1` * `c2`. The arguments + are sequences of coefficients, from lowest order "term" to highest, + e.g., [1,2,3] represents the series ``P_0 + 2*P_1 + 3*P_2``. + + Parameters + ---------- + c1, c2 : array_like + 1-D arrays of Hermite series coefficients ordered from low to + high. + + Returns + ------- + out : ndarray + Of Hermite series coefficients representing their product. + + See Also + -------- + hermeadd, hermesub, hermemulx, hermediv, hermepow + + Notes + ----- + In general, the (polynomial) product of two C-series results in terms + that are not in the Hermite polynomial basis set. Thus, to express + the product as a Hermite series, it is necessary to "reproject" the + product onto said basis set, which may produce "unintuitive" (but + correct) results; see Examples section below. + + Examples + -------- + >>> from numpy.polynomial.hermite_e import hermemul + >>> hermemul([1, 2, 3], [0, 1, 2]) + array([14., 15., 28., 7., 6.]) + + """ + # s1, s2 are trimmed copies + [c1, c2] = pu.as_series([c1, c2]) + + if len(c1) > len(c2): + c = c2 + xs = c1 + else: + c = c1 + xs = c2 + + if len(c) == 1: + c0 = c[0]*xs + c1 = 0 + elif len(c) == 2: + c0 = c[0]*xs + c1 = c[1]*xs + else: + nd = len(c) + c0 = c[-2]*xs + c1 = c[-1]*xs + for i in range(3, len(c) + 1): + tmp = c0 + nd = nd - 1 + c0 = hermesub(c[-i]*xs, c1*(nd - 1)) + c1 = hermeadd(tmp, hermemulx(c1)) + return hermeadd(c0, hermemulx(c1)) + + +def hermediv(c1, c2): + """ + Divide one Hermite series by another. + + Returns the quotient-with-remainder of two Hermite series + `c1` / `c2`. The arguments are sequences of coefficients from lowest + order "term" to highest, e.g., [1,2,3] represents the series + ``P_0 + 2*P_1 + 3*P_2``. + + Parameters + ---------- + c1, c2 : array_like + 1-D arrays of Hermite series coefficients ordered from low to + high. + + Returns + ------- + [quo, rem] : ndarrays + Of Hermite series coefficients representing the quotient and + remainder. + + See Also + -------- + hermeadd, hermesub, hermemulx, hermemul, hermepow + + Notes + ----- + In general, the (polynomial) division of one Hermite series by another + results in quotient and remainder terms that are not in the Hermite + polynomial basis set. Thus, to express these results as a Hermite + series, it is necessary to "reproject" the results onto the Hermite + basis set, which may produce "unintuitive" (but correct) results; see + Examples section below. + + Examples + -------- + >>> from numpy.polynomial.hermite_e import hermediv + >>> hermediv([ 14., 15., 28., 7., 6.], [0, 1, 2]) + (array([1., 2., 3.]), array([0.])) + >>> hermediv([ 15., 17., 28., 7., 6.], [0, 1, 2]) + (array([1., 2., 3.]), array([1., 2.])) + + """ + return pu._div(hermemul, c1, c2) + + +def hermepow(c, pow, maxpower=16): + """Raise a Hermite series to a power. + + Returns the Hermite series `c` raised to the power `pow`. The + argument `c` is a sequence of coefficients ordered from low to high. + i.e., [1,2,3] is the series ``P_0 + 2*P_1 + 3*P_2.`` + + Parameters + ---------- + c : array_like + 1-D array of Hermite series coefficients ordered from low to + high. + pow : integer + Power to which the series will be raised + maxpower : integer, optional + Maximum power allowed. This is mainly to limit growth of the series + to unmanageable size. Default is 16 + + Returns + ------- + coef : ndarray + Hermite series of power. + + See Also + -------- + hermeadd, hermesub, hermemulx, hermemul, hermediv + + Examples + -------- + >>> from numpy.polynomial.hermite_e import hermepow + >>> hermepow([1, 2, 3], 2) + array([23., 28., 46., 12., 9.]) + + """ + return pu._pow(hermemul, c, pow, maxpower) + + +def hermeder(c, m=1, scl=1, axis=0): + """ + Differentiate a Hermite_e series. + + Returns the series coefficients `c` differentiated `m` times along + `axis`. At each iteration the result is multiplied by `scl` (the + scaling factor is for use in a linear change of variable). The argument + `c` is an array of coefficients from low to high degree along each + axis, e.g., [1,2,3] represents the series ``1*He_0 + 2*He_1 + 3*He_2`` + while [[1,2],[1,2]] represents ``1*He_0(x)*He_0(y) + 1*He_1(x)*He_0(y) + + 2*He_0(x)*He_1(y) + 2*He_1(x)*He_1(y)`` if axis=0 is ``x`` and axis=1 + is ``y``. + + Parameters + ---------- + c : array_like + Array of Hermite_e series coefficients. If `c` is multidimensional + the different axis correspond to different variables with the + degree in each axis given by the corresponding index. + m : int, optional + Number of derivatives taken, must be non-negative. (Default: 1) + scl : scalar, optional + Each differentiation is multiplied by `scl`. The end result is + multiplication by ``scl**m``. This is for use in a linear change of + variable. (Default: 1) + axis : int, optional + Axis over which the derivative is taken. (Default: 0). + + .. versionadded:: 1.7.0 + + Returns + ------- + der : ndarray + Hermite series of the derivative. + + See Also + -------- + hermeint + + Notes + ----- + In general, the result of differentiating a Hermite series does not + resemble the same operation on a power series. Thus the result of this + function may be "unintuitive," albeit correct; see Examples section + below. + + Examples + -------- + >>> from numpy.polynomial.hermite_e import hermeder + >>> hermeder([ 1., 1., 1., 1.]) + array([1., 2., 3.]) + >>> hermeder([-0.25, 1., 1./2., 1./3., 1./4 ], m=2) + array([1., 2., 3.]) + + """ + c = np.array(c, ndmin=1, copy=True) + if c.dtype.char in '?bBhHiIlLqQpP': + c = c.astype(np.double) + cnt = pu._deprecate_as_int(m, "the order of derivation") + iaxis = pu._deprecate_as_int(axis, "the axis") + if cnt < 0: + raise ValueError("The order of derivation must be non-negative") + iaxis = normalize_axis_index(iaxis, c.ndim) + + if cnt == 0: + return c + + c = np.moveaxis(c, iaxis, 0) + n = len(c) + if cnt >= n: + return c[:1]*0 + else: + for i in range(cnt): + n = n - 1 + c *= scl + der = np.empty((n,) + c.shape[1:], dtype=c.dtype) + for j in range(n, 0, -1): + der[j - 1] = j*c[j] + c = der + c = np.moveaxis(c, 0, iaxis) + return c + + +def hermeint(c, m=1, k=[], lbnd=0, scl=1, axis=0): + """ + Integrate a Hermite_e series. + + Returns the Hermite_e series coefficients `c` integrated `m` times from + `lbnd` along `axis`. At each iteration the resulting series is + **multiplied** by `scl` and an integration constant, `k`, is added. + The scaling factor is for use in a linear change of variable. ("Buyer + beware": note that, depending on what one is doing, one may want `scl` + to be the reciprocal of what one might expect; for more information, + see the Notes section below.) The argument `c` is an array of + coefficients from low to high degree along each axis, e.g., [1,2,3] + represents the series ``H_0 + 2*H_1 + 3*H_2`` while [[1,2],[1,2]] + represents ``1*H_0(x)*H_0(y) + 1*H_1(x)*H_0(y) + 2*H_0(x)*H_1(y) + + 2*H_1(x)*H_1(y)`` if axis=0 is ``x`` and axis=1 is ``y``. + + Parameters + ---------- + c : array_like + Array of Hermite_e series coefficients. If c is multidimensional + the different axis correspond to different variables with the + degree in each axis given by the corresponding index. + m : int, optional + Order of integration, must be positive. (Default: 1) + k : {[], list, scalar}, optional + Integration constant(s). The value of the first integral at + ``lbnd`` is the first value in the list, the value of the second + integral at ``lbnd`` is the second value, etc. If ``k == []`` (the + default), all constants are set to zero. If ``m == 1``, a single + scalar can be given instead of a list. + lbnd : scalar, optional + The lower bound of the integral. (Default: 0) + scl : scalar, optional + Following each integration the result is *multiplied* by `scl` + before the integration constant is added. (Default: 1) + axis : int, optional + Axis over which the integral is taken. (Default: 0). + + .. versionadded:: 1.7.0 + + Returns + ------- + S : ndarray + Hermite_e series coefficients of the integral. + + Raises + ------ + ValueError + If ``m < 0``, ``len(k) > m``, ``np.ndim(lbnd) != 0``, or + ``np.ndim(scl) != 0``. + + See Also + -------- + hermeder + + Notes + ----- + Note that the result of each integration is *multiplied* by `scl`. + Why is this important to note? Say one is making a linear change of + variable :math:`u = ax + b` in an integral relative to `x`. Then + :math:`dx = du/a`, so one will need to set `scl` equal to + :math:`1/a` - perhaps not what one would have first thought. + + Also note that, in general, the result of integrating a C-series needs + to be "reprojected" onto the C-series basis set. Thus, typically, + the result of this function is "unintuitive," albeit correct; see + Examples section below. + + Examples + -------- + >>> from numpy.polynomial.hermite_e import hermeint + >>> hermeint([1, 2, 3]) # integrate once, value 0 at 0. + array([1., 1., 1., 1.]) + >>> hermeint([1, 2, 3], m=2) # integrate twice, value & deriv 0 at 0 + array([-0.25 , 1. , 0.5 , 0.33333333, 0.25 ]) # may vary + >>> hermeint([1, 2, 3], k=1) # integrate once, value 1 at 0. + array([2., 1., 1., 1.]) + >>> hermeint([1, 2, 3], lbnd=-1) # integrate once, value 0 at -1 + array([-1., 1., 1., 1.]) + >>> hermeint([1, 2, 3], m=2, k=[1, 2], lbnd=-1) + array([ 1.83333333, 0. , 0.5 , 0.33333333, 0.25 ]) # may vary + + """ + c = np.array(c, ndmin=1, copy=True) + if c.dtype.char in '?bBhHiIlLqQpP': + c = c.astype(np.double) + if not np.iterable(k): + k = [k] + cnt = pu._deprecate_as_int(m, "the order of integration") + iaxis = pu._deprecate_as_int(axis, "the axis") + if cnt < 0: + raise ValueError("The order of integration must be non-negative") + if len(k) > cnt: + raise ValueError("Too many integration constants") + if np.ndim(lbnd) != 0: + raise ValueError("lbnd must be a scalar.") + if np.ndim(scl) != 0: + raise ValueError("scl must be a scalar.") + iaxis = normalize_axis_index(iaxis, c.ndim) + + if cnt == 0: + return c + + c = np.moveaxis(c, iaxis, 0) + k = list(k) + [0]*(cnt - len(k)) + for i in range(cnt): + n = len(c) + c *= scl + if n == 1 and np.all(c[0] == 0): + c[0] += k[i] + else: + tmp = np.empty((n + 1,) + c.shape[1:], dtype=c.dtype) + tmp[0] = c[0]*0 + tmp[1] = c[0] + for j in range(1, n): + tmp[j + 1] = c[j]/(j + 1) + tmp[0] += k[i] - hermeval(lbnd, tmp) + c = tmp + c = np.moveaxis(c, 0, iaxis) + return c + + +def hermeval(x, c, tensor=True): + """ + Evaluate an HermiteE series at points x. + + If `c` is of length `n + 1`, this function returns the value: + + .. math:: p(x) = c_0 * He_0(x) + c_1 * He_1(x) + ... + c_n * He_n(x) + + The parameter `x` is converted to an array only if it is a tuple or a + list, otherwise it is treated as a scalar. In either case, either `x` + or its elements must support multiplication and addition both with + themselves and with the elements of `c`. + + If `c` is a 1-D array, then `p(x)` will have the same shape as `x`. If + `c` is multidimensional, then the shape of the result depends on the + value of `tensor`. If `tensor` is true the shape will be c.shape[1:] + + x.shape. If `tensor` is false the shape will be c.shape[1:]. Note that + scalars have shape (,). + + Trailing zeros in the coefficients will be used in the evaluation, so + they should be avoided if efficiency is a concern. + + Parameters + ---------- + x : array_like, compatible object + If `x` is a list or tuple, it is converted to an ndarray, otherwise + it is left unchanged and treated as a scalar. In either case, `x` + or its elements must support addition and multiplication with + with themselves and with the elements of `c`. + c : array_like + Array of coefficients ordered so that the coefficients for terms of + degree n are contained in c[n]. If `c` is multidimensional the + remaining indices enumerate multiple polynomials. In the two + dimensional case the coefficients may be thought of as stored in + the columns of `c`. + tensor : boolean, optional + If True, the shape of the coefficient array is extended with ones + on the right, one for each dimension of `x`. Scalars have dimension 0 + for this action. The result is that every column of coefficients in + `c` is evaluated for every element of `x`. If False, `x` is broadcast + over the columns of `c` for the evaluation. This keyword is useful + when `c` is multidimensional. The default value is True. + + .. versionadded:: 1.7.0 + + Returns + ------- + values : ndarray, algebra_like + The shape of the return value is described above. + + See Also + -------- + hermeval2d, hermegrid2d, hermeval3d, hermegrid3d + + Notes + ----- + The evaluation uses Clenshaw recursion, aka synthetic division. + + Examples + -------- + >>> from numpy.polynomial.hermite_e import hermeval + >>> coef = [1,2,3] + >>> hermeval(1, coef) + 3.0 + >>> hermeval([[1,2],[3,4]], coef) + array([[ 3., 14.], + [31., 54.]]) + + """ + c = np.array(c, ndmin=1, copy=False) + if c.dtype.char in '?bBhHiIlLqQpP': + c = c.astype(np.double) + if isinstance(x, (tuple, list)): + x = np.asarray(x) + if isinstance(x, np.ndarray) and tensor: + c = c.reshape(c.shape + (1,)*x.ndim) + + if len(c) == 1: + c0 = c[0] + c1 = 0 + elif len(c) == 2: + c0 = c[0] + c1 = c[1] + else: + nd = len(c) + c0 = c[-2] + c1 = c[-1] + for i in range(3, len(c) + 1): + tmp = c0 + nd = nd - 1 + c0 = c[-i] - c1*(nd - 1) + c1 = tmp + c1*x + return c0 + c1*x + + +def hermeval2d(x, y, c): + """ + Evaluate a 2-D HermiteE series at points (x, y). + + This function returns the values: + + .. math:: p(x,y) = \\sum_{i,j} c_{i,j} * He_i(x) * He_j(y) + + The parameters `x` and `y` are converted to arrays only if they are + tuples or a lists, otherwise they are treated as a scalars and they + must have the same shape after conversion. In either case, either `x` + and `y` or their elements must support multiplication and addition both + with themselves and with the elements of `c`. + + If `c` is a 1-D array a one is implicitly appended to its shape to make + it 2-D. The shape of the result will be c.shape[2:] + x.shape. + + Parameters + ---------- + x, y : array_like, compatible objects + The two dimensional series is evaluated at the points `(x, y)`, + where `x` and `y` must have the same shape. If `x` or `y` is a list + or tuple, it is first converted to an ndarray, otherwise it is left + unchanged and if it isn't an ndarray it is treated as a scalar. + c : array_like + Array of coefficients ordered so that the coefficient of the term + of multi-degree i,j is contained in ``c[i,j]``. If `c` has + dimension greater than two the remaining indices enumerate multiple + sets of coefficients. + + Returns + ------- + values : ndarray, compatible object + The values of the two dimensional polynomial at points formed with + pairs of corresponding values from `x` and `y`. + + See Also + -------- + hermeval, hermegrid2d, hermeval3d, hermegrid3d + + Notes + ----- + + .. versionadded:: 1.7.0 + + """ + return pu._valnd(hermeval, c, x, y) + + +def hermegrid2d(x, y, c): + """ + Evaluate a 2-D HermiteE series on the Cartesian product of x and y. + + This function returns the values: + + .. math:: p(a,b) = \\sum_{i,j} c_{i,j} * H_i(a) * H_j(b) + + where the points `(a, b)` consist of all pairs formed by taking + `a` from `x` and `b` from `y`. The resulting points form a grid with + `x` in the first dimension and `y` in the second. + + The parameters `x` and `y` are converted to arrays only if they are + tuples or a lists, otherwise they are treated as a scalars. In either + case, either `x` and `y` or their elements must support multiplication + and addition both with themselves and with the elements of `c`. + + If `c` has fewer than two dimensions, ones are implicitly appended to + its shape to make it 2-D. The shape of the result will be c.shape[2:] + + x.shape. + + Parameters + ---------- + x, y : array_like, compatible objects + The two dimensional series is evaluated at the points in the + Cartesian product of `x` and `y`. If `x` or `y` is a list or + tuple, it is first converted to an ndarray, otherwise it is left + unchanged and, if it isn't an ndarray, it is treated as a scalar. + c : array_like + Array of coefficients ordered so that the coefficients for terms of + degree i,j are contained in ``c[i,j]``. If `c` has dimension + greater than two the remaining indices enumerate multiple sets of + coefficients. + + Returns + ------- + values : ndarray, compatible object + The values of the two dimensional polynomial at points in the Cartesian + product of `x` and `y`. + + See Also + -------- + hermeval, hermeval2d, hermeval3d, hermegrid3d + + Notes + ----- + + .. versionadded:: 1.7.0 + + """ + return pu._gridnd(hermeval, c, x, y) + + +def hermeval3d(x, y, z, c): + """ + Evaluate a 3-D Hermite_e series at points (x, y, z). + + This function returns the values: + + .. math:: p(x,y,z) = \\sum_{i,j,k} c_{i,j,k} * He_i(x) * He_j(y) * He_k(z) + + The parameters `x`, `y`, and `z` are converted to arrays only if + they are tuples or a lists, otherwise they are treated as a scalars and + they must have the same shape after conversion. In either case, either + `x`, `y`, and `z` or their elements must support multiplication and + addition both with themselves and with the elements of `c`. + + If `c` has fewer than 3 dimensions, ones are implicitly appended to its + shape to make it 3-D. The shape of the result will be c.shape[3:] + + x.shape. + + Parameters + ---------- + x, y, z : array_like, compatible object + The three dimensional series is evaluated at the points + `(x, y, z)`, where `x`, `y`, and `z` must have the same shape. If + any of `x`, `y`, or `z` is a list or tuple, it is first converted + to an ndarray, otherwise it is left unchanged and if it isn't an + ndarray it is treated as a scalar. + c : array_like + Array of coefficients ordered so that the coefficient of the term of + multi-degree i,j,k is contained in ``c[i,j,k]``. If `c` has dimension + greater than 3 the remaining indices enumerate multiple sets of + coefficients. + + Returns + ------- + values : ndarray, compatible object + The values of the multidimensional polynomial on points formed with + triples of corresponding values from `x`, `y`, and `z`. + + See Also + -------- + hermeval, hermeval2d, hermegrid2d, hermegrid3d + + Notes + ----- + + .. versionadded:: 1.7.0 + + """ + return pu._valnd(hermeval, c, x, y, z) + + +def hermegrid3d(x, y, z, c): + """ + Evaluate a 3-D HermiteE series on the Cartesian product of x, y, and z. + + This function returns the values: + + .. math:: p(a,b,c) = \\sum_{i,j,k} c_{i,j,k} * He_i(a) * He_j(b) * He_k(c) + + where the points `(a, b, c)` consist of all triples formed by taking + `a` from `x`, `b` from `y`, and `c` from `z`. The resulting points form + a grid with `x` in the first dimension, `y` in the second, and `z` in + the third. + + The parameters `x`, `y`, and `z` are converted to arrays only if they + are tuples or a lists, otherwise they are treated as a scalars. In + either case, either `x`, `y`, and `z` or their elements must support + multiplication and addition both with themselves and with the elements + of `c`. + + If `c` has fewer than three dimensions, ones are implicitly appended to + its shape to make it 3-D. The shape of the result will be c.shape[3:] + + x.shape + y.shape + z.shape. + + Parameters + ---------- + x, y, z : array_like, compatible objects + The three dimensional series is evaluated at the points in the + Cartesian product of `x`, `y`, and `z`. If `x`,`y`, or `z` is a + list or tuple, it is first converted to an ndarray, otherwise it is + left unchanged and, if it isn't an ndarray, it is treated as a + scalar. + c : array_like + Array of coefficients ordered so that the coefficients for terms of + degree i,j are contained in ``c[i,j]``. If `c` has dimension + greater than two the remaining indices enumerate multiple sets of + coefficients. + + Returns + ------- + values : ndarray, compatible object + The values of the two dimensional polynomial at points in the Cartesian + product of `x` and `y`. + + See Also + -------- + hermeval, hermeval2d, hermegrid2d, hermeval3d + + Notes + ----- + + .. versionadded:: 1.7.0 + + """ + return pu._gridnd(hermeval, c, x, y, z) + + +def hermevander(x, deg): + """Pseudo-Vandermonde matrix of given degree. + + Returns the pseudo-Vandermonde matrix of degree `deg` and sample points + `x`. The pseudo-Vandermonde matrix is defined by + + .. math:: V[..., i] = He_i(x), + + where `0 <= i <= deg`. The leading indices of `V` index the elements of + `x` and the last index is the degree of the HermiteE polynomial. + + If `c` is a 1-D array of coefficients of length `n + 1` and `V` is the + array ``V = hermevander(x, n)``, then ``np.dot(V, c)`` and + ``hermeval(x, c)`` are the same up to roundoff. This equivalence is + useful both for least squares fitting and for the evaluation of a large + number of HermiteE series of the same degree and sample points. + + Parameters + ---------- + x : array_like + Array of points. The dtype is converted to float64 or complex128 + depending on whether any of the elements are complex. If `x` is + scalar it is converted to a 1-D array. + deg : int + Degree of the resulting matrix. + + Returns + ------- + vander : ndarray + The pseudo-Vandermonde matrix. The shape of the returned matrix is + ``x.shape + (deg + 1,)``, where The last index is the degree of the + corresponding HermiteE polynomial. The dtype will be the same as + the converted `x`. + + Examples + -------- + >>> from numpy.polynomial.hermite_e import hermevander + >>> x = np.array([-1, 0, 1]) + >>> hermevander(x, 3) + array([[ 1., -1., 0., 2.], + [ 1., 0., -1., -0.], + [ 1., 1., 0., -2.]]) + + """ + ideg = pu._deprecate_as_int(deg, "deg") + if ideg < 0: + raise ValueError("deg must be non-negative") + + x = np.array(x, copy=False, ndmin=1) + 0.0 + dims = (ideg + 1,) + x.shape + dtyp = x.dtype + v = np.empty(dims, dtype=dtyp) + v[0] = x*0 + 1 + if ideg > 0: + v[1] = x + for i in range(2, ideg + 1): + v[i] = (v[i-1]*x - v[i-2]*(i - 1)) + return np.moveaxis(v, 0, -1) + + +def hermevander2d(x, y, deg): + """Pseudo-Vandermonde matrix of given degrees. + + Returns the pseudo-Vandermonde matrix of degrees `deg` and sample + points `(x, y)`. The pseudo-Vandermonde matrix is defined by + + .. math:: V[..., (deg[1] + 1)*i + j] = He_i(x) * He_j(y), + + where `0 <= i <= deg[0]` and `0 <= j <= deg[1]`. The leading indices of + `V` index the points `(x, y)` and the last index encodes the degrees of + the HermiteE polynomials. + + If ``V = hermevander2d(x, y, [xdeg, ydeg])``, then the columns of `V` + correspond to the elements of a 2-D coefficient array `c` of shape + (xdeg + 1, ydeg + 1) in the order + + .. math:: c_{00}, c_{01}, c_{02} ... , c_{10}, c_{11}, c_{12} ... + + and ``np.dot(V, c.flat)`` and ``hermeval2d(x, y, c)`` will be the same + up to roundoff. This equivalence is useful both for least squares + fitting and for the evaluation of a large number of 2-D HermiteE + series of the same degrees and sample points. + + Parameters + ---------- + x, y : array_like + Arrays of point coordinates, all of the same shape. The dtypes + will be converted to either float64 or complex128 depending on + whether any of the elements are complex. Scalars are converted to + 1-D arrays. + deg : list of ints + List of maximum degrees of the form [x_deg, y_deg]. + + Returns + ------- + vander2d : ndarray + The shape of the returned matrix is ``x.shape + (order,)``, where + :math:`order = (deg[0]+1)*(deg[1]+1)`. The dtype will be the same + as the converted `x` and `y`. + + See Also + -------- + hermevander, hermevander3d, hermeval2d, hermeval3d + + Notes + ----- + + .. versionadded:: 1.7.0 + + """ + return pu._vander_nd_flat((hermevander, hermevander), (x, y), deg) + + +def hermevander3d(x, y, z, deg): + """Pseudo-Vandermonde matrix of given degrees. + + Returns the pseudo-Vandermonde matrix of degrees `deg` and sample + points `(x, y, z)`. If `l, m, n` are the given degrees in `x, y, z`, + then Hehe pseudo-Vandermonde matrix is defined by + + .. math:: V[..., (m+1)(n+1)i + (n+1)j + k] = He_i(x)*He_j(y)*He_k(z), + + where `0 <= i <= l`, `0 <= j <= m`, and `0 <= j <= n`. The leading + indices of `V` index the points `(x, y, z)` and the last index encodes + the degrees of the HermiteE polynomials. + + If ``V = hermevander3d(x, y, z, [xdeg, ydeg, zdeg])``, then the columns + of `V` correspond to the elements of a 3-D coefficient array `c` of + shape (xdeg + 1, ydeg + 1, zdeg + 1) in the order + + .. math:: c_{000}, c_{001}, c_{002},... , c_{010}, c_{011}, c_{012},... + + and ``np.dot(V, c.flat)`` and ``hermeval3d(x, y, z, c)`` will be the + same up to roundoff. This equivalence is useful both for least squares + fitting and for the evaluation of a large number of 3-D HermiteE + series of the same degrees and sample points. + + Parameters + ---------- + x, y, z : array_like + Arrays of point coordinates, all of the same shape. The dtypes will + be converted to either float64 or complex128 depending on whether + any of the elements are complex. Scalars are converted to 1-D + arrays. + deg : list of ints + List of maximum degrees of the form [x_deg, y_deg, z_deg]. + + Returns + ------- + vander3d : ndarray + The shape of the returned matrix is ``x.shape + (order,)``, where + :math:`order = (deg[0]+1)*(deg[1]+1)*(deg[2]+1)`. The dtype will + be the same as the converted `x`, `y`, and `z`. + + See Also + -------- + hermevander, hermevander3d, hermeval2d, hermeval3d + + Notes + ----- + + .. versionadded:: 1.7.0 + + """ + return pu._vander_nd_flat((hermevander, hermevander, hermevander), (x, y, z), deg) + + +def hermefit(x, y, deg, rcond=None, full=False, w=None): + """ + Least squares fit of Hermite series to data. + + Return the coefficients of a HermiteE series of degree `deg` that is + the least squares fit to the data values `y` given at points `x`. If + `y` is 1-D the returned coefficients will also be 1-D. If `y` is 2-D + multiple fits are done, one for each column of `y`, and the resulting + coefficients are stored in the corresponding columns of a 2-D return. + The fitted polynomial(s) are in the form + + .. math:: p(x) = c_0 + c_1 * He_1(x) + ... + c_n * He_n(x), + + where `n` is `deg`. + + Parameters + ---------- + x : array_like, shape (M,) + x-coordinates of the M sample points ``(x[i], y[i])``. + y : array_like, shape (M,) or (M, K) + y-coordinates of the sample points. Several data sets of sample + points sharing the same x-coordinates can be fitted at once by + passing in a 2D-array that contains one dataset per column. + deg : int or 1-D array_like + Degree(s) of the fitting polynomials. If `deg` is a single integer + all terms up to and including the `deg`'th term are included in the + fit. For NumPy versions >= 1.11.0 a list of integers specifying the + degrees of the terms to include may be used instead. + rcond : float, optional + Relative condition number of the fit. Singular values smaller than + this relative to the largest singular value will be ignored. The + default value is len(x)*eps, where eps is the relative precision of + the float type, about 2e-16 in most cases. + full : bool, optional + Switch determining nature of return value. When it is False (the + default) just the coefficients are returned, when True diagnostic + information from the singular value decomposition is also returned. + w : array_like, shape (`M`,), optional + Weights. If not None, the weight ``w[i]`` applies to the unsquared + residual ``y[i] - y_hat[i]`` at ``x[i]``. Ideally the weights are + chosen so that the errors of the products ``w[i]*y[i]`` all have the + same variance. When using inverse-variance weighting, use + ``w[i] = 1/sigma(y[i])``. The default value is None. + + Returns + ------- + coef : ndarray, shape (M,) or (M, K) + Hermite coefficients ordered from low to high. If `y` was 2-D, + the coefficients for the data in column k of `y` are in column + `k`. + + [residuals, rank, singular_values, rcond] : list + These values are only returned if ``full == True`` + + - residuals -- sum of squared residuals of the least squares fit + - rank -- the numerical rank of the scaled Vandermonde matrix + - singular_values -- singular values of the scaled Vandermonde matrix + - rcond -- value of `rcond`. + + For more details, see `numpy.linalg.lstsq`. + + Warns + ----- + RankWarning + The rank of the coefficient matrix in the least-squares fit is + deficient. The warning is only raised if ``full = False``. The + warnings can be turned off by + + >>> import warnings + >>> warnings.simplefilter('ignore', np.RankWarning) + + See Also + -------- + numpy.polynomial.chebyshev.chebfit + numpy.polynomial.legendre.legfit + numpy.polynomial.polynomial.polyfit + numpy.polynomial.hermite.hermfit + numpy.polynomial.laguerre.lagfit + hermeval : Evaluates a Hermite series. + hermevander : pseudo Vandermonde matrix of Hermite series. + hermeweight : HermiteE weight function. + numpy.linalg.lstsq : Computes a least-squares fit from the matrix. + scipy.interpolate.UnivariateSpline : Computes spline fits. + + Notes + ----- + The solution is the coefficients of the HermiteE series `p` that + minimizes the sum of the weighted squared errors + + .. math:: E = \\sum_j w_j^2 * |y_j - p(x_j)|^2, + + where the :math:`w_j` are the weights. This problem is solved by + setting up the (typically) overdetermined matrix equation + + .. math:: V(x) * c = w * y, + + where `V` is the pseudo Vandermonde matrix of `x`, the elements of `c` + are the coefficients to be solved for, and the elements of `y` are the + observed values. This equation is then solved using the singular value + decomposition of `V`. + + If some of the singular values of `V` are so small that they are + neglected, then a `RankWarning` will be issued. This means that the + coefficient values may be poorly determined. Using a lower order fit + will usually get rid of the warning. The `rcond` parameter can also be + set to a value smaller than its default, but the resulting fit may be + spurious and have large contributions from roundoff error. + + Fits using HermiteE series are probably most useful when the data can + be approximated by ``sqrt(w(x)) * p(x)``, where `w(x)` is the HermiteE + weight. In that case the weight ``sqrt(w(x[i]))`` should be used + together with data values ``y[i]/sqrt(w(x[i]))``. The weight function is + available as `hermeweight`. + + References + ---------- + .. [1] Wikipedia, "Curve fitting", + https://en.wikipedia.org/wiki/Curve_fitting + + Examples + -------- + >>> from numpy.polynomial.hermite_e import hermefit, hermeval + >>> x = np.linspace(-10, 10) + >>> np.random.seed(123) + >>> err = np.random.randn(len(x))/10 + >>> y = hermeval(x, [1, 2, 3]) + err + >>> hermefit(x, y, 2) + array([ 1.01690445, 1.99951418, 2.99948696]) # may vary + + """ + return pu._fit(hermevander, x, y, deg, rcond, full, w) + + +def hermecompanion(c): + """ + Return the scaled companion matrix of c. + + The basis polynomials are scaled so that the companion matrix is + symmetric when `c` is an HermiteE basis polynomial. This provides + better eigenvalue estimates than the unscaled case and for basis + polynomials the eigenvalues are guaranteed to be real if + `numpy.linalg.eigvalsh` is used to obtain them. + + Parameters + ---------- + c : array_like + 1-D array of HermiteE series coefficients ordered from low to high + degree. + + Returns + ------- + mat : ndarray + Scaled companion matrix of dimensions (deg, deg). + + Notes + ----- + + .. versionadded:: 1.7.0 + + """ + # c is a trimmed copy + [c] = pu.as_series([c]) + if len(c) < 2: + raise ValueError('Series must have maximum degree of at least 1.') + if len(c) == 2: + return np.array([[-c[0]/c[1]]]) + + n = len(c) - 1 + mat = np.zeros((n, n), dtype=c.dtype) + scl = np.hstack((1., 1./np.sqrt(np.arange(n - 1, 0, -1)))) + scl = np.multiply.accumulate(scl)[::-1] + top = mat.reshape(-1)[1::n+1] + bot = mat.reshape(-1)[n::n+1] + top[...] = np.sqrt(np.arange(1, n)) + bot[...] = top + mat[:, -1] -= scl*c[:-1]/c[-1] + return mat + + +def hermeroots(c): + """ + Compute the roots of a HermiteE series. + + Return the roots (a.k.a. "zeros") of the polynomial + + .. math:: p(x) = \\sum_i c[i] * He_i(x). + + Parameters + ---------- + c : 1-D array_like + 1-D array of coefficients. + + Returns + ------- + out : ndarray + Array of the roots of the series. If all the roots are real, + then `out` is also real, otherwise it is complex. + + See Also + -------- + numpy.polynomial.polynomial.polyroots + numpy.polynomial.legendre.legroots + numpy.polynomial.laguerre.lagroots + numpy.polynomial.hermite.hermroots + numpy.polynomial.chebyshev.chebroots + + Notes + ----- + The root estimates are obtained as the eigenvalues of the companion + matrix, Roots far from the origin of the complex plane may have large + errors due to the numerical instability of the series for such + values. Roots with multiplicity greater than 1 will also show larger + errors as the value of the series near such points is relatively + insensitive to errors in the roots. Isolated roots near the origin can + be improved by a few iterations of Newton's method. + + The HermiteE series basis polynomials aren't powers of `x` so the + results of this function may seem unintuitive. + + Examples + -------- + >>> from numpy.polynomial.hermite_e import hermeroots, hermefromroots + >>> coef = hermefromroots([-1, 0, 1]) + >>> coef + array([0., 2., 0., 1.]) + >>> hermeroots(coef) + array([-1., 0., 1.]) # may vary + + """ + # c is a trimmed copy + [c] = pu.as_series([c]) + if len(c) <= 1: + return np.array([], dtype=c.dtype) + if len(c) == 2: + return np.array([-c[0]/c[1]]) + + # rotated companion matrix reduces error + m = hermecompanion(c)[::-1,::-1] + r = la.eigvals(m) + r.sort() + return r + + +def _normed_hermite_e_n(x, n): + """ + Evaluate a normalized HermiteE polynomial. + + Compute the value of the normalized HermiteE polynomial of degree ``n`` + at the points ``x``. + + + Parameters + ---------- + x : ndarray of double. + Points at which to evaluate the function + n : int + Degree of the normalized HermiteE function to be evaluated. + + Returns + ------- + values : ndarray + The shape of the return value is described above. + + Notes + ----- + .. versionadded:: 1.10.0 + + This function is needed for finding the Gauss points and integration + weights for high degrees. The values of the standard HermiteE functions + overflow when n >= 207. + + """ + if n == 0: + return np.full(x.shape, 1/np.sqrt(np.sqrt(2*np.pi))) + + c0 = 0. + c1 = 1./np.sqrt(np.sqrt(2*np.pi)) + nd = float(n) + for i in range(n - 1): + tmp = c0 + c0 = -c1*np.sqrt((nd - 1.)/nd) + c1 = tmp + c1*x*np.sqrt(1./nd) + nd = nd - 1.0 + return c0 + c1*x + + +def hermegauss(deg): + """ + Gauss-HermiteE quadrature. + + Computes the sample points and weights for Gauss-HermiteE quadrature. + These sample points and weights will correctly integrate polynomials of + degree :math:`2*deg - 1` or less over the interval :math:`[-\\inf, \\inf]` + with the weight function :math:`f(x) = \\exp(-x^2/2)`. + + Parameters + ---------- + deg : int + Number of sample points and weights. It must be >= 1. + + Returns + ------- + x : ndarray + 1-D ndarray containing the sample points. + y : ndarray + 1-D ndarray containing the weights. + + Notes + ----- + + .. versionadded:: 1.7.0 + + The results have only been tested up to degree 100, higher degrees may + be problematic. The weights are determined by using the fact that + + .. math:: w_k = c / (He'_n(x_k) * He_{n-1}(x_k)) + + where :math:`c` is a constant independent of :math:`k` and :math:`x_k` + is the k'th root of :math:`He_n`, and then scaling the results to get + the right value when integrating 1. + + """ + ideg = pu._deprecate_as_int(deg, "deg") + if ideg <= 0: + raise ValueError("deg must be a positive integer") + + # first approximation of roots. We use the fact that the companion + # matrix is symmetric in this case in order to obtain better zeros. + c = np.array([0]*deg + [1]) + m = hermecompanion(c) + x = la.eigvalsh(m) + + # improve roots by one application of Newton + dy = _normed_hermite_e_n(x, ideg) + df = _normed_hermite_e_n(x, ideg - 1) * np.sqrt(ideg) + x -= dy/df + + # compute the weights. We scale the factor to avoid possible numerical + # overflow. + fm = _normed_hermite_e_n(x, ideg - 1) + fm /= np.abs(fm).max() + w = 1/(fm * fm) + + # for Hermite_e we can also symmetrize + w = (w + w[::-1])/2 + x = (x - x[::-1])/2 + + # scale w to get the right value + w *= np.sqrt(2*np.pi) / w.sum() + + return x, w + + +def hermeweight(x): + """Weight function of the Hermite_e polynomials. + + The weight function is :math:`\\exp(-x^2/2)` and the interval of + integration is :math:`[-\\inf, \\inf]`. the HermiteE polynomials are + orthogonal, but not normalized, with respect to this weight function. + + Parameters + ---------- + x : array_like + Values at which the weight function will be computed. + + Returns + ------- + w : ndarray + The weight function at `x`. + + Notes + ----- + + .. versionadded:: 1.7.0 + + """ + w = np.exp(-.5*x**2) + return w + + +# +# HermiteE series class +# + +class HermiteE(ABCPolyBase): + """An HermiteE series class. + + The HermiteE class provides the standard Python numerical methods + '+', '-', '*', '//', '%', 'divmod', '**', and '()' as well as the + attributes and methods listed in the `ABCPolyBase` documentation. + + Parameters + ---------- + coef : array_like + HermiteE coefficients in order of increasing degree, i.e, + ``(1, 2, 3)`` gives ``1*He_0(x) + 2*He_1(X) + 3*He_2(x)``. + domain : (2,) array_like, optional + Domain to use. The interval ``[domain[0], domain[1]]`` is mapped + to the interval ``[window[0], window[1]]`` by shifting and scaling. + The default value is [-1, 1]. + window : (2,) array_like, optional + Window, see `domain` for its use. The default value is [-1, 1]. + + .. versionadded:: 1.6.0 + symbol : str, optional + Symbol used to represent the independent variable in string + representations of the polynomial expression, e.g. for printing. + The symbol must be a valid Python identifier. Default value is 'x'. + + .. versionadded:: 1.24 + + """ + # Virtual Functions + _add = staticmethod(hermeadd) + _sub = staticmethod(hermesub) + _mul = staticmethod(hermemul) + _div = staticmethod(hermediv) + _pow = staticmethod(hermepow) + _val = staticmethod(hermeval) + _int = staticmethod(hermeint) + _der = staticmethod(hermeder) + _fit = staticmethod(hermefit) + _line = staticmethod(hermeline) + _roots = staticmethod(hermeroots) + _fromroots = staticmethod(hermefromroots) + + # Virtual properties + domain = np.array(hermedomain) + window = np.array(hermedomain) + basis_name = 'He' diff --git a/deepseekvl2/lib/python3.10/site-packages/numpy/polynomial/hermite_e.pyi b/deepseekvl2/lib/python3.10/site-packages/numpy/polynomial/hermite_e.pyi new file mode 100644 index 0000000000000000000000000000000000000000..0b7152a253b654da2c069711a1bfdbd4e084cf6f --- /dev/null +++ b/deepseekvl2/lib/python3.10/site-packages/numpy/polynomial/hermite_e.pyi @@ -0,0 +1,46 @@ +from typing import Any + +from numpy import ndarray, dtype, int_ +from numpy.polynomial._polybase import ABCPolyBase +from numpy.polynomial.polyutils import trimcoef + +__all__: list[str] + +hermetrim = trimcoef + +def poly2herme(pol): ... +def herme2poly(c): ... + +hermedomain: ndarray[Any, dtype[int_]] +hermezero: ndarray[Any, dtype[int_]] +hermeone: ndarray[Any, dtype[int_]] +hermex: ndarray[Any, dtype[int_]] + +def hermeline(off, scl): ... +def hermefromroots(roots): ... +def hermeadd(c1, c2): ... +def hermesub(c1, c2): ... +def hermemulx(c): ... +def hermemul(c1, c2): ... +def hermediv(c1, c2): ... +def hermepow(c, pow, maxpower=...): ... +def hermeder(c, m=..., scl=..., axis=...): ... +def hermeint(c, m=..., k = ..., lbnd=..., scl=..., axis=...): ... +def hermeval(x, c, tensor=...): ... +def hermeval2d(x, y, c): ... +def hermegrid2d(x, y, c): ... +def hermeval3d(x, y, z, c): ... +def hermegrid3d(x, y, z, c): ... +def hermevander(x, deg): ... +def hermevander2d(x, y, deg): ... +def hermevander3d(x, y, z, deg): ... +def hermefit(x, y, deg, rcond=..., full=..., w=...): ... +def hermecompanion(c): ... +def hermeroots(c): ... +def hermegauss(deg): ... +def hermeweight(x): ... + +class HermiteE(ABCPolyBase): + domain: Any + window: Any + basis_name: Any diff --git a/deepseekvl2/lib/python3.10/site-packages/numpy/polynomial/legendre.py b/deepseekvl2/lib/python3.10/site-packages/numpy/polynomial/legendre.py new file mode 100644 index 0000000000000000000000000000000000000000..8e9c19d94ff60c7d314231e8bfbc1c200f12653e --- /dev/null +++ b/deepseekvl2/lib/python3.10/site-packages/numpy/polynomial/legendre.py @@ -0,0 +1,1664 @@ +""" +================================================== +Legendre Series (:mod:`numpy.polynomial.legendre`) +================================================== + +This module provides a number of objects (mostly functions) useful for +dealing with Legendre series, including a `Legendre` class that +encapsulates the usual arithmetic operations. (General information +on how this module represents and works with such polynomials is in the +docstring for its "parent" sub-package, `numpy.polynomial`). + +Classes +------- +.. autosummary:: + :toctree: generated/ + + Legendre + +Constants +--------- + +.. autosummary:: + :toctree: generated/ + + legdomain + legzero + legone + legx + +Arithmetic +---------- + +.. autosummary:: + :toctree: generated/ + + legadd + legsub + legmulx + legmul + legdiv + legpow + legval + legval2d + legval3d + leggrid2d + leggrid3d + +Calculus +-------- + +.. autosummary:: + :toctree: generated/ + + legder + legint + +Misc Functions +-------------- + +.. autosummary:: + :toctree: generated/ + + legfromroots + legroots + legvander + legvander2d + legvander3d + leggauss + legweight + legcompanion + legfit + legtrim + legline + leg2poly + poly2leg + +See also +-------- +numpy.polynomial + +""" +import numpy as np +import numpy.linalg as la +from numpy.core.multiarray import normalize_axis_index + +from . import polyutils as pu +from ._polybase import ABCPolyBase + +__all__ = [ + 'legzero', 'legone', 'legx', 'legdomain', 'legline', 'legadd', + 'legsub', 'legmulx', 'legmul', 'legdiv', 'legpow', 'legval', 'legder', + 'legint', 'leg2poly', 'poly2leg', 'legfromroots', 'legvander', + 'legfit', 'legtrim', 'legroots', 'Legendre', 'legval2d', 'legval3d', + 'leggrid2d', 'leggrid3d', 'legvander2d', 'legvander3d', 'legcompanion', + 'leggauss', 'legweight'] + +legtrim = pu.trimcoef + + +def poly2leg(pol): + """ + Convert a polynomial to a Legendre series. + + Convert an array representing the coefficients of a polynomial (relative + to the "standard" basis) ordered from lowest degree to highest, to an + array of the coefficients of the equivalent Legendre series, ordered + from lowest to highest degree. + + Parameters + ---------- + pol : array_like + 1-D array containing the polynomial coefficients + + Returns + ------- + c : ndarray + 1-D array containing the coefficients of the equivalent Legendre + series. + + See Also + -------- + leg2poly + + Notes + ----- + The easy way to do conversions between polynomial basis sets + is to use the convert method of a class instance. + + Examples + -------- + >>> from numpy import polynomial as P + >>> p = P.Polynomial(np.arange(4)) + >>> p + Polynomial([0., 1., 2., 3.], domain=[-1, 1], window=[-1, 1]) + >>> c = P.Legendre(P.legendre.poly2leg(p.coef)) + >>> c + Legendre([ 1. , 3.25, 1. , 0.75], domain=[-1, 1], window=[-1, 1]) # may vary + + """ + [pol] = pu.as_series([pol]) + deg = len(pol) - 1 + res = 0 + for i in range(deg, -1, -1): + res = legadd(legmulx(res), pol[i]) + return res + + +def leg2poly(c): + """ + Convert a Legendre series to a polynomial. + + Convert an array representing the coefficients of a Legendre series, + ordered from lowest degree to highest, to an array of the coefficients + of the equivalent polynomial (relative to the "standard" basis) ordered + from lowest to highest degree. + + Parameters + ---------- + c : array_like + 1-D array containing the Legendre series coefficients, ordered + from lowest order term to highest. + + Returns + ------- + pol : ndarray + 1-D array containing the coefficients of the equivalent polynomial + (relative to the "standard" basis) ordered from lowest order term + to highest. + + See Also + -------- + poly2leg + + Notes + ----- + The easy way to do conversions between polynomial basis sets + is to use the convert method of a class instance. + + Examples + -------- + >>> from numpy import polynomial as P + >>> c = P.Legendre(range(4)) + >>> c + Legendre([0., 1., 2., 3.], domain=[-1, 1], window=[-1, 1]) + >>> p = c.convert(kind=P.Polynomial) + >>> p + Polynomial([-1. , -3.5, 3. , 7.5], domain=[-1., 1.], window=[-1., 1.]) + >>> P.legendre.leg2poly(range(4)) + array([-1. , -3.5, 3. , 7.5]) + + + """ + from .polynomial import polyadd, polysub, polymulx + + [c] = pu.as_series([c]) + n = len(c) + if n < 3: + return c + else: + c0 = c[-2] + c1 = c[-1] + # i is the current degree of c1 + for i in range(n - 1, 1, -1): + tmp = c0 + c0 = polysub(c[i - 2], (c1*(i - 1))/i) + c1 = polyadd(tmp, (polymulx(c1)*(2*i - 1))/i) + return polyadd(c0, polymulx(c1)) + +# +# These are constant arrays are of integer type so as to be compatible +# with the widest range of other types, such as Decimal. +# + +# Legendre +legdomain = np.array([-1, 1]) + +# Legendre coefficients representing zero. +legzero = np.array([0]) + +# Legendre coefficients representing one. +legone = np.array([1]) + +# Legendre coefficients representing the identity x. +legx = np.array([0, 1]) + + +def legline(off, scl): + """ + Legendre series whose graph is a straight line. + + + + Parameters + ---------- + off, scl : scalars + The specified line is given by ``off + scl*x``. + + Returns + ------- + y : ndarray + This module's representation of the Legendre series for + ``off + scl*x``. + + See Also + -------- + numpy.polynomial.polynomial.polyline + numpy.polynomial.chebyshev.chebline + numpy.polynomial.laguerre.lagline + numpy.polynomial.hermite.hermline + numpy.polynomial.hermite_e.hermeline + + Examples + -------- + >>> import numpy.polynomial.legendre as L + >>> L.legline(3,2) + array([3, 2]) + >>> L.legval(-3, L.legline(3,2)) # should be -3 + -3.0 + + """ + if scl != 0: + return np.array([off, scl]) + else: + return np.array([off]) + + +def legfromroots(roots): + """ + Generate a Legendre series with given roots. + + The function returns the coefficients of the polynomial + + .. math:: p(x) = (x - r_0) * (x - r_1) * ... * (x - r_n), + + in Legendre form, where the `r_n` are the roots specified in `roots`. + If a zero has multiplicity n, then it must appear in `roots` n times. + For instance, if 2 is a root of multiplicity three and 3 is a root of + multiplicity 2, then `roots` looks something like [2, 2, 2, 3, 3]. The + roots can appear in any order. + + If the returned coefficients are `c`, then + + .. math:: p(x) = c_0 + c_1 * L_1(x) + ... + c_n * L_n(x) + + The coefficient of the last term is not generally 1 for monic + polynomials in Legendre form. + + Parameters + ---------- + roots : array_like + Sequence containing the roots. + + Returns + ------- + out : ndarray + 1-D array of coefficients. If all roots are real then `out` is a + real array, if some of the roots are complex, then `out` is complex + even if all the coefficients in the result are real (see Examples + below). + + See Also + -------- + numpy.polynomial.polynomial.polyfromroots + numpy.polynomial.chebyshev.chebfromroots + numpy.polynomial.laguerre.lagfromroots + numpy.polynomial.hermite.hermfromroots + numpy.polynomial.hermite_e.hermefromroots + + Examples + -------- + >>> import numpy.polynomial.legendre as L + >>> L.legfromroots((-1,0,1)) # x^3 - x relative to the standard basis + array([ 0. , -0.4, 0. , 0.4]) + >>> j = complex(0,1) + >>> L.legfromroots((-j,j)) # x^2 + 1 relative to the standard basis + array([ 1.33333333+0.j, 0.00000000+0.j, 0.66666667+0.j]) # may vary + + """ + return pu._fromroots(legline, legmul, roots) + + +def legadd(c1, c2): + """ + Add one Legendre series to another. + + Returns the sum of two Legendre series `c1` + `c2`. The arguments + are sequences of coefficients ordered from lowest order term to + highest, i.e., [1,2,3] represents the series ``P_0 + 2*P_1 + 3*P_2``. + + Parameters + ---------- + c1, c2 : array_like + 1-D arrays of Legendre series coefficients ordered from low to + high. + + Returns + ------- + out : ndarray + Array representing the Legendre series of their sum. + + See Also + -------- + legsub, legmulx, legmul, legdiv, legpow + + Notes + ----- + Unlike multiplication, division, etc., the sum of two Legendre series + is a Legendre series (without having to "reproject" the result onto + the basis set) so addition, just like that of "standard" polynomials, + is simply "component-wise." + + Examples + -------- + >>> from numpy.polynomial import legendre as L + >>> c1 = (1,2,3) + >>> c2 = (3,2,1) + >>> L.legadd(c1,c2) + array([4., 4., 4.]) + + """ + return pu._add(c1, c2) + + +def legsub(c1, c2): + """ + Subtract one Legendre series from another. + + Returns the difference of two Legendre series `c1` - `c2`. The + sequences of coefficients are from lowest order term to highest, i.e., + [1,2,3] represents the series ``P_0 + 2*P_1 + 3*P_2``. + + Parameters + ---------- + c1, c2 : array_like + 1-D arrays of Legendre series coefficients ordered from low to + high. + + Returns + ------- + out : ndarray + Of Legendre series coefficients representing their difference. + + See Also + -------- + legadd, legmulx, legmul, legdiv, legpow + + Notes + ----- + Unlike multiplication, division, etc., the difference of two Legendre + series is a Legendre series (without having to "reproject" the result + onto the basis set) so subtraction, just like that of "standard" + polynomials, is simply "component-wise." + + Examples + -------- + >>> from numpy.polynomial import legendre as L + >>> c1 = (1,2,3) + >>> c2 = (3,2,1) + >>> L.legsub(c1,c2) + array([-2., 0., 2.]) + >>> L.legsub(c2,c1) # -C.legsub(c1,c2) + array([ 2., 0., -2.]) + + """ + return pu._sub(c1, c2) + + +def legmulx(c): + """Multiply a Legendre series by x. + + Multiply the Legendre series `c` by x, where x is the independent + variable. + + + Parameters + ---------- + c : array_like + 1-D array of Legendre series coefficients ordered from low to + high. + + Returns + ------- + out : ndarray + Array representing the result of the multiplication. + + See Also + -------- + legadd, legmul, legdiv, legpow + + Notes + ----- + The multiplication uses the recursion relationship for Legendre + polynomials in the form + + .. math:: + + xP_i(x) = ((i + 1)*P_{i + 1}(x) + i*P_{i - 1}(x))/(2i + 1) + + Examples + -------- + >>> from numpy.polynomial import legendre as L + >>> L.legmulx([1,2,3]) + array([ 0.66666667, 2.2, 1.33333333, 1.8]) # may vary + + """ + # c is a trimmed copy + [c] = pu.as_series([c]) + # The zero series needs special treatment + if len(c) == 1 and c[0] == 0: + return c + + prd = np.empty(len(c) + 1, dtype=c.dtype) + prd[0] = c[0]*0 + prd[1] = c[0] + for i in range(1, len(c)): + j = i + 1 + k = i - 1 + s = i + j + prd[j] = (c[i]*j)/s + prd[k] += (c[i]*i)/s + return prd + + +def legmul(c1, c2): + """ + Multiply one Legendre series by another. + + Returns the product of two Legendre series `c1` * `c2`. The arguments + are sequences of coefficients, from lowest order "term" to highest, + e.g., [1,2,3] represents the series ``P_0 + 2*P_1 + 3*P_2``. + + Parameters + ---------- + c1, c2 : array_like + 1-D arrays of Legendre series coefficients ordered from low to + high. + + Returns + ------- + out : ndarray + Of Legendre series coefficients representing their product. + + See Also + -------- + legadd, legsub, legmulx, legdiv, legpow + + Notes + ----- + In general, the (polynomial) product of two C-series results in terms + that are not in the Legendre polynomial basis set. Thus, to express + the product as a Legendre series, it is necessary to "reproject" the + product onto said basis set, which may produce "unintuitive" (but + correct) results; see Examples section below. + + Examples + -------- + >>> from numpy.polynomial import legendre as L + >>> c1 = (1,2,3) + >>> c2 = (3,2) + >>> L.legmul(c1,c2) # multiplication requires "reprojection" + array([ 4.33333333, 10.4 , 11.66666667, 3.6 ]) # may vary + + """ + # s1, s2 are trimmed copies + [c1, c2] = pu.as_series([c1, c2]) + + if len(c1) > len(c2): + c = c2 + xs = c1 + else: + c = c1 + xs = c2 + + if len(c) == 1: + c0 = c[0]*xs + c1 = 0 + elif len(c) == 2: + c0 = c[0]*xs + c1 = c[1]*xs + else: + nd = len(c) + c0 = c[-2]*xs + c1 = c[-1]*xs + for i in range(3, len(c) + 1): + tmp = c0 + nd = nd - 1 + c0 = legsub(c[-i]*xs, (c1*(nd - 1))/nd) + c1 = legadd(tmp, (legmulx(c1)*(2*nd - 1))/nd) + return legadd(c0, legmulx(c1)) + + +def legdiv(c1, c2): + """ + Divide one Legendre series by another. + + Returns the quotient-with-remainder of two Legendre series + `c1` / `c2`. The arguments are sequences of coefficients from lowest + order "term" to highest, e.g., [1,2,3] represents the series + ``P_0 + 2*P_1 + 3*P_2``. + + Parameters + ---------- + c1, c2 : array_like + 1-D arrays of Legendre series coefficients ordered from low to + high. + + Returns + ------- + quo, rem : ndarrays + Of Legendre series coefficients representing the quotient and + remainder. + + See Also + -------- + legadd, legsub, legmulx, legmul, legpow + + Notes + ----- + In general, the (polynomial) division of one Legendre series by another + results in quotient and remainder terms that are not in the Legendre + polynomial basis set. Thus, to express these results as a Legendre + series, it is necessary to "reproject" the results onto the Legendre + basis set, which may produce "unintuitive" (but correct) results; see + Examples section below. + + Examples + -------- + >>> from numpy.polynomial import legendre as L + >>> c1 = (1,2,3) + >>> c2 = (3,2,1) + >>> L.legdiv(c1,c2) # quotient "intuitive," remainder not + (array([3.]), array([-8., -4.])) + >>> c2 = (0,1,2,3) + >>> L.legdiv(c2,c1) # neither "intuitive" + (array([-0.07407407, 1.66666667]), array([-1.03703704, -2.51851852])) # may vary + + """ + return pu._div(legmul, c1, c2) + + +def legpow(c, pow, maxpower=16): + """Raise a Legendre series to a power. + + Returns the Legendre series `c` raised to the power `pow`. The + argument `c` is a sequence of coefficients ordered from low to high. + i.e., [1,2,3] is the series ``P_0 + 2*P_1 + 3*P_2.`` + + Parameters + ---------- + c : array_like + 1-D array of Legendre series coefficients ordered from low to + high. + pow : integer + Power to which the series will be raised + maxpower : integer, optional + Maximum power allowed. This is mainly to limit growth of the series + to unmanageable size. Default is 16 + + Returns + ------- + coef : ndarray + Legendre series of power. + + See Also + -------- + legadd, legsub, legmulx, legmul, legdiv + + """ + return pu._pow(legmul, c, pow, maxpower) + + +def legder(c, m=1, scl=1, axis=0): + """ + Differentiate a Legendre series. + + Returns the Legendre series coefficients `c` differentiated `m` times + along `axis`. At each iteration the result is multiplied by `scl` (the + scaling factor is for use in a linear change of variable). The argument + `c` is an array of coefficients from low to high degree along each + axis, e.g., [1,2,3] represents the series ``1*L_0 + 2*L_1 + 3*L_2`` + while [[1,2],[1,2]] represents ``1*L_0(x)*L_0(y) + 1*L_1(x)*L_0(y) + + 2*L_0(x)*L_1(y) + 2*L_1(x)*L_1(y)`` if axis=0 is ``x`` and axis=1 is + ``y``. + + Parameters + ---------- + c : array_like + Array of Legendre series coefficients. If c is multidimensional the + different axis correspond to different variables with the degree in + each axis given by the corresponding index. + m : int, optional + Number of derivatives taken, must be non-negative. (Default: 1) + scl : scalar, optional + Each differentiation is multiplied by `scl`. The end result is + multiplication by ``scl**m``. This is for use in a linear change of + variable. (Default: 1) + axis : int, optional + Axis over which the derivative is taken. (Default: 0). + + .. versionadded:: 1.7.0 + + Returns + ------- + der : ndarray + Legendre series of the derivative. + + See Also + -------- + legint + + Notes + ----- + In general, the result of differentiating a Legendre series does not + resemble the same operation on a power series. Thus the result of this + function may be "unintuitive," albeit correct; see Examples section + below. + + Examples + -------- + >>> from numpy.polynomial import legendre as L + >>> c = (1,2,3,4) + >>> L.legder(c) + array([ 6., 9., 20.]) + >>> L.legder(c, 3) + array([60.]) + >>> L.legder(c, scl=-1) + array([ -6., -9., -20.]) + >>> L.legder(c, 2,-1) + array([ 9., 60.]) + + """ + c = np.array(c, ndmin=1, copy=True) + if c.dtype.char in '?bBhHiIlLqQpP': + c = c.astype(np.double) + cnt = pu._deprecate_as_int(m, "the order of derivation") + iaxis = pu._deprecate_as_int(axis, "the axis") + if cnt < 0: + raise ValueError("The order of derivation must be non-negative") + iaxis = normalize_axis_index(iaxis, c.ndim) + + if cnt == 0: + return c + + c = np.moveaxis(c, iaxis, 0) + n = len(c) + if cnt >= n: + c = c[:1]*0 + else: + for i in range(cnt): + n = n - 1 + c *= scl + der = np.empty((n,) + c.shape[1:], dtype=c.dtype) + for j in range(n, 2, -1): + der[j - 1] = (2*j - 1)*c[j] + c[j - 2] += c[j] + if n > 1: + der[1] = 3*c[2] + der[0] = c[1] + c = der + c = np.moveaxis(c, 0, iaxis) + return c + + +def legint(c, m=1, k=[], lbnd=0, scl=1, axis=0): + """ + Integrate a Legendre series. + + Returns the Legendre series coefficients `c` integrated `m` times from + `lbnd` along `axis`. At each iteration the resulting series is + **multiplied** by `scl` and an integration constant, `k`, is added. + The scaling factor is for use in a linear change of variable. ("Buyer + beware": note that, depending on what one is doing, one may want `scl` + to be the reciprocal of what one might expect; for more information, + see the Notes section below.) The argument `c` is an array of + coefficients from low to high degree along each axis, e.g., [1,2,3] + represents the series ``L_0 + 2*L_1 + 3*L_2`` while [[1,2],[1,2]] + represents ``1*L_0(x)*L_0(y) + 1*L_1(x)*L_0(y) + 2*L_0(x)*L_1(y) + + 2*L_1(x)*L_1(y)`` if axis=0 is ``x`` and axis=1 is ``y``. + + Parameters + ---------- + c : array_like + Array of Legendre series coefficients. If c is multidimensional the + different axis correspond to different variables with the degree in + each axis given by the corresponding index. + m : int, optional + Order of integration, must be positive. (Default: 1) + k : {[], list, scalar}, optional + Integration constant(s). The value of the first integral at + ``lbnd`` is the first value in the list, the value of the second + integral at ``lbnd`` is the second value, etc. If ``k == []`` (the + default), all constants are set to zero. If ``m == 1``, a single + scalar can be given instead of a list. + lbnd : scalar, optional + The lower bound of the integral. (Default: 0) + scl : scalar, optional + Following each integration the result is *multiplied* by `scl` + before the integration constant is added. (Default: 1) + axis : int, optional + Axis over which the integral is taken. (Default: 0). + + .. versionadded:: 1.7.0 + + Returns + ------- + S : ndarray + Legendre series coefficient array of the integral. + + Raises + ------ + ValueError + If ``m < 0``, ``len(k) > m``, ``np.ndim(lbnd) != 0``, or + ``np.ndim(scl) != 0``. + + See Also + -------- + legder + + Notes + ----- + Note that the result of each integration is *multiplied* by `scl`. + Why is this important to note? Say one is making a linear change of + variable :math:`u = ax + b` in an integral relative to `x`. Then + :math:`dx = du/a`, so one will need to set `scl` equal to + :math:`1/a` - perhaps not what one would have first thought. + + Also note that, in general, the result of integrating a C-series needs + to be "reprojected" onto the C-series basis set. Thus, typically, + the result of this function is "unintuitive," albeit correct; see + Examples section below. + + Examples + -------- + >>> from numpy.polynomial import legendre as L + >>> c = (1,2,3) + >>> L.legint(c) + array([ 0.33333333, 0.4 , 0.66666667, 0.6 ]) # may vary + >>> L.legint(c, 3) + array([ 1.66666667e-02, -1.78571429e-02, 4.76190476e-02, # may vary + -1.73472348e-18, 1.90476190e-02, 9.52380952e-03]) + >>> L.legint(c, k=3) + array([ 3.33333333, 0.4 , 0.66666667, 0.6 ]) # may vary + >>> L.legint(c, lbnd=-2) + array([ 7.33333333, 0.4 , 0.66666667, 0.6 ]) # may vary + >>> L.legint(c, scl=2) + array([ 0.66666667, 0.8 , 1.33333333, 1.2 ]) # may vary + + """ + c = np.array(c, ndmin=1, copy=True) + if c.dtype.char in '?bBhHiIlLqQpP': + c = c.astype(np.double) + if not np.iterable(k): + k = [k] + cnt = pu._deprecate_as_int(m, "the order of integration") + iaxis = pu._deprecate_as_int(axis, "the axis") + if cnt < 0: + raise ValueError("The order of integration must be non-negative") + if len(k) > cnt: + raise ValueError("Too many integration constants") + if np.ndim(lbnd) != 0: + raise ValueError("lbnd must be a scalar.") + if np.ndim(scl) != 0: + raise ValueError("scl must be a scalar.") + iaxis = normalize_axis_index(iaxis, c.ndim) + + if cnt == 0: + return c + + c = np.moveaxis(c, iaxis, 0) + k = list(k) + [0]*(cnt - len(k)) + for i in range(cnt): + n = len(c) + c *= scl + if n == 1 and np.all(c[0] == 0): + c[0] += k[i] + else: + tmp = np.empty((n + 1,) + c.shape[1:], dtype=c.dtype) + tmp[0] = c[0]*0 + tmp[1] = c[0] + if n > 1: + tmp[2] = c[1]/3 + for j in range(2, n): + t = c[j]/(2*j + 1) + tmp[j + 1] = t + tmp[j - 1] -= t + tmp[0] += k[i] - legval(lbnd, tmp) + c = tmp + c = np.moveaxis(c, 0, iaxis) + return c + + +def legval(x, c, tensor=True): + """ + Evaluate a Legendre series at points x. + + If `c` is of length `n + 1`, this function returns the value: + + .. math:: p(x) = c_0 * L_0(x) + c_1 * L_1(x) + ... + c_n * L_n(x) + + The parameter `x` is converted to an array only if it is a tuple or a + list, otherwise it is treated as a scalar. In either case, either `x` + or its elements must support multiplication and addition both with + themselves and with the elements of `c`. + + If `c` is a 1-D array, then `p(x)` will have the same shape as `x`. If + `c` is multidimensional, then the shape of the result depends on the + value of `tensor`. If `tensor` is true the shape will be c.shape[1:] + + x.shape. If `tensor` is false the shape will be c.shape[1:]. Note that + scalars have shape (,). + + Trailing zeros in the coefficients will be used in the evaluation, so + they should be avoided if efficiency is a concern. + + Parameters + ---------- + x : array_like, compatible object + If `x` is a list or tuple, it is converted to an ndarray, otherwise + it is left unchanged and treated as a scalar. In either case, `x` + or its elements must support addition and multiplication with + themselves and with the elements of `c`. + c : array_like + Array of coefficients ordered so that the coefficients for terms of + degree n are contained in c[n]. If `c` is multidimensional the + remaining indices enumerate multiple polynomials. In the two + dimensional case the coefficients may be thought of as stored in + the columns of `c`. + tensor : boolean, optional + If True, the shape of the coefficient array is extended with ones + on the right, one for each dimension of `x`. Scalars have dimension 0 + for this action. The result is that every column of coefficients in + `c` is evaluated for every element of `x`. If False, `x` is broadcast + over the columns of `c` for the evaluation. This keyword is useful + when `c` is multidimensional. The default value is True. + + .. versionadded:: 1.7.0 + + Returns + ------- + values : ndarray, algebra_like + The shape of the return value is described above. + + See Also + -------- + legval2d, leggrid2d, legval3d, leggrid3d + + Notes + ----- + The evaluation uses Clenshaw recursion, aka synthetic division. + + """ + c = np.array(c, ndmin=1, copy=False) + if c.dtype.char in '?bBhHiIlLqQpP': + c = c.astype(np.double) + if isinstance(x, (tuple, list)): + x = np.asarray(x) + if isinstance(x, np.ndarray) and tensor: + c = c.reshape(c.shape + (1,)*x.ndim) + + if len(c) == 1: + c0 = c[0] + c1 = 0 + elif len(c) == 2: + c0 = c[0] + c1 = c[1] + else: + nd = len(c) + c0 = c[-2] + c1 = c[-1] + for i in range(3, len(c) + 1): + tmp = c0 + nd = nd - 1 + c0 = c[-i] - (c1*(nd - 1))/nd + c1 = tmp + (c1*x*(2*nd - 1))/nd + return c0 + c1*x + + +def legval2d(x, y, c): + """ + Evaluate a 2-D Legendre series at points (x, y). + + This function returns the values: + + .. math:: p(x,y) = \\sum_{i,j} c_{i,j} * L_i(x) * L_j(y) + + The parameters `x` and `y` are converted to arrays only if they are + tuples or a lists, otherwise they are treated as a scalars and they + must have the same shape after conversion. In either case, either `x` + and `y` or their elements must support multiplication and addition both + with themselves and with the elements of `c`. + + If `c` is a 1-D array a one is implicitly appended to its shape to make + it 2-D. The shape of the result will be c.shape[2:] + x.shape. + + Parameters + ---------- + x, y : array_like, compatible objects + The two dimensional series is evaluated at the points `(x, y)`, + where `x` and `y` must have the same shape. If `x` or `y` is a list + or tuple, it is first converted to an ndarray, otherwise it is left + unchanged and if it isn't an ndarray it is treated as a scalar. + c : array_like + Array of coefficients ordered so that the coefficient of the term + of multi-degree i,j is contained in ``c[i,j]``. If `c` has + dimension greater than two the remaining indices enumerate multiple + sets of coefficients. + + Returns + ------- + values : ndarray, compatible object + The values of the two dimensional Legendre series at points formed + from pairs of corresponding values from `x` and `y`. + + See Also + -------- + legval, leggrid2d, legval3d, leggrid3d + + Notes + ----- + + .. versionadded:: 1.7.0 + + """ + return pu._valnd(legval, c, x, y) + + +def leggrid2d(x, y, c): + """ + Evaluate a 2-D Legendre series on the Cartesian product of x and y. + + This function returns the values: + + .. math:: p(a,b) = \\sum_{i,j} c_{i,j} * L_i(a) * L_j(b) + + where the points `(a, b)` consist of all pairs formed by taking + `a` from `x` and `b` from `y`. The resulting points form a grid with + `x` in the first dimension and `y` in the second. + + The parameters `x` and `y` are converted to arrays only if they are + tuples or a lists, otherwise they are treated as a scalars. In either + case, either `x` and `y` or their elements must support multiplication + and addition both with themselves and with the elements of `c`. + + If `c` has fewer than two dimensions, ones are implicitly appended to + its shape to make it 2-D. The shape of the result will be c.shape[2:] + + x.shape + y.shape. + + Parameters + ---------- + x, y : array_like, compatible objects + The two dimensional series is evaluated at the points in the + Cartesian product of `x` and `y`. If `x` or `y` is a list or + tuple, it is first converted to an ndarray, otherwise it is left + unchanged and, if it isn't an ndarray, it is treated as a scalar. + c : array_like + Array of coefficients ordered so that the coefficient of the term of + multi-degree i,j is contained in `c[i,j]`. If `c` has dimension + greater than two the remaining indices enumerate multiple sets of + coefficients. + + Returns + ------- + values : ndarray, compatible object + The values of the two dimensional Chebyshev series at points in the + Cartesian product of `x` and `y`. + + See Also + -------- + legval, legval2d, legval3d, leggrid3d + + Notes + ----- + + .. versionadded:: 1.7.0 + + """ + return pu._gridnd(legval, c, x, y) + + +def legval3d(x, y, z, c): + """ + Evaluate a 3-D Legendre series at points (x, y, z). + + This function returns the values: + + .. math:: p(x,y,z) = \\sum_{i,j,k} c_{i,j,k} * L_i(x) * L_j(y) * L_k(z) + + The parameters `x`, `y`, and `z` are converted to arrays only if + they are tuples or a lists, otherwise they are treated as a scalars and + they must have the same shape after conversion. In either case, either + `x`, `y`, and `z` or their elements must support multiplication and + addition both with themselves and with the elements of `c`. + + If `c` has fewer than 3 dimensions, ones are implicitly appended to its + shape to make it 3-D. The shape of the result will be c.shape[3:] + + x.shape. + + Parameters + ---------- + x, y, z : array_like, compatible object + The three dimensional series is evaluated at the points + `(x, y, z)`, where `x`, `y`, and `z` must have the same shape. If + any of `x`, `y`, or `z` is a list or tuple, it is first converted + to an ndarray, otherwise it is left unchanged and if it isn't an + ndarray it is treated as a scalar. + c : array_like + Array of coefficients ordered so that the coefficient of the term of + multi-degree i,j,k is contained in ``c[i,j,k]``. If `c` has dimension + greater than 3 the remaining indices enumerate multiple sets of + coefficients. + + Returns + ------- + values : ndarray, compatible object + The values of the multidimensional polynomial on points formed with + triples of corresponding values from `x`, `y`, and `z`. + + See Also + -------- + legval, legval2d, leggrid2d, leggrid3d + + Notes + ----- + + .. versionadded:: 1.7.0 + + """ + return pu._valnd(legval, c, x, y, z) + + +def leggrid3d(x, y, z, c): + """ + Evaluate a 3-D Legendre series on the Cartesian product of x, y, and z. + + This function returns the values: + + .. math:: p(a,b,c) = \\sum_{i,j,k} c_{i,j,k} * L_i(a) * L_j(b) * L_k(c) + + where the points `(a, b, c)` consist of all triples formed by taking + `a` from `x`, `b` from `y`, and `c` from `z`. The resulting points form + a grid with `x` in the first dimension, `y` in the second, and `z` in + the third. + + The parameters `x`, `y`, and `z` are converted to arrays only if they + are tuples or a lists, otherwise they are treated as a scalars. In + either case, either `x`, `y`, and `z` or their elements must support + multiplication and addition both with themselves and with the elements + of `c`. + + If `c` has fewer than three dimensions, ones are implicitly appended to + its shape to make it 3-D. The shape of the result will be c.shape[3:] + + x.shape + y.shape + z.shape. + + Parameters + ---------- + x, y, z : array_like, compatible objects + The three dimensional series is evaluated at the points in the + Cartesian product of `x`, `y`, and `z`. If `x`,`y`, or `z` is a + list or tuple, it is first converted to an ndarray, otherwise it is + left unchanged and, if it isn't an ndarray, it is treated as a + scalar. + c : array_like + Array of coefficients ordered so that the coefficients for terms of + degree i,j are contained in ``c[i,j]``. If `c` has dimension + greater than two the remaining indices enumerate multiple sets of + coefficients. + + Returns + ------- + values : ndarray, compatible object + The values of the two dimensional polynomial at points in the Cartesian + product of `x` and `y`. + + See Also + -------- + legval, legval2d, leggrid2d, legval3d + + Notes + ----- + + .. versionadded:: 1.7.0 + + """ + return pu._gridnd(legval, c, x, y, z) + + +def legvander(x, deg): + """Pseudo-Vandermonde matrix of given degree. + + Returns the pseudo-Vandermonde matrix of degree `deg` and sample points + `x`. The pseudo-Vandermonde matrix is defined by + + .. math:: V[..., i] = L_i(x) + + where `0 <= i <= deg`. The leading indices of `V` index the elements of + `x` and the last index is the degree of the Legendre polynomial. + + If `c` is a 1-D array of coefficients of length `n + 1` and `V` is the + array ``V = legvander(x, n)``, then ``np.dot(V, c)`` and + ``legval(x, c)`` are the same up to roundoff. This equivalence is + useful both for least squares fitting and for the evaluation of a large + number of Legendre series of the same degree and sample points. + + Parameters + ---------- + x : array_like + Array of points. The dtype is converted to float64 or complex128 + depending on whether any of the elements are complex. If `x` is + scalar it is converted to a 1-D array. + deg : int + Degree of the resulting matrix. + + Returns + ------- + vander : ndarray + The pseudo-Vandermonde matrix. The shape of the returned matrix is + ``x.shape + (deg + 1,)``, where The last index is the degree of the + corresponding Legendre polynomial. The dtype will be the same as + the converted `x`. + + """ + ideg = pu._deprecate_as_int(deg, "deg") + if ideg < 0: + raise ValueError("deg must be non-negative") + + x = np.array(x, copy=False, ndmin=1) + 0.0 + dims = (ideg + 1,) + x.shape + dtyp = x.dtype + v = np.empty(dims, dtype=dtyp) + # Use forward recursion to generate the entries. This is not as accurate + # as reverse recursion in this application but it is more efficient. + v[0] = x*0 + 1 + if ideg > 0: + v[1] = x + for i in range(2, ideg + 1): + v[i] = (v[i-1]*x*(2*i - 1) - v[i-2]*(i - 1))/i + return np.moveaxis(v, 0, -1) + + +def legvander2d(x, y, deg): + """Pseudo-Vandermonde matrix of given degrees. + + Returns the pseudo-Vandermonde matrix of degrees `deg` and sample + points `(x, y)`. The pseudo-Vandermonde matrix is defined by + + .. math:: V[..., (deg[1] + 1)*i + j] = L_i(x) * L_j(y), + + where `0 <= i <= deg[0]` and `0 <= j <= deg[1]`. The leading indices of + `V` index the points `(x, y)` and the last index encodes the degrees of + the Legendre polynomials. + + If ``V = legvander2d(x, y, [xdeg, ydeg])``, then the columns of `V` + correspond to the elements of a 2-D coefficient array `c` of shape + (xdeg + 1, ydeg + 1) in the order + + .. math:: c_{00}, c_{01}, c_{02} ... , c_{10}, c_{11}, c_{12} ... + + and ``np.dot(V, c.flat)`` and ``legval2d(x, y, c)`` will be the same + up to roundoff. This equivalence is useful both for least squares + fitting and for the evaluation of a large number of 2-D Legendre + series of the same degrees and sample points. + + Parameters + ---------- + x, y : array_like + Arrays of point coordinates, all of the same shape. The dtypes + will be converted to either float64 or complex128 depending on + whether any of the elements are complex. Scalars are converted to + 1-D arrays. + deg : list of ints + List of maximum degrees of the form [x_deg, y_deg]. + + Returns + ------- + vander2d : ndarray + The shape of the returned matrix is ``x.shape + (order,)``, where + :math:`order = (deg[0]+1)*(deg[1]+1)`. The dtype will be the same + as the converted `x` and `y`. + + See Also + -------- + legvander, legvander3d, legval2d, legval3d + + Notes + ----- + + .. versionadded:: 1.7.0 + + """ + return pu._vander_nd_flat((legvander, legvander), (x, y), deg) + + +def legvander3d(x, y, z, deg): + """Pseudo-Vandermonde matrix of given degrees. + + Returns the pseudo-Vandermonde matrix of degrees `deg` and sample + points `(x, y, z)`. If `l, m, n` are the given degrees in `x, y, z`, + then The pseudo-Vandermonde matrix is defined by + + .. math:: V[..., (m+1)(n+1)i + (n+1)j + k] = L_i(x)*L_j(y)*L_k(z), + + where `0 <= i <= l`, `0 <= j <= m`, and `0 <= j <= n`. The leading + indices of `V` index the points `(x, y, z)` and the last index encodes + the degrees of the Legendre polynomials. + + If ``V = legvander3d(x, y, z, [xdeg, ydeg, zdeg])``, then the columns + of `V` correspond to the elements of a 3-D coefficient array `c` of + shape (xdeg + 1, ydeg + 1, zdeg + 1) in the order + + .. math:: c_{000}, c_{001}, c_{002},... , c_{010}, c_{011}, c_{012},... + + and ``np.dot(V, c.flat)`` and ``legval3d(x, y, z, c)`` will be the + same up to roundoff. This equivalence is useful both for least squares + fitting and for the evaluation of a large number of 3-D Legendre + series of the same degrees and sample points. + + Parameters + ---------- + x, y, z : array_like + Arrays of point coordinates, all of the same shape. The dtypes will + be converted to either float64 or complex128 depending on whether + any of the elements are complex. Scalars are converted to 1-D + arrays. + deg : list of ints + List of maximum degrees of the form [x_deg, y_deg, z_deg]. + + Returns + ------- + vander3d : ndarray + The shape of the returned matrix is ``x.shape + (order,)``, where + :math:`order = (deg[0]+1)*(deg[1]+1)*(deg[2]+1)`. The dtype will + be the same as the converted `x`, `y`, and `z`. + + See Also + -------- + legvander, legvander3d, legval2d, legval3d + + Notes + ----- + + .. versionadded:: 1.7.0 + + """ + return pu._vander_nd_flat((legvander, legvander, legvander), (x, y, z), deg) + + +def legfit(x, y, deg, rcond=None, full=False, w=None): + """ + Least squares fit of Legendre series to data. + + Return the coefficients of a Legendre series of degree `deg` that is the + least squares fit to the data values `y` given at points `x`. If `y` is + 1-D the returned coefficients will also be 1-D. If `y` is 2-D multiple + fits are done, one for each column of `y`, and the resulting + coefficients are stored in the corresponding columns of a 2-D return. + The fitted polynomial(s) are in the form + + .. math:: p(x) = c_0 + c_1 * L_1(x) + ... + c_n * L_n(x), + + where `n` is `deg`. + + Parameters + ---------- + x : array_like, shape (M,) + x-coordinates of the M sample points ``(x[i], y[i])``. + y : array_like, shape (M,) or (M, K) + y-coordinates of the sample points. Several data sets of sample + points sharing the same x-coordinates can be fitted at once by + passing in a 2D-array that contains one dataset per column. + deg : int or 1-D array_like + Degree(s) of the fitting polynomials. If `deg` is a single integer + all terms up to and including the `deg`'th term are included in the + fit. For NumPy versions >= 1.11.0 a list of integers specifying the + degrees of the terms to include may be used instead. + rcond : float, optional + Relative condition number of the fit. Singular values smaller than + this relative to the largest singular value will be ignored. The + default value is len(x)*eps, where eps is the relative precision of + the float type, about 2e-16 in most cases. + full : bool, optional + Switch determining nature of return value. When it is False (the + default) just the coefficients are returned, when True diagnostic + information from the singular value decomposition is also returned. + w : array_like, shape (`M`,), optional + Weights. If not None, the weight ``w[i]`` applies to the unsquared + residual ``y[i] - y_hat[i]`` at ``x[i]``. Ideally the weights are + chosen so that the errors of the products ``w[i]*y[i]`` all have the + same variance. When using inverse-variance weighting, use + ``w[i] = 1/sigma(y[i])``. The default value is None. + + .. versionadded:: 1.5.0 + + Returns + ------- + coef : ndarray, shape (M,) or (M, K) + Legendre coefficients ordered from low to high. If `y` was + 2-D, the coefficients for the data in column k of `y` are in + column `k`. If `deg` is specified as a list, coefficients for + terms not included in the fit are set equal to zero in the + returned `coef`. + + [residuals, rank, singular_values, rcond] : list + These values are only returned if ``full == True`` + + - residuals -- sum of squared residuals of the least squares fit + - rank -- the numerical rank of the scaled Vandermonde matrix + - singular_values -- singular values of the scaled Vandermonde matrix + - rcond -- value of `rcond`. + + For more details, see `numpy.linalg.lstsq`. + + Warns + ----- + RankWarning + The rank of the coefficient matrix in the least-squares fit is + deficient. The warning is only raised if ``full == False``. The + warnings can be turned off by + + >>> import warnings + >>> warnings.simplefilter('ignore', np.RankWarning) + + See Also + -------- + numpy.polynomial.polynomial.polyfit + numpy.polynomial.chebyshev.chebfit + numpy.polynomial.laguerre.lagfit + numpy.polynomial.hermite.hermfit + numpy.polynomial.hermite_e.hermefit + legval : Evaluates a Legendre series. + legvander : Vandermonde matrix of Legendre series. + legweight : Legendre weight function (= 1). + numpy.linalg.lstsq : Computes a least-squares fit from the matrix. + scipy.interpolate.UnivariateSpline : Computes spline fits. + + Notes + ----- + The solution is the coefficients of the Legendre series `p` that + minimizes the sum of the weighted squared errors + + .. math:: E = \\sum_j w_j^2 * |y_j - p(x_j)|^2, + + where :math:`w_j` are the weights. This problem is solved by setting up + as the (typically) overdetermined matrix equation + + .. math:: V(x) * c = w * y, + + where `V` is the weighted pseudo Vandermonde matrix of `x`, `c` are the + coefficients to be solved for, `w` are the weights, and `y` are the + observed values. This equation is then solved using the singular value + decomposition of `V`. + + If some of the singular values of `V` are so small that they are + neglected, then a `RankWarning` will be issued. This means that the + coefficient values may be poorly determined. Using a lower order fit + will usually get rid of the warning. The `rcond` parameter can also be + set to a value smaller than its default, but the resulting fit may be + spurious and have large contributions from roundoff error. + + Fits using Legendre series are usually better conditioned than fits + using power series, but much can depend on the distribution of the + sample points and the smoothness of the data. If the quality of the fit + is inadequate splines may be a good alternative. + + References + ---------- + .. [1] Wikipedia, "Curve fitting", + https://en.wikipedia.org/wiki/Curve_fitting + + Examples + -------- + + """ + return pu._fit(legvander, x, y, deg, rcond, full, w) + + +def legcompanion(c): + """Return the scaled companion matrix of c. + + The basis polynomials are scaled so that the companion matrix is + symmetric when `c` is an Legendre basis polynomial. This provides + better eigenvalue estimates than the unscaled case and for basis + polynomials the eigenvalues are guaranteed to be real if + `numpy.linalg.eigvalsh` is used to obtain them. + + Parameters + ---------- + c : array_like + 1-D array of Legendre series coefficients ordered from low to high + degree. + + Returns + ------- + mat : ndarray + Scaled companion matrix of dimensions (deg, deg). + + Notes + ----- + + .. versionadded:: 1.7.0 + + """ + # c is a trimmed copy + [c] = pu.as_series([c]) + if len(c) < 2: + raise ValueError('Series must have maximum degree of at least 1.') + if len(c) == 2: + return np.array([[-c[0]/c[1]]]) + + n = len(c) - 1 + mat = np.zeros((n, n), dtype=c.dtype) + scl = 1./np.sqrt(2*np.arange(n) + 1) + top = mat.reshape(-1)[1::n+1] + bot = mat.reshape(-1)[n::n+1] + top[...] = np.arange(1, n)*scl[:n-1]*scl[1:n] + bot[...] = top + mat[:, -1] -= (c[:-1]/c[-1])*(scl/scl[-1])*(n/(2*n - 1)) + return mat + + +def legroots(c): + """ + Compute the roots of a Legendre series. + + Return the roots (a.k.a. "zeros") of the polynomial + + .. math:: p(x) = \\sum_i c[i] * L_i(x). + + Parameters + ---------- + c : 1-D array_like + 1-D array of coefficients. + + Returns + ------- + out : ndarray + Array of the roots of the series. If all the roots are real, + then `out` is also real, otherwise it is complex. + + See Also + -------- + numpy.polynomial.polynomial.polyroots + numpy.polynomial.chebyshev.chebroots + numpy.polynomial.laguerre.lagroots + numpy.polynomial.hermite.hermroots + numpy.polynomial.hermite_e.hermeroots + + Notes + ----- + The root estimates are obtained as the eigenvalues of the companion + matrix, Roots far from the origin of the complex plane may have large + errors due to the numerical instability of the series for such values. + Roots with multiplicity greater than 1 will also show larger errors as + the value of the series near such points is relatively insensitive to + errors in the roots. Isolated roots near the origin can be improved by + a few iterations of Newton's method. + + The Legendre series basis polynomials aren't powers of ``x`` so the + results of this function may seem unintuitive. + + Examples + -------- + >>> import numpy.polynomial.legendre as leg + >>> leg.legroots((1, 2, 3, 4)) # 4L_3 + 3L_2 + 2L_1 + 1L_0, all real roots + array([-0.85099543, -0.11407192, 0.51506735]) # may vary + + """ + # c is a trimmed copy + [c] = pu.as_series([c]) + if len(c) < 2: + return np.array([], dtype=c.dtype) + if len(c) == 2: + return np.array([-c[0]/c[1]]) + + # rotated companion matrix reduces error + m = legcompanion(c)[::-1,::-1] + r = la.eigvals(m) + r.sort() + return r + + +def leggauss(deg): + """ + Gauss-Legendre quadrature. + + Computes the sample points and weights for Gauss-Legendre quadrature. + These sample points and weights will correctly integrate polynomials of + degree :math:`2*deg - 1` or less over the interval :math:`[-1, 1]` with + the weight function :math:`f(x) = 1`. + + Parameters + ---------- + deg : int + Number of sample points and weights. It must be >= 1. + + Returns + ------- + x : ndarray + 1-D ndarray containing the sample points. + y : ndarray + 1-D ndarray containing the weights. + + Notes + ----- + + .. versionadded:: 1.7.0 + + The results have only been tested up to degree 100, higher degrees may + be problematic. The weights are determined by using the fact that + + .. math:: w_k = c / (L'_n(x_k) * L_{n-1}(x_k)) + + where :math:`c` is a constant independent of :math:`k` and :math:`x_k` + is the k'th root of :math:`L_n`, and then scaling the results to get + the right value when integrating 1. + + """ + ideg = pu._deprecate_as_int(deg, "deg") + if ideg <= 0: + raise ValueError("deg must be a positive integer") + + # first approximation of roots. We use the fact that the companion + # matrix is symmetric in this case in order to obtain better zeros. + c = np.array([0]*deg + [1]) + m = legcompanion(c) + x = la.eigvalsh(m) + + # improve roots by one application of Newton + dy = legval(x, c) + df = legval(x, legder(c)) + x -= dy/df + + # compute the weights. We scale the factor to avoid possible numerical + # overflow. + fm = legval(x, c[1:]) + fm /= np.abs(fm).max() + df /= np.abs(df).max() + w = 1/(fm * df) + + # for Legendre we can also symmetrize + w = (w + w[::-1])/2 + x = (x - x[::-1])/2 + + # scale w to get the right value + w *= 2. / w.sum() + + return x, w + + +def legweight(x): + """ + Weight function of the Legendre polynomials. + + The weight function is :math:`1` and the interval of integration is + :math:`[-1, 1]`. The Legendre polynomials are orthogonal, but not + normalized, with respect to this weight function. + + Parameters + ---------- + x : array_like + Values at which the weight function will be computed. + + Returns + ------- + w : ndarray + The weight function at `x`. + + Notes + ----- + + .. versionadded:: 1.7.0 + + """ + w = x*0.0 + 1.0 + return w + +# +# Legendre series class +# + +class Legendre(ABCPolyBase): + """A Legendre series class. + + The Legendre class provides the standard Python numerical methods + '+', '-', '*', '//', '%', 'divmod', '**', and '()' as well as the + attributes and methods listed in the `ABCPolyBase` documentation. + + Parameters + ---------- + coef : array_like + Legendre coefficients in order of increasing degree, i.e., + ``(1, 2, 3)`` gives ``1*P_0(x) + 2*P_1(x) + 3*P_2(x)``. + domain : (2,) array_like, optional + Domain to use. The interval ``[domain[0], domain[1]]`` is mapped + to the interval ``[window[0], window[1]]`` by shifting and scaling. + The default value is [-1, 1]. + window : (2,) array_like, optional + Window, see `domain` for its use. The default value is [-1, 1]. + + .. versionadded:: 1.6.0 + symbol : str, optional + Symbol used to represent the independent variable in string + representations of the polynomial expression, e.g. for printing. + The symbol must be a valid Python identifier. Default value is 'x'. + + .. versionadded:: 1.24 + + """ + # Virtual Functions + _add = staticmethod(legadd) + _sub = staticmethod(legsub) + _mul = staticmethod(legmul) + _div = staticmethod(legdiv) + _pow = staticmethod(legpow) + _val = staticmethod(legval) + _int = staticmethod(legint) + _der = staticmethod(legder) + _fit = staticmethod(legfit) + _line = staticmethod(legline) + _roots = staticmethod(legroots) + _fromroots = staticmethod(legfromroots) + + # Virtual properties + domain = np.array(legdomain) + window = np.array(legdomain) + basis_name = 'P' diff --git a/deepseekvl2/lib/python3.10/site-packages/numpy/polynomial/legendre.pyi b/deepseekvl2/lib/python3.10/site-packages/numpy/polynomial/legendre.pyi new file mode 100644 index 0000000000000000000000000000000000000000..63a1c3f3a1f89c2c2da61e385f7dba1e7be16c06 --- /dev/null +++ b/deepseekvl2/lib/python3.10/site-packages/numpy/polynomial/legendre.pyi @@ -0,0 +1,46 @@ +from typing import Any + +from numpy import ndarray, dtype, int_ +from numpy.polynomial._polybase import ABCPolyBase +from numpy.polynomial.polyutils import trimcoef + +__all__: list[str] + +legtrim = trimcoef + +def poly2leg(pol): ... +def leg2poly(c): ... + +legdomain: ndarray[Any, dtype[int_]] +legzero: ndarray[Any, dtype[int_]] +legone: ndarray[Any, dtype[int_]] +legx: ndarray[Any, dtype[int_]] + +def legline(off, scl): ... +def legfromroots(roots): ... +def legadd(c1, c2): ... +def legsub(c1, c2): ... +def legmulx(c): ... +def legmul(c1, c2): ... +def legdiv(c1, c2): ... +def legpow(c, pow, maxpower=...): ... +def legder(c, m=..., scl=..., axis=...): ... +def legint(c, m=..., k = ..., lbnd=..., scl=..., axis=...): ... +def legval(x, c, tensor=...): ... +def legval2d(x, y, c): ... +def leggrid2d(x, y, c): ... +def legval3d(x, y, z, c): ... +def leggrid3d(x, y, z, c): ... +def legvander(x, deg): ... +def legvander2d(x, y, deg): ... +def legvander3d(x, y, z, deg): ... +def legfit(x, y, deg, rcond=..., full=..., w=...): ... +def legcompanion(c): ... +def legroots(c): ... +def leggauss(deg): ... +def legweight(x): ... + +class Legendre(ABCPolyBase): + domain: Any + window: Any + basis_name: Any diff --git a/deepseekvl2/lib/python3.10/site-packages/numpy/polynomial/polynomial.py b/deepseekvl2/lib/python3.10/site-packages/numpy/polynomial/polynomial.py new file mode 100644 index 0000000000000000000000000000000000000000..ceadff0bf4ed32f8bbbb9f208bf4d84946efe195 --- /dev/null +++ b/deepseekvl2/lib/python3.10/site-packages/numpy/polynomial/polynomial.py @@ -0,0 +1,1542 @@ +""" +================================================= +Power Series (:mod:`numpy.polynomial.polynomial`) +================================================= + +This module provides a number of objects (mostly functions) useful for +dealing with polynomials, including a `Polynomial` class that +encapsulates the usual arithmetic operations. (General information +on how this module represents and works with polynomial objects is in +the docstring for its "parent" sub-package, `numpy.polynomial`). + +Classes +------- +.. autosummary:: + :toctree: generated/ + + Polynomial + +Constants +--------- +.. autosummary:: + :toctree: generated/ + + polydomain + polyzero + polyone + polyx + +Arithmetic +---------- +.. autosummary:: + :toctree: generated/ + + polyadd + polysub + polymulx + polymul + polydiv + polypow + polyval + polyval2d + polyval3d + polygrid2d + polygrid3d + +Calculus +-------- +.. autosummary:: + :toctree: generated/ + + polyder + polyint + +Misc Functions +-------------- +.. autosummary:: + :toctree: generated/ + + polyfromroots + polyroots + polyvalfromroots + polyvander + polyvander2d + polyvander3d + polycompanion + polyfit + polytrim + polyline + +See Also +-------- +`numpy.polynomial` + +""" +__all__ = [ + 'polyzero', 'polyone', 'polyx', 'polydomain', 'polyline', 'polyadd', + 'polysub', 'polymulx', 'polymul', 'polydiv', 'polypow', 'polyval', + 'polyvalfromroots', 'polyder', 'polyint', 'polyfromroots', 'polyvander', + 'polyfit', 'polytrim', 'polyroots', 'Polynomial', 'polyval2d', 'polyval3d', + 'polygrid2d', 'polygrid3d', 'polyvander2d', 'polyvander3d'] + +import numpy as np +import numpy.linalg as la +from numpy.core.multiarray import normalize_axis_index + +from . import polyutils as pu +from ._polybase import ABCPolyBase + +polytrim = pu.trimcoef + +# +# These are constant arrays are of integer type so as to be compatible +# with the widest range of other types, such as Decimal. +# + +# Polynomial default domain. +polydomain = np.array([-1, 1]) + +# Polynomial coefficients representing zero. +polyzero = np.array([0]) + +# Polynomial coefficients representing one. +polyone = np.array([1]) + +# Polynomial coefficients representing the identity x. +polyx = np.array([0, 1]) + +# +# Polynomial series functions +# + + +def polyline(off, scl): + """ + Returns an array representing a linear polynomial. + + Parameters + ---------- + off, scl : scalars + The "y-intercept" and "slope" of the line, respectively. + + Returns + ------- + y : ndarray + This module's representation of the linear polynomial ``off + + scl*x``. + + See Also + -------- + numpy.polynomial.chebyshev.chebline + numpy.polynomial.legendre.legline + numpy.polynomial.laguerre.lagline + numpy.polynomial.hermite.hermline + numpy.polynomial.hermite_e.hermeline + + Examples + -------- + >>> from numpy.polynomial import polynomial as P + >>> P.polyline(1,-1) + array([ 1, -1]) + >>> P.polyval(1, P.polyline(1,-1)) # should be 0 + 0.0 + + """ + if scl != 0: + return np.array([off, scl]) + else: + return np.array([off]) + + +def polyfromroots(roots): + """ + Generate a monic polynomial with given roots. + + Return the coefficients of the polynomial + + .. math:: p(x) = (x - r_0) * (x - r_1) * ... * (x - r_n), + + where the ``r_n`` are the roots specified in `roots`. If a zero has + multiplicity n, then it must appear in `roots` n times. For instance, + if 2 is a root of multiplicity three and 3 is a root of multiplicity 2, + then `roots` looks something like [2, 2, 2, 3, 3]. The roots can appear + in any order. + + If the returned coefficients are `c`, then + + .. math:: p(x) = c_0 + c_1 * x + ... + x^n + + The coefficient of the last term is 1 for monic polynomials in this + form. + + Parameters + ---------- + roots : array_like + Sequence containing the roots. + + Returns + ------- + out : ndarray + 1-D array of the polynomial's coefficients If all the roots are + real, then `out` is also real, otherwise it is complex. (see + Examples below). + + See Also + -------- + numpy.polynomial.chebyshev.chebfromroots + numpy.polynomial.legendre.legfromroots + numpy.polynomial.laguerre.lagfromroots + numpy.polynomial.hermite.hermfromroots + numpy.polynomial.hermite_e.hermefromroots + + Notes + ----- + The coefficients are determined by multiplying together linear factors + of the form ``(x - r_i)``, i.e. + + .. math:: p(x) = (x - r_0) (x - r_1) ... (x - r_n) + + where ``n == len(roots) - 1``; note that this implies that ``1`` is always + returned for :math:`a_n`. + + Examples + -------- + >>> from numpy.polynomial import polynomial as P + >>> P.polyfromroots((-1,0,1)) # x(x - 1)(x + 1) = x^3 - x + array([ 0., -1., 0., 1.]) + >>> j = complex(0,1) + >>> P.polyfromroots((-j,j)) # complex returned, though values are real + array([1.+0.j, 0.+0.j, 1.+0.j]) + + """ + return pu._fromroots(polyline, polymul, roots) + + +def polyadd(c1, c2): + """ + Add one polynomial to another. + + Returns the sum of two polynomials `c1` + `c2`. The arguments are + sequences of coefficients from lowest order term to highest, i.e., + [1,2,3] represents the polynomial ``1 + 2*x + 3*x**2``. + + Parameters + ---------- + c1, c2 : array_like + 1-D arrays of polynomial coefficients ordered from low to high. + + Returns + ------- + out : ndarray + The coefficient array representing their sum. + + See Also + -------- + polysub, polymulx, polymul, polydiv, polypow + + Examples + -------- + >>> from numpy.polynomial import polynomial as P + >>> c1 = (1,2,3) + >>> c2 = (3,2,1) + >>> sum = P.polyadd(c1,c2); sum + array([4., 4., 4.]) + >>> P.polyval(2, sum) # 4 + 4(2) + 4(2**2) + 28.0 + + """ + return pu._add(c1, c2) + + +def polysub(c1, c2): + """ + Subtract one polynomial from another. + + Returns the difference of two polynomials `c1` - `c2`. The arguments + are sequences of coefficients from lowest order term to highest, i.e., + [1,2,3] represents the polynomial ``1 + 2*x + 3*x**2``. + + Parameters + ---------- + c1, c2 : array_like + 1-D arrays of polynomial coefficients ordered from low to + high. + + Returns + ------- + out : ndarray + Of coefficients representing their difference. + + See Also + -------- + polyadd, polymulx, polymul, polydiv, polypow + + Examples + -------- + >>> from numpy.polynomial import polynomial as P + >>> c1 = (1,2,3) + >>> c2 = (3,2,1) + >>> P.polysub(c1,c2) + array([-2., 0., 2.]) + >>> P.polysub(c2,c1) # -P.polysub(c1,c2) + array([ 2., 0., -2.]) + + """ + return pu._sub(c1, c2) + + +def polymulx(c): + """Multiply a polynomial by x. + + Multiply the polynomial `c` by x, where x is the independent + variable. + + + Parameters + ---------- + c : array_like + 1-D array of polynomial coefficients ordered from low to + high. + + Returns + ------- + out : ndarray + Array representing the result of the multiplication. + + See Also + -------- + polyadd, polysub, polymul, polydiv, polypow + + Notes + ----- + + .. versionadded:: 1.5.0 + + """ + # c is a trimmed copy + [c] = pu.as_series([c]) + # The zero series needs special treatment + if len(c) == 1 and c[0] == 0: + return c + + prd = np.empty(len(c) + 1, dtype=c.dtype) + prd[0] = c[0]*0 + prd[1:] = c + return prd + + +def polymul(c1, c2): + """ + Multiply one polynomial by another. + + Returns the product of two polynomials `c1` * `c2`. The arguments are + sequences of coefficients, from lowest order term to highest, e.g., + [1,2,3] represents the polynomial ``1 + 2*x + 3*x**2.`` + + Parameters + ---------- + c1, c2 : array_like + 1-D arrays of coefficients representing a polynomial, relative to the + "standard" basis, and ordered from lowest order term to highest. + + Returns + ------- + out : ndarray + Of the coefficients of their product. + + See Also + -------- + polyadd, polysub, polymulx, polydiv, polypow + + Examples + -------- + >>> from numpy.polynomial import polynomial as P + >>> c1 = (1,2,3) + >>> c2 = (3,2,1) + >>> P.polymul(c1,c2) + array([ 3., 8., 14., 8., 3.]) + + """ + # c1, c2 are trimmed copies + [c1, c2] = pu.as_series([c1, c2]) + ret = np.convolve(c1, c2) + return pu.trimseq(ret) + + +def polydiv(c1, c2): + """ + Divide one polynomial by another. + + Returns the quotient-with-remainder of two polynomials `c1` / `c2`. + The arguments are sequences of coefficients, from lowest order term + to highest, e.g., [1,2,3] represents ``1 + 2*x + 3*x**2``. + + Parameters + ---------- + c1, c2 : array_like + 1-D arrays of polynomial coefficients ordered from low to high. + + Returns + ------- + [quo, rem] : ndarrays + Of coefficient series representing the quotient and remainder. + + See Also + -------- + polyadd, polysub, polymulx, polymul, polypow + + Examples + -------- + >>> from numpy.polynomial import polynomial as P + >>> c1 = (1,2,3) + >>> c2 = (3,2,1) + >>> P.polydiv(c1,c2) + (array([3.]), array([-8., -4.])) + >>> P.polydiv(c2,c1) + (array([ 0.33333333]), array([ 2.66666667, 1.33333333])) # may vary + + """ + # c1, c2 are trimmed copies + [c1, c2] = pu.as_series([c1, c2]) + if c2[-1] == 0: + raise ZeroDivisionError() + + # note: this is more efficient than `pu._div(polymul, c1, c2)` + lc1 = len(c1) + lc2 = len(c2) + if lc1 < lc2: + return c1[:1]*0, c1 + elif lc2 == 1: + return c1/c2[-1], c1[:1]*0 + else: + dlen = lc1 - lc2 + scl = c2[-1] + c2 = c2[:-1]/scl + i = dlen + j = lc1 - 1 + while i >= 0: + c1[i:j] -= c2*c1[j] + i -= 1 + j -= 1 + return c1[j+1:]/scl, pu.trimseq(c1[:j+1]) + + +def polypow(c, pow, maxpower=None): + """Raise a polynomial to a power. + + Returns the polynomial `c` raised to the power `pow`. The argument + `c` is a sequence of coefficients ordered from low to high. i.e., + [1,2,3] is the series ``1 + 2*x + 3*x**2.`` + + Parameters + ---------- + c : array_like + 1-D array of array of series coefficients ordered from low to + high degree. + pow : integer + Power to which the series will be raised + maxpower : integer, optional + Maximum power allowed. This is mainly to limit growth of the series + to unmanageable size. Default is 16 + + Returns + ------- + coef : ndarray + Power series of power. + + See Also + -------- + polyadd, polysub, polymulx, polymul, polydiv + + Examples + -------- + >>> from numpy.polynomial import polynomial as P + >>> P.polypow([1,2,3], 2) + array([ 1., 4., 10., 12., 9.]) + + """ + # note: this is more efficient than `pu._pow(polymul, c1, c2)`, as it + # avoids calling `as_series` repeatedly + return pu._pow(np.convolve, c, pow, maxpower) + + +def polyder(c, m=1, scl=1, axis=0): + """ + Differentiate a polynomial. + + Returns the polynomial coefficients `c` differentiated `m` times along + `axis`. At each iteration the result is multiplied by `scl` (the + scaling factor is for use in a linear change of variable). The + argument `c` is an array of coefficients from low to high degree along + each axis, e.g., [1,2,3] represents the polynomial ``1 + 2*x + 3*x**2`` + while [[1,2],[1,2]] represents ``1 + 1*x + 2*y + 2*x*y`` if axis=0 is + ``x`` and axis=1 is ``y``. + + Parameters + ---------- + c : array_like + Array of polynomial coefficients. If c is multidimensional the + different axis correspond to different variables with the degree + in each axis given by the corresponding index. + m : int, optional + Number of derivatives taken, must be non-negative. (Default: 1) + scl : scalar, optional + Each differentiation is multiplied by `scl`. The end result is + multiplication by ``scl**m``. This is for use in a linear change + of variable. (Default: 1) + axis : int, optional + Axis over which the derivative is taken. (Default: 0). + + .. versionadded:: 1.7.0 + + Returns + ------- + der : ndarray + Polynomial coefficients of the derivative. + + See Also + -------- + polyint + + Examples + -------- + >>> from numpy.polynomial import polynomial as P + >>> c = (1,2,3,4) # 1 + 2x + 3x**2 + 4x**3 + >>> P.polyder(c) # (d/dx)(c) = 2 + 6x + 12x**2 + array([ 2., 6., 12.]) + >>> P.polyder(c,3) # (d**3/dx**3)(c) = 24 + array([24.]) + >>> P.polyder(c,scl=-1) # (d/d(-x))(c) = -2 - 6x - 12x**2 + array([ -2., -6., -12.]) + >>> P.polyder(c,2,-1) # (d**2/d(-x)**2)(c) = 6 + 24x + array([ 6., 24.]) + + """ + c = np.array(c, ndmin=1, copy=True) + if c.dtype.char in '?bBhHiIlLqQpP': + # astype fails with NA + c = c + 0.0 + cdt = c.dtype + cnt = pu._deprecate_as_int(m, "the order of derivation") + iaxis = pu._deprecate_as_int(axis, "the axis") + if cnt < 0: + raise ValueError("The order of derivation must be non-negative") + iaxis = normalize_axis_index(iaxis, c.ndim) + + if cnt == 0: + return c + + c = np.moveaxis(c, iaxis, 0) + n = len(c) + if cnt >= n: + c = c[:1]*0 + else: + for i in range(cnt): + n = n - 1 + c *= scl + der = np.empty((n,) + c.shape[1:], dtype=cdt) + for j in range(n, 0, -1): + der[j - 1] = j*c[j] + c = der + c = np.moveaxis(c, 0, iaxis) + return c + + +def polyint(c, m=1, k=[], lbnd=0, scl=1, axis=0): + """ + Integrate a polynomial. + + Returns the polynomial coefficients `c` integrated `m` times from + `lbnd` along `axis`. At each iteration the resulting series is + **multiplied** by `scl` and an integration constant, `k`, is added. + The scaling factor is for use in a linear change of variable. ("Buyer + beware": note that, depending on what one is doing, one may want `scl` + to be the reciprocal of what one might expect; for more information, + see the Notes section below.) The argument `c` is an array of + coefficients, from low to high degree along each axis, e.g., [1,2,3] + represents the polynomial ``1 + 2*x + 3*x**2`` while [[1,2],[1,2]] + represents ``1 + 1*x + 2*y + 2*x*y`` if axis=0 is ``x`` and axis=1 is + ``y``. + + Parameters + ---------- + c : array_like + 1-D array of polynomial coefficients, ordered from low to high. + m : int, optional + Order of integration, must be positive. (Default: 1) + k : {[], list, scalar}, optional + Integration constant(s). The value of the first integral at zero + is the first value in the list, the value of the second integral + at zero is the second value, etc. If ``k == []`` (the default), + all constants are set to zero. If ``m == 1``, a single scalar can + be given instead of a list. + lbnd : scalar, optional + The lower bound of the integral. (Default: 0) + scl : scalar, optional + Following each integration the result is *multiplied* by `scl` + before the integration constant is added. (Default: 1) + axis : int, optional + Axis over which the integral is taken. (Default: 0). + + .. versionadded:: 1.7.0 + + Returns + ------- + S : ndarray + Coefficient array of the integral. + + Raises + ------ + ValueError + If ``m < 1``, ``len(k) > m``, ``np.ndim(lbnd) != 0``, or + ``np.ndim(scl) != 0``. + + See Also + -------- + polyder + + Notes + ----- + Note that the result of each integration is *multiplied* by `scl`. Why + is this important to note? Say one is making a linear change of + variable :math:`u = ax + b` in an integral relative to `x`. Then + :math:`dx = du/a`, so one will need to set `scl` equal to + :math:`1/a` - perhaps not what one would have first thought. + + Examples + -------- + >>> from numpy.polynomial import polynomial as P + >>> c = (1,2,3) + >>> P.polyint(c) # should return array([0, 1, 1, 1]) + array([0., 1., 1., 1.]) + >>> P.polyint(c,3) # should return array([0, 0, 0, 1/6, 1/12, 1/20]) + array([ 0. , 0. , 0. , 0.16666667, 0.08333333, # may vary + 0.05 ]) + >>> P.polyint(c,k=3) # should return array([3, 1, 1, 1]) + array([3., 1., 1., 1.]) + >>> P.polyint(c,lbnd=-2) # should return array([6, 1, 1, 1]) + array([6., 1., 1., 1.]) + >>> P.polyint(c,scl=-2) # should return array([0, -2, -2, -2]) + array([ 0., -2., -2., -2.]) + + """ + c = np.array(c, ndmin=1, copy=True) + if c.dtype.char in '?bBhHiIlLqQpP': + # astype doesn't preserve mask attribute. + c = c + 0.0 + cdt = c.dtype + if not np.iterable(k): + k = [k] + cnt = pu._deprecate_as_int(m, "the order of integration") + iaxis = pu._deprecate_as_int(axis, "the axis") + if cnt < 0: + raise ValueError("The order of integration must be non-negative") + if len(k) > cnt: + raise ValueError("Too many integration constants") + if np.ndim(lbnd) != 0: + raise ValueError("lbnd must be a scalar.") + if np.ndim(scl) != 0: + raise ValueError("scl must be a scalar.") + iaxis = normalize_axis_index(iaxis, c.ndim) + + if cnt == 0: + return c + + k = list(k) + [0]*(cnt - len(k)) + c = np.moveaxis(c, iaxis, 0) + for i in range(cnt): + n = len(c) + c *= scl + if n == 1 and np.all(c[0] == 0): + c[0] += k[i] + else: + tmp = np.empty((n + 1,) + c.shape[1:], dtype=cdt) + tmp[0] = c[0]*0 + tmp[1] = c[0] + for j in range(1, n): + tmp[j + 1] = c[j]/(j + 1) + tmp[0] += k[i] - polyval(lbnd, tmp) + c = tmp + c = np.moveaxis(c, 0, iaxis) + return c + + +def polyval(x, c, tensor=True): + """ + Evaluate a polynomial at points x. + + If `c` is of length `n + 1`, this function returns the value + + .. math:: p(x) = c_0 + c_1 * x + ... + c_n * x^n + + The parameter `x` is converted to an array only if it is a tuple or a + list, otherwise it is treated as a scalar. In either case, either `x` + or its elements must support multiplication and addition both with + themselves and with the elements of `c`. + + If `c` is a 1-D array, then `p(x)` will have the same shape as `x`. If + `c` is multidimensional, then the shape of the result depends on the + value of `tensor`. If `tensor` is true the shape will be c.shape[1:] + + x.shape. If `tensor` is false the shape will be c.shape[1:]. Note that + scalars have shape (,). + + Trailing zeros in the coefficients will be used in the evaluation, so + they should be avoided if efficiency is a concern. + + Parameters + ---------- + x : array_like, compatible object + If `x` is a list or tuple, it is converted to an ndarray, otherwise + it is left unchanged and treated as a scalar. In either case, `x` + or its elements must support addition and multiplication with + with themselves and with the elements of `c`. + c : array_like + Array of coefficients ordered so that the coefficients for terms of + degree n are contained in c[n]. If `c` is multidimensional the + remaining indices enumerate multiple polynomials. In the two + dimensional case the coefficients may be thought of as stored in + the columns of `c`. + tensor : boolean, optional + If True, the shape of the coefficient array is extended with ones + on the right, one for each dimension of `x`. Scalars have dimension 0 + for this action. The result is that every column of coefficients in + `c` is evaluated for every element of `x`. If False, `x` is broadcast + over the columns of `c` for the evaluation. This keyword is useful + when `c` is multidimensional. The default value is True. + + .. versionadded:: 1.7.0 + + Returns + ------- + values : ndarray, compatible object + The shape of the returned array is described above. + + See Also + -------- + polyval2d, polygrid2d, polyval3d, polygrid3d + + Notes + ----- + The evaluation uses Horner's method. + + Examples + -------- + >>> from numpy.polynomial.polynomial import polyval + >>> polyval(1, [1,2,3]) + 6.0 + >>> a = np.arange(4).reshape(2,2) + >>> a + array([[0, 1], + [2, 3]]) + >>> polyval(a, [1,2,3]) + array([[ 1., 6.], + [17., 34.]]) + >>> coef = np.arange(4).reshape(2,2) # multidimensional coefficients + >>> coef + array([[0, 1], + [2, 3]]) + >>> polyval([1,2], coef, tensor=True) + array([[2., 4.], + [4., 7.]]) + >>> polyval([1,2], coef, tensor=False) + array([2., 7.]) + + """ + c = np.array(c, ndmin=1, copy=False) + if c.dtype.char in '?bBhHiIlLqQpP': + # astype fails with NA + c = c + 0.0 + if isinstance(x, (tuple, list)): + x = np.asarray(x) + if isinstance(x, np.ndarray) and tensor: + c = c.reshape(c.shape + (1,)*x.ndim) + + c0 = c[-1] + x*0 + for i in range(2, len(c) + 1): + c0 = c[-i] + c0*x + return c0 + + +def polyvalfromroots(x, r, tensor=True): + """ + Evaluate a polynomial specified by its roots at points x. + + If `r` is of length `N`, this function returns the value + + .. math:: p(x) = \\prod_{n=1}^{N} (x - r_n) + + The parameter `x` is converted to an array only if it is a tuple or a + list, otherwise it is treated as a scalar. In either case, either `x` + or its elements must support multiplication and addition both with + themselves and with the elements of `r`. + + If `r` is a 1-D array, then `p(x)` will have the same shape as `x`. If `r` + is multidimensional, then the shape of the result depends on the value of + `tensor`. If `tensor` is ``True`` the shape will be r.shape[1:] + x.shape; + that is, each polynomial is evaluated at every value of `x`. If `tensor` is + ``False``, the shape will be r.shape[1:]; that is, each polynomial is + evaluated only for the corresponding broadcast value of `x`. Note that + scalars have shape (,). + + .. versionadded:: 1.12 + + Parameters + ---------- + x : array_like, compatible object + If `x` is a list or tuple, it is converted to an ndarray, otherwise + it is left unchanged and treated as a scalar. In either case, `x` + or its elements must support addition and multiplication with + with themselves and with the elements of `r`. + r : array_like + Array of roots. If `r` is multidimensional the first index is the + root index, while the remaining indices enumerate multiple + polynomials. For instance, in the two dimensional case the roots + of each polynomial may be thought of as stored in the columns of `r`. + tensor : boolean, optional + If True, the shape of the roots array is extended with ones on the + right, one for each dimension of `x`. Scalars have dimension 0 for this + action. The result is that every column of coefficients in `r` is + evaluated for every element of `x`. If False, `x` is broadcast over the + columns of `r` for the evaluation. This keyword is useful when `r` is + multidimensional. The default value is True. + + Returns + ------- + values : ndarray, compatible object + The shape of the returned array is described above. + + See Also + -------- + polyroots, polyfromroots, polyval + + Examples + -------- + >>> from numpy.polynomial.polynomial import polyvalfromroots + >>> polyvalfromroots(1, [1,2,3]) + 0.0 + >>> a = np.arange(4).reshape(2,2) + >>> a + array([[0, 1], + [2, 3]]) + >>> polyvalfromroots(a, [-1, 0, 1]) + array([[-0., 0.], + [ 6., 24.]]) + >>> r = np.arange(-2, 2).reshape(2,2) # multidimensional coefficients + >>> r # each column of r defines one polynomial + array([[-2, -1], + [ 0, 1]]) + >>> b = [-2, 1] + >>> polyvalfromroots(b, r, tensor=True) + array([[-0., 3.], + [ 3., 0.]]) + >>> polyvalfromroots(b, r, tensor=False) + array([-0., 0.]) + """ + r = np.array(r, ndmin=1, copy=False) + if r.dtype.char in '?bBhHiIlLqQpP': + r = r.astype(np.double) + if isinstance(x, (tuple, list)): + x = np.asarray(x) + if isinstance(x, np.ndarray): + if tensor: + r = r.reshape(r.shape + (1,)*x.ndim) + elif x.ndim >= r.ndim: + raise ValueError("x.ndim must be < r.ndim when tensor == False") + return np.prod(x - r, axis=0) + + +def polyval2d(x, y, c): + """ + Evaluate a 2-D polynomial at points (x, y). + + This function returns the value + + .. math:: p(x,y) = \\sum_{i,j} c_{i,j} * x^i * y^j + + The parameters `x` and `y` are converted to arrays only if they are + tuples or a lists, otherwise they are treated as a scalars and they + must have the same shape after conversion. In either case, either `x` + and `y` or their elements must support multiplication and addition both + with themselves and with the elements of `c`. + + If `c` has fewer than two dimensions, ones are implicitly appended to + its shape to make it 2-D. The shape of the result will be c.shape[2:] + + x.shape. + + Parameters + ---------- + x, y : array_like, compatible objects + The two dimensional series is evaluated at the points `(x, y)`, + where `x` and `y` must have the same shape. If `x` or `y` is a list + or tuple, it is first converted to an ndarray, otherwise it is left + unchanged and, if it isn't an ndarray, it is treated as a scalar. + c : array_like + Array of coefficients ordered so that the coefficient of the term + of multi-degree i,j is contained in `c[i,j]`. If `c` has + dimension greater than two the remaining indices enumerate multiple + sets of coefficients. + + Returns + ------- + values : ndarray, compatible object + The values of the two dimensional polynomial at points formed with + pairs of corresponding values from `x` and `y`. + + See Also + -------- + polyval, polygrid2d, polyval3d, polygrid3d + + Notes + ----- + + .. versionadded:: 1.7.0 + + """ + return pu._valnd(polyval, c, x, y) + + +def polygrid2d(x, y, c): + """ + Evaluate a 2-D polynomial on the Cartesian product of x and y. + + This function returns the values: + + .. math:: p(a,b) = \\sum_{i,j} c_{i,j} * a^i * b^j + + where the points `(a, b)` consist of all pairs formed by taking + `a` from `x` and `b` from `y`. The resulting points form a grid with + `x` in the first dimension and `y` in the second. + + The parameters `x` and `y` are converted to arrays only if they are + tuples or a lists, otherwise they are treated as a scalars. In either + case, either `x` and `y` or their elements must support multiplication + and addition both with themselves and with the elements of `c`. + + If `c` has fewer than two dimensions, ones are implicitly appended to + its shape to make it 2-D. The shape of the result will be c.shape[2:] + + x.shape + y.shape. + + Parameters + ---------- + x, y : array_like, compatible objects + The two dimensional series is evaluated at the points in the + Cartesian product of `x` and `y`. If `x` or `y` is a list or + tuple, it is first converted to an ndarray, otherwise it is left + unchanged and, if it isn't an ndarray, it is treated as a scalar. + c : array_like + Array of coefficients ordered so that the coefficients for terms of + degree i,j are contained in ``c[i,j]``. If `c` has dimension + greater than two the remaining indices enumerate multiple sets of + coefficients. + + Returns + ------- + values : ndarray, compatible object + The values of the two dimensional polynomial at points in the Cartesian + product of `x` and `y`. + + See Also + -------- + polyval, polyval2d, polyval3d, polygrid3d + + Notes + ----- + + .. versionadded:: 1.7.0 + + """ + return pu._gridnd(polyval, c, x, y) + + +def polyval3d(x, y, z, c): + """ + Evaluate a 3-D polynomial at points (x, y, z). + + This function returns the values: + + .. math:: p(x,y,z) = \\sum_{i,j,k} c_{i,j,k} * x^i * y^j * z^k + + The parameters `x`, `y`, and `z` are converted to arrays only if + they are tuples or a lists, otherwise they are treated as a scalars and + they must have the same shape after conversion. In either case, either + `x`, `y`, and `z` or their elements must support multiplication and + addition both with themselves and with the elements of `c`. + + If `c` has fewer than 3 dimensions, ones are implicitly appended to its + shape to make it 3-D. The shape of the result will be c.shape[3:] + + x.shape. + + Parameters + ---------- + x, y, z : array_like, compatible object + The three dimensional series is evaluated at the points + `(x, y, z)`, where `x`, `y`, and `z` must have the same shape. If + any of `x`, `y`, or `z` is a list or tuple, it is first converted + to an ndarray, otherwise it is left unchanged and if it isn't an + ndarray it is treated as a scalar. + c : array_like + Array of coefficients ordered so that the coefficient of the term of + multi-degree i,j,k is contained in ``c[i,j,k]``. If `c` has dimension + greater than 3 the remaining indices enumerate multiple sets of + coefficients. + + Returns + ------- + values : ndarray, compatible object + The values of the multidimensional polynomial on points formed with + triples of corresponding values from `x`, `y`, and `z`. + + See Also + -------- + polyval, polyval2d, polygrid2d, polygrid3d + + Notes + ----- + + .. versionadded:: 1.7.0 + + """ + return pu._valnd(polyval, c, x, y, z) + + +def polygrid3d(x, y, z, c): + """ + Evaluate a 3-D polynomial on the Cartesian product of x, y and z. + + This function returns the values: + + .. math:: p(a,b,c) = \\sum_{i,j,k} c_{i,j,k} * a^i * b^j * c^k + + where the points `(a, b, c)` consist of all triples formed by taking + `a` from `x`, `b` from `y`, and `c` from `z`. The resulting points form + a grid with `x` in the first dimension, `y` in the second, and `z` in + the third. + + The parameters `x`, `y`, and `z` are converted to arrays only if they + are tuples or a lists, otherwise they are treated as a scalars. In + either case, either `x`, `y`, and `z` or their elements must support + multiplication and addition both with themselves and with the elements + of `c`. + + If `c` has fewer than three dimensions, ones are implicitly appended to + its shape to make it 3-D. The shape of the result will be c.shape[3:] + + x.shape + y.shape + z.shape. + + Parameters + ---------- + x, y, z : array_like, compatible objects + The three dimensional series is evaluated at the points in the + Cartesian product of `x`, `y`, and `z`. If `x`,`y`, or `z` is a + list or tuple, it is first converted to an ndarray, otherwise it is + left unchanged and, if it isn't an ndarray, it is treated as a + scalar. + c : array_like + Array of coefficients ordered so that the coefficients for terms of + degree i,j are contained in ``c[i,j]``. If `c` has dimension + greater than two the remaining indices enumerate multiple sets of + coefficients. + + Returns + ------- + values : ndarray, compatible object + The values of the two dimensional polynomial at points in the Cartesian + product of `x` and `y`. + + See Also + -------- + polyval, polyval2d, polygrid2d, polyval3d + + Notes + ----- + + .. versionadded:: 1.7.0 + + """ + return pu._gridnd(polyval, c, x, y, z) + + +def polyvander(x, deg): + """Vandermonde matrix of given degree. + + Returns the Vandermonde matrix of degree `deg` and sample points + `x`. The Vandermonde matrix is defined by + + .. math:: V[..., i] = x^i, + + where `0 <= i <= deg`. The leading indices of `V` index the elements of + `x` and the last index is the power of `x`. + + If `c` is a 1-D array of coefficients of length `n + 1` and `V` is the + matrix ``V = polyvander(x, n)``, then ``np.dot(V, c)`` and + ``polyval(x, c)`` are the same up to roundoff. This equivalence is + useful both for least squares fitting and for the evaluation of a large + number of polynomials of the same degree and sample points. + + Parameters + ---------- + x : array_like + Array of points. The dtype is converted to float64 or complex128 + depending on whether any of the elements are complex. If `x` is + scalar it is converted to a 1-D array. + deg : int + Degree of the resulting matrix. + + Returns + ------- + vander : ndarray. + The Vandermonde matrix. The shape of the returned matrix is + ``x.shape + (deg + 1,)``, where the last index is the power of `x`. + The dtype will be the same as the converted `x`. + + See Also + -------- + polyvander2d, polyvander3d + + """ + ideg = pu._deprecate_as_int(deg, "deg") + if ideg < 0: + raise ValueError("deg must be non-negative") + + x = np.array(x, copy=False, ndmin=1) + 0.0 + dims = (ideg + 1,) + x.shape + dtyp = x.dtype + v = np.empty(dims, dtype=dtyp) + v[0] = x*0 + 1 + if ideg > 0: + v[1] = x + for i in range(2, ideg + 1): + v[i] = v[i-1]*x + return np.moveaxis(v, 0, -1) + + +def polyvander2d(x, y, deg): + """Pseudo-Vandermonde matrix of given degrees. + + Returns the pseudo-Vandermonde matrix of degrees `deg` and sample + points `(x, y)`. The pseudo-Vandermonde matrix is defined by + + .. math:: V[..., (deg[1] + 1)*i + j] = x^i * y^j, + + where `0 <= i <= deg[0]` and `0 <= j <= deg[1]`. The leading indices of + `V` index the points `(x, y)` and the last index encodes the powers of + `x` and `y`. + + If ``V = polyvander2d(x, y, [xdeg, ydeg])``, then the columns of `V` + correspond to the elements of a 2-D coefficient array `c` of shape + (xdeg + 1, ydeg + 1) in the order + + .. math:: c_{00}, c_{01}, c_{02} ... , c_{10}, c_{11}, c_{12} ... + + and ``np.dot(V, c.flat)`` and ``polyval2d(x, y, c)`` will be the same + up to roundoff. This equivalence is useful both for least squares + fitting and for the evaluation of a large number of 2-D polynomials + of the same degrees and sample points. + + Parameters + ---------- + x, y : array_like + Arrays of point coordinates, all of the same shape. The dtypes + will be converted to either float64 or complex128 depending on + whether any of the elements are complex. Scalars are converted to + 1-D arrays. + deg : list of ints + List of maximum degrees of the form [x_deg, y_deg]. + + Returns + ------- + vander2d : ndarray + The shape of the returned matrix is ``x.shape + (order,)``, where + :math:`order = (deg[0]+1)*(deg([1]+1)`. The dtype will be the same + as the converted `x` and `y`. + + See Also + -------- + polyvander, polyvander3d, polyval2d, polyval3d + + """ + return pu._vander_nd_flat((polyvander, polyvander), (x, y), deg) + + +def polyvander3d(x, y, z, deg): + """Pseudo-Vandermonde matrix of given degrees. + + Returns the pseudo-Vandermonde matrix of degrees `deg` and sample + points `(x, y, z)`. If `l, m, n` are the given degrees in `x, y, z`, + then The pseudo-Vandermonde matrix is defined by + + .. math:: V[..., (m+1)(n+1)i + (n+1)j + k] = x^i * y^j * z^k, + + where `0 <= i <= l`, `0 <= j <= m`, and `0 <= j <= n`. The leading + indices of `V` index the points `(x, y, z)` and the last index encodes + the powers of `x`, `y`, and `z`. + + If ``V = polyvander3d(x, y, z, [xdeg, ydeg, zdeg])``, then the columns + of `V` correspond to the elements of a 3-D coefficient array `c` of + shape (xdeg + 1, ydeg + 1, zdeg + 1) in the order + + .. math:: c_{000}, c_{001}, c_{002},... , c_{010}, c_{011}, c_{012},... + + and ``np.dot(V, c.flat)`` and ``polyval3d(x, y, z, c)`` will be the + same up to roundoff. This equivalence is useful both for least squares + fitting and for the evaluation of a large number of 3-D polynomials + of the same degrees and sample points. + + Parameters + ---------- + x, y, z : array_like + Arrays of point coordinates, all of the same shape. The dtypes will + be converted to either float64 or complex128 depending on whether + any of the elements are complex. Scalars are converted to 1-D + arrays. + deg : list of ints + List of maximum degrees of the form [x_deg, y_deg, z_deg]. + + Returns + ------- + vander3d : ndarray + The shape of the returned matrix is ``x.shape + (order,)``, where + :math:`order = (deg[0]+1)*(deg([1]+1)*(deg[2]+1)`. The dtype will + be the same as the converted `x`, `y`, and `z`. + + See Also + -------- + polyvander, polyvander3d, polyval2d, polyval3d + + Notes + ----- + + .. versionadded:: 1.7.0 + + """ + return pu._vander_nd_flat((polyvander, polyvander, polyvander), (x, y, z), deg) + + +def polyfit(x, y, deg, rcond=None, full=False, w=None): + """ + Least-squares fit of a polynomial to data. + + Return the coefficients of a polynomial of degree `deg` that is the + least squares fit to the data values `y` given at points `x`. If `y` is + 1-D the returned coefficients will also be 1-D. If `y` is 2-D multiple + fits are done, one for each column of `y`, and the resulting + coefficients are stored in the corresponding columns of a 2-D return. + The fitted polynomial(s) are in the form + + .. math:: p(x) = c_0 + c_1 * x + ... + c_n * x^n, + + where `n` is `deg`. + + Parameters + ---------- + x : array_like, shape (`M`,) + x-coordinates of the `M` sample (data) points ``(x[i], y[i])``. + y : array_like, shape (`M`,) or (`M`, `K`) + y-coordinates of the sample points. Several sets of sample points + sharing the same x-coordinates can be (independently) fit with one + call to `polyfit` by passing in for `y` a 2-D array that contains + one data set per column. + deg : int or 1-D array_like + Degree(s) of the fitting polynomials. If `deg` is a single integer + all terms up to and including the `deg`'th term are included in the + fit. For NumPy versions >= 1.11.0 a list of integers specifying the + degrees of the terms to include may be used instead. + rcond : float, optional + Relative condition number of the fit. Singular values smaller + than `rcond`, relative to the largest singular value, will be + ignored. The default value is ``len(x)*eps``, where `eps` is the + relative precision of the platform's float type, about 2e-16 in + most cases. + full : bool, optional + Switch determining the nature of the return value. When ``False`` + (the default) just the coefficients are returned; when ``True``, + diagnostic information from the singular value decomposition (used + to solve the fit's matrix equation) is also returned. + w : array_like, shape (`M`,), optional + Weights. If not None, the weight ``w[i]`` applies to the unsquared + residual ``y[i] - y_hat[i]`` at ``x[i]``. Ideally the weights are + chosen so that the errors of the products ``w[i]*y[i]`` all have the + same variance. When using inverse-variance weighting, use + ``w[i] = 1/sigma(y[i])``. The default value is None. + + .. versionadded:: 1.5.0 + + Returns + ------- + coef : ndarray, shape (`deg` + 1,) or (`deg` + 1, `K`) + Polynomial coefficients ordered from low to high. If `y` was 2-D, + the coefficients in column `k` of `coef` represent the polynomial + fit to the data in `y`'s `k`-th column. + + [residuals, rank, singular_values, rcond] : list + These values are only returned if ``full == True`` + + - residuals -- sum of squared residuals of the least squares fit + - rank -- the numerical rank of the scaled Vandermonde matrix + - singular_values -- singular values of the scaled Vandermonde matrix + - rcond -- value of `rcond`. + + For more details, see `numpy.linalg.lstsq`. + + Raises + ------ + RankWarning + Raised if the matrix in the least-squares fit is rank deficient. + The warning is only raised if ``full == False``. The warnings can + be turned off by: + + >>> import warnings + >>> warnings.simplefilter('ignore', np.RankWarning) + + See Also + -------- + numpy.polynomial.chebyshev.chebfit + numpy.polynomial.legendre.legfit + numpy.polynomial.laguerre.lagfit + numpy.polynomial.hermite.hermfit + numpy.polynomial.hermite_e.hermefit + polyval : Evaluates a polynomial. + polyvander : Vandermonde matrix for powers. + numpy.linalg.lstsq : Computes a least-squares fit from the matrix. + scipy.interpolate.UnivariateSpline : Computes spline fits. + + Notes + ----- + The solution is the coefficients of the polynomial `p` that minimizes + the sum of the weighted squared errors + + .. math:: E = \\sum_j w_j^2 * |y_j - p(x_j)|^2, + + where the :math:`w_j` are the weights. This problem is solved by + setting up the (typically) over-determined matrix equation: + + .. math:: V(x) * c = w * y, + + where `V` is the weighted pseudo Vandermonde matrix of `x`, `c` are the + coefficients to be solved for, `w` are the weights, and `y` are the + observed values. This equation is then solved using the singular value + decomposition of `V`. + + If some of the singular values of `V` are so small that they are + neglected (and `full` == ``False``), a `RankWarning` will be raised. + This means that the coefficient values may be poorly determined. + Fitting to a lower order polynomial will usually get rid of the warning + (but may not be what you want, of course; if you have independent + reason(s) for choosing the degree which isn't working, you may have to: + a) reconsider those reasons, and/or b) reconsider the quality of your + data). The `rcond` parameter can also be set to a value smaller than + its default, but the resulting fit may be spurious and have large + contributions from roundoff error. + + Polynomial fits using double precision tend to "fail" at about + (polynomial) degree 20. Fits using Chebyshev or Legendre series are + generally better conditioned, but much can still depend on the + distribution of the sample points and the smoothness of the data. If + the quality of the fit is inadequate, splines may be a good + alternative. + + Examples + -------- + >>> np.random.seed(123) + >>> from numpy.polynomial import polynomial as P + >>> x = np.linspace(-1,1,51) # x "data": [-1, -0.96, ..., 0.96, 1] + >>> y = x**3 - x + np.random.randn(len(x)) # x^3 - x + Gaussian noise + >>> c, stats = P.polyfit(x,y,3,full=True) + >>> np.random.seed(123) + >>> c # c[0], c[2] should be approx. 0, c[1] approx. -1, c[3] approx. 1 + array([ 0.01909725, -1.30598256, -0.00577963, 1.02644286]) # may vary + >>> stats # note the large SSR, explaining the rather poor results + [array([ 38.06116253]), 4, array([ 1.38446749, 1.32119158, 0.50443316, # may vary + 0.28853036]), 1.1324274851176597e-014] + + Same thing without the added noise + + >>> y = x**3 - x + >>> c, stats = P.polyfit(x,y,3,full=True) + >>> c # c[0], c[2] should be "very close to 0", c[1] ~= -1, c[3] ~= 1 + array([-6.36925336e-18, -1.00000000e+00, -4.08053781e-16, 1.00000000e+00]) + >>> stats # note the minuscule SSR + [array([ 7.46346754e-31]), 4, array([ 1.38446749, 1.32119158, # may vary + 0.50443316, 0.28853036]), 1.1324274851176597e-014] + + """ + return pu._fit(polyvander, x, y, deg, rcond, full, w) + + +def polycompanion(c): + """ + Return the companion matrix of c. + + The companion matrix for power series cannot be made symmetric by + scaling the basis, so this function differs from those for the + orthogonal polynomials. + + Parameters + ---------- + c : array_like + 1-D array of polynomial coefficients ordered from low to high + degree. + + Returns + ------- + mat : ndarray + Companion matrix of dimensions (deg, deg). + + Notes + ----- + + .. versionadded:: 1.7.0 + + """ + # c is a trimmed copy + [c] = pu.as_series([c]) + if len(c) < 2: + raise ValueError('Series must have maximum degree of at least 1.') + if len(c) == 2: + return np.array([[-c[0]/c[1]]]) + + n = len(c) - 1 + mat = np.zeros((n, n), dtype=c.dtype) + bot = mat.reshape(-1)[n::n+1] + bot[...] = 1 + mat[:, -1] -= c[:-1]/c[-1] + return mat + + +def polyroots(c): + """ + Compute the roots of a polynomial. + + Return the roots (a.k.a. "zeros") of the polynomial + + .. math:: p(x) = \\sum_i c[i] * x^i. + + Parameters + ---------- + c : 1-D array_like + 1-D array of polynomial coefficients. + + Returns + ------- + out : ndarray + Array of the roots of the polynomial. If all the roots are real, + then `out` is also real, otherwise it is complex. + + See Also + -------- + numpy.polynomial.chebyshev.chebroots + numpy.polynomial.legendre.legroots + numpy.polynomial.laguerre.lagroots + numpy.polynomial.hermite.hermroots + numpy.polynomial.hermite_e.hermeroots + + Notes + ----- + The root estimates are obtained as the eigenvalues of the companion + matrix, Roots far from the origin of the complex plane may have large + errors due to the numerical instability of the power series for such + values. Roots with multiplicity greater than 1 will also show larger + errors as the value of the series near such points is relatively + insensitive to errors in the roots. Isolated roots near the origin can + be improved by a few iterations of Newton's method. + + Examples + -------- + >>> import numpy.polynomial.polynomial as poly + >>> poly.polyroots(poly.polyfromroots((-1,0,1))) + array([-1., 0., 1.]) + >>> poly.polyroots(poly.polyfromroots((-1,0,1))).dtype + dtype('float64') + >>> j = complex(0,1) + >>> poly.polyroots(poly.polyfromroots((-j,0,j))) + array([ 0.00000000e+00+0.j, 0.00000000e+00+1.j, 2.77555756e-17-1.j]) # may vary + + """ + # c is a trimmed copy + [c] = pu.as_series([c]) + if len(c) < 2: + return np.array([], dtype=c.dtype) + if len(c) == 2: + return np.array([-c[0]/c[1]]) + + # rotated companion matrix reduces error + m = polycompanion(c)[::-1,::-1] + r = la.eigvals(m) + r.sort() + return r + + +# +# polynomial class +# + +class Polynomial(ABCPolyBase): + """A power series class. + + The Polynomial class provides the standard Python numerical methods + '+', '-', '*', '//', '%', 'divmod', '**', and '()' as well as the + attributes and methods listed in the `ABCPolyBase` documentation. + + Parameters + ---------- + coef : array_like + Polynomial coefficients in order of increasing degree, i.e., + ``(1, 2, 3)`` give ``1 + 2*x + 3*x**2``. + domain : (2,) array_like, optional + Domain to use. The interval ``[domain[0], domain[1]]`` is mapped + to the interval ``[window[0], window[1]]`` by shifting and scaling. + The default value is [-1, 1]. + window : (2,) array_like, optional + Window, see `domain` for its use. The default value is [-1, 1]. + + .. versionadded:: 1.6.0 + symbol : str, optional + Symbol used to represent the independent variable in string + representations of the polynomial expression, e.g. for printing. + The symbol must be a valid Python identifier. Default value is 'x'. + + .. versionadded:: 1.24 + + """ + # Virtual Functions + _add = staticmethod(polyadd) + _sub = staticmethod(polysub) + _mul = staticmethod(polymul) + _div = staticmethod(polydiv) + _pow = staticmethod(polypow) + _val = staticmethod(polyval) + _int = staticmethod(polyint) + _der = staticmethod(polyder) + _fit = staticmethod(polyfit) + _line = staticmethod(polyline) + _roots = staticmethod(polyroots) + _fromroots = staticmethod(polyfromroots) + + # Virtual properties + domain = np.array(polydomain) + window = np.array(polydomain) + basis_name = None + + @classmethod + def _str_term_unicode(cls, i, arg_str): + if i == '1': + return f"·{arg_str}" + else: + return f"·{arg_str}{i.translate(cls._superscript_mapping)}" + + @staticmethod + def _str_term_ascii(i, arg_str): + if i == '1': + return f" {arg_str}" + else: + return f" {arg_str}**{i}" + + @staticmethod + def _repr_latex_term(i, arg_str, needs_parens): + if needs_parens: + arg_str = rf"\left({arg_str}\right)" + if i == 0: + return '1' + elif i == 1: + return arg_str + else: + return f"{arg_str}^{{{i}}}" diff --git a/deepseekvl2/lib/python3.10/site-packages/numpy/polynomial/polyutils.pyi b/deepseekvl2/lib/python3.10/site-packages/numpy/polynomial/polyutils.pyi new file mode 100644 index 0000000000000000000000000000000000000000..c0bcc67847f6b466c8d4fcf6f9b323df736c1c5f --- /dev/null +++ b/deepseekvl2/lib/python3.10/site-packages/numpy/polynomial/polyutils.pyi @@ -0,0 +1,11 @@ +__all__: list[str] + +class RankWarning(UserWarning): ... + +def trimseq(seq): ... +def as_series(alist, trim=...): ... +def trimcoef(c, tol=...): ... +def getdomain(x): ... +def mapparms(old, new): ... +def mapdomain(x, old, new): ... +def format_float(x, parens=...): ... diff --git a/deepseekvl2/lib/python3.10/site-packages/numpy/polynomial/setup.py b/deepseekvl2/lib/python3.10/site-packages/numpy/polynomial/setup.py new file mode 100644 index 0000000000000000000000000000000000000000..b58e867a133f804fbaf0d31099258a11e29058aa --- /dev/null +++ b/deepseekvl2/lib/python3.10/site-packages/numpy/polynomial/setup.py @@ -0,0 +1,10 @@ +def configuration(parent_package='',top_path=None): + from numpy.distutils.misc_util import Configuration + config = Configuration('polynomial', parent_package, top_path) + config.add_subpackage('tests') + config.add_data_files('*.pyi') + return config + +if __name__ == '__main__': + from numpy.distutils.core import setup + setup(configuration=configuration) diff --git a/deepseekvl2/lib/python3.10/site-packages/numpy/polynomial/tests/__init__.py b/deepseekvl2/lib/python3.10/site-packages/numpy/polynomial/tests/__init__.py new file mode 100644 index 0000000000000000000000000000000000000000..e69de29bb2d1d6434b8b29ae775ad8c2e48c5391 diff --git a/deepseekvl2/lib/python3.10/site-packages/numpy/polynomial/tests/__pycache__/__init__.cpython-310.pyc b/deepseekvl2/lib/python3.10/site-packages/numpy/polynomial/tests/__pycache__/__init__.cpython-310.pyc new file mode 100644 index 0000000000000000000000000000000000000000..6344d5e19ddd2bf6c6bb88094ab80b0b68fdb7a3 Binary files /dev/null and b/deepseekvl2/lib/python3.10/site-packages/numpy/polynomial/tests/__pycache__/__init__.cpython-310.pyc differ diff --git a/deepseekvl2/lib/python3.10/site-packages/numpy/polynomial/tests/__pycache__/test_chebyshev.cpython-310.pyc b/deepseekvl2/lib/python3.10/site-packages/numpy/polynomial/tests/__pycache__/test_chebyshev.cpython-310.pyc new file mode 100644 index 0000000000000000000000000000000000000000..65b0d5299a5e17516ed467d439bd233f7d4c0da1 Binary files /dev/null and b/deepseekvl2/lib/python3.10/site-packages/numpy/polynomial/tests/__pycache__/test_chebyshev.cpython-310.pyc differ diff --git a/deepseekvl2/lib/python3.10/site-packages/numpy/polynomial/tests/__pycache__/test_classes.cpython-310.pyc b/deepseekvl2/lib/python3.10/site-packages/numpy/polynomial/tests/__pycache__/test_classes.cpython-310.pyc new file mode 100644 index 0000000000000000000000000000000000000000..a6e7b4dfcbdceb0f246f821897ac68bb2cd36aae Binary files /dev/null and b/deepseekvl2/lib/python3.10/site-packages/numpy/polynomial/tests/__pycache__/test_classes.cpython-310.pyc differ diff --git a/deepseekvl2/lib/python3.10/site-packages/numpy/polynomial/tests/__pycache__/test_hermite.cpython-310.pyc b/deepseekvl2/lib/python3.10/site-packages/numpy/polynomial/tests/__pycache__/test_hermite.cpython-310.pyc new file mode 100644 index 0000000000000000000000000000000000000000..f7e7a1d0110da27373581b6d351875ac83520723 Binary files /dev/null and b/deepseekvl2/lib/python3.10/site-packages/numpy/polynomial/tests/__pycache__/test_hermite.cpython-310.pyc differ diff --git a/deepseekvl2/lib/python3.10/site-packages/numpy/polynomial/tests/__pycache__/test_hermite_e.cpython-310.pyc b/deepseekvl2/lib/python3.10/site-packages/numpy/polynomial/tests/__pycache__/test_hermite_e.cpython-310.pyc new file mode 100644 index 0000000000000000000000000000000000000000..3778e6cd50a2c4d21a99fd0cda6494155fc32228 Binary files /dev/null and b/deepseekvl2/lib/python3.10/site-packages/numpy/polynomial/tests/__pycache__/test_hermite_e.cpython-310.pyc differ diff --git a/deepseekvl2/lib/python3.10/site-packages/numpy/polynomial/tests/__pycache__/test_laguerre.cpython-310.pyc b/deepseekvl2/lib/python3.10/site-packages/numpy/polynomial/tests/__pycache__/test_laguerre.cpython-310.pyc new file mode 100644 index 0000000000000000000000000000000000000000..a3210f47eed8141eb8d2c84b2db7c7be52602277 Binary files /dev/null and b/deepseekvl2/lib/python3.10/site-packages/numpy/polynomial/tests/__pycache__/test_laguerre.cpython-310.pyc differ diff --git a/deepseekvl2/lib/python3.10/site-packages/numpy/polynomial/tests/__pycache__/test_legendre.cpython-310.pyc b/deepseekvl2/lib/python3.10/site-packages/numpy/polynomial/tests/__pycache__/test_legendre.cpython-310.pyc new file mode 100644 index 0000000000000000000000000000000000000000..e0dc74df6fadbca61ea1bac6813e23984df28d17 Binary files /dev/null and b/deepseekvl2/lib/python3.10/site-packages/numpy/polynomial/tests/__pycache__/test_legendre.cpython-310.pyc differ diff --git a/deepseekvl2/lib/python3.10/site-packages/numpy/polynomial/tests/__pycache__/test_polynomial.cpython-310.pyc b/deepseekvl2/lib/python3.10/site-packages/numpy/polynomial/tests/__pycache__/test_polynomial.cpython-310.pyc new file mode 100644 index 0000000000000000000000000000000000000000..673221eaac1d4cb351176e70bc3d03212c7e7141 Binary files /dev/null and b/deepseekvl2/lib/python3.10/site-packages/numpy/polynomial/tests/__pycache__/test_polynomial.cpython-310.pyc differ diff --git a/deepseekvl2/lib/python3.10/site-packages/numpy/polynomial/tests/__pycache__/test_polyutils.cpython-310.pyc b/deepseekvl2/lib/python3.10/site-packages/numpy/polynomial/tests/__pycache__/test_polyutils.cpython-310.pyc new file mode 100644 index 0000000000000000000000000000000000000000..6602acd699be9336a8d67bd87769c9c3106d96e2 Binary files /dev/null and b/deepseekvl2/lib/python3.10/site-packages/numpy/polynomial/tests/__pycache__/test_polyutils.cpython-310.pyc differ diff --git a/deepseekvl2/lib/python3.10/site-packages/numpy/polynomial/tests/__pycache__/test_printing.cpython-310.pyc b/deepseekvl2/lib/python3.10/site-packages/numpy/polynomial/tests/__pycache__/test_printing.cpython-310.pyc new file mode 100644 index 0000000000000000000000000000000000000000..cceddb1ec01d2d867141fba9f8277f8346f54f50 Binary files /dev/null and b/deepseekvl2/lib/python3.10/site-packages/numpy/polynomial/tests/__pycache__/test_printing.cpython-310.pyc differ diff --git a/deepseekvl2/lib/python3.10/site-packages/numpy/polynomial/tests/__pycache__/test_symbol.cpython-310.pyc b/deepseekvl2/lib/python3.10/site-packages/numpy/polynomial/tests/__pycache__/test_symbol.cpython-310.pyc new file mode 100644 index 0000000000000000000000000000000000000000..69ebb69dcbf36307f7d8a93bf46f91b2dd4d7741 Binary files /dev/null and b/deepseekvl2/lib/python3.10/site-packages/numpy/polynomial/tests/__pycache__/test_symbol.cpython-310.pyc differ diff --git a/deepseekvl2/lib/python3.10/site-packages/numpy/polynomial/tests/test_chebyshev.py b/deepseekvl2/lib/python3.10/site-packages/numpy/polynomial/tests/test_chebyshev.py new file mode 100644 index 0000000000000000000000000000000000000000..2f54bebfdb27d54f436378e4ab6d6c8f2426dd90 --- /dev/null +++ b/deepseekvl2/lib/python3.10/site-packages/numpy/polynomial/tests/test_chebyshev.py @@ -0,0 +1,619 @@ +"""Tests for chebyshev module. + +""" +from functools import reduce + +import numpy as np +import numpy.polynomial.chebyshev as cheb +from numpy.polynomial.polynomial import polyval +from numpy.testing import ( + assert_almost_equal, assert_raises, assert_equal, assert_, + ) + + +def trim(x): + return cheb.chebtrim(x, tol=1e-6) + +T0 = [1] +T1 = [0, 1] +T2 = [-1, 0, 2] +T3 = [0, -3, 0, 4] +T4 = [1, 0, -8, 0, 8] +T5 = [0, 5, 0, -20, 0, 16] +T6 = [-1, 0, 18, 0, -48, 0, 32] +T7 = [0, -7, 0, 56, 0, -112, 0, 64] +T8 = [1, 0, -32, 0, 160, 0, -256, 0, 128] +T9 = [0, 9, 0, -120, 0, 432, 0, -576, 0, 256] + +Tlist = [T0, T1, T2, T3, T4, T5, T6, T7, T8, T9] + + +class TestPrivate: + + def test__cseries_to_zseries(self): + for i in range(5): + inp = np.array([2] + [1]*i, np.double) + tgt = np.array([.5]*i + [2] + [.5]*i, np.double) + res = cheb._cseries_to_zseries(inp) + assert_equal(res, tgt) + + def test__zseries_to_cseries(self): + for i in range(5): + inp = np.array([.5]*i + [2] + [.5]*i, np.double) + tgt = np.array([2] + [1]*i, np.double) + res = cheb._zseries_to_cseries(inp) + assert_equal(res, tgt) + + +class TestConstants: + + def test_chebdomain(self): + assert_equal(cheb.chebdomain, [-1, 1]) + + def test_chebzero(self): + assert_equal(cheb.chebzero, [0]) + + def test_chebone(self): + assert_equal(cheb.chebone, [1]) + + def test_chebx(self): + assert_equal(cheb.chebx, [0, 1]) + + +class TestArithmetic: + + def test_chebadd(self): + for i in range(5): + for j in range(5): + msg = f"At i={i}, j={j}" + tgt = np.zeros(max(i, j) + 1) + tgt[i] += 1 + tgt[j] += 1 + res = cheb.chebadd([0]*i + [1], [0]*j + [1]) + assert_equal(trim(res), trim(tgt), err_msg=msg) + + def test_chebsub(self): + for i in range(5): + for j in range(5): + msg = f"At i={i}, j={j}" + tgt = np.zeros(max(i, j) + 1) + tgt[i] += 1 + tgt[j] -= 1 + res = cheb.chebsub([0]*i + [1], [0]*j + [1]) + assert_equal(trim(res), trim(tgt), err_msg=msg) + + def test_chebmulx(self): + assert_equal(cheb.chebmulx([0]), [0]) + assert_equal(cheb.chebmulx([1]), [0, 1]) + for i in range(1, 5): + ser = [0]*i + [1] + tgt = [0]*(i - 1) + [.5, 0, .5] + assert_equal(cheb.chebmulx(ser), tgt) + + def test_chebmul(self): + for i in range(5): + for j in range(5): + msg = f"At i={i}, j={j}" + tgt = np.zeros(i + j + 1) + tgt[i + j] += .5 + tgt[abs(i - j)] += .5 + res = cheb.chebmul([0]*i + [1], [0]*j + [1]) + assert_equal(trim(res), trim(tgt), err_msg=msg) + + def test_chebdiv(self): + for i in range(5): + for j in range(5): + msg = f"At i={i}, j={j}" + ci = [0]*i + [1] + cj = [0]*j + [1] + tgt = cheb.chebadd(ci, cj) + quo, rem = cheb.chebdiv(tgt, ci) + res = cheb.chebadd(cheb.chebmul(quo, ci), rem) + assert_equal(trim(res), trim(tgt), err_msg=msg) + + def test_chebpow(self): + for i in range(5): + for j in range(5): + msg = f"At i={i}, j={j}" + c = np.arange(i + 1) + tgt = reduce(cheb.chebmul, [c]*j, np.array([1])) + res = cheb.chebpow(c, j) + assert_equal(trim(res), trim(tgt), err_msg=msg) + + +class TestEvaluation: + # coefficients of 1 + 2*x + 3*x**2 + c1d = np.array([2.5, 2., 1.5]) + c2d = np.einsum('i,j->ij', c1d, c1d) + c3d = np.einsum('i,j,k->ijk', c1d, c1d, c1d) + + # some random values in [-1, 1) + x = np.random.random((3, 5))*2 - 1 + y = polyval(x, [1., 2., 3.]) + + def test_chebval(self): + #check empty input + assert_equal(cheb.chebval([], [1]).size, 0) + + #check normal input) + x = np.linspace(-1, 1) + y = [polyval(x, c) for c in Tlist] + for i in range(10): + msg = f"At i={i}" + tgt = y[i] + res = cheb.chebval(x, [0]*i + [1]) + assert_almost_equal(res, tgt, err_msg=msg) + + #check that shape is preserved + for i in range(3): + dims = [2]*i + x = np.zeros(dims) + assert_equal(cheb.chebval(x, [1]).shape, dims) + assert_equal(cheb.chebval(x, [1, 0]).shape, dims) + assert_equal(cheb.chebval(x, [1, 0, 0]).shape, dims) + + def test_chebval2d(self): + x1, x2, x3 = self.x + y1, y2, y3 = self.y + + #test exceptions + assert_raises(ValueError, cheb.chebval2d, x1, x2[:2], self.c2d) + + #test values + tgt = y1*y2 + res = cheb.chebval2d(x1, x2, self.c2d) + assert_almost_equal(res, tgt) + + #test shape + z = np.ones((2, 3)) + res = cheb.chebval2d(z, z, self.c2d) + assert_(res.shape == (2, 3)) + + def test_chebval3d(self): + x1, x2, x3 = self.x + y1, y2, y3 = self.y + + #test exceptions + assert_raises(ValueError, cheb.chebval3d, x1, x2, x3[:2], self.c3d) + + #test values + tgt = y1*y2*y3 + res = cheb.chebval3d(x1, x2, x3, self.c3d) + assert_almost_equal(res, tgt) + + #test shape + z = np.ones((2, 3)) + res = cheb.chebval3d(z, z, z, self.c3d) + assert_(res.shape == (2, 3)) + + def test_chebgrid2d(self): + x1, x2, x3 = self.x + y1, y2, y3 = self.y + + #test values + tgt = np.einsum('i,j->ij', y1, y2) + res = cheb.chebgrid2d(x1, x2, self.c2d) + assert_almost_equal(res, tgt) + + #test shape + z = np.ones((2, 3)) + res = cheb.chebgrid2d(z, z, self.c2d) + assert_(res.shape == (2, 3)*2) + + def test_chebgrid3d(self): + x1, x2, x3 = self.x + y1, y2, y3 = self.y + + #test values + tgt = np.einsum('i,j,k->ijk', y1, y2, y3) + res = cheb.chebgrid3d(x1, x2, x3, self.c3d) + assert_almost_equal(res, tgt) + + #test shape + z = np.ones((2, 3)) + res = cheb.chebgrid3d(z, z, z, self.c3d) + assert_(res.shape == (2, 3)*3) + + +class TestIntegral: + + def test_chebint(self): + # check exceptions + assert_raises(TypeError, cheb.chebint, [0], .5) + assert_raises(ValueError, cheb.chebint, [0], -1) + assert_raises(ValueError, cheb.chebint, [0], 1, [0, 0]) + assert_raises(ValueError, cheb.chebint, [0], lbnd=[0]) + assert_raises(ValueError, cheb.chebint, [0], scl=[0]) + assert_raises(TypeError, cheb.chebint, [0], axis=.5) + + # test integration of zero polynomial + for i in range(2, 5): + k = [0]*(i - 2) + [1] + res = cheb.chebint([0], m=i, k=k) + assert_almost_equal(res, [0, 1]) + + # check single integration with integration constant + for i in range(5): + scl = i + 1 + pol = [0]*i + [1] + tgt = [i] + [0]*i + [1/scl] + chebpol = cheb.poly2cheb(pol) + chebint = cheb.chebint(chebpol, m=1, k=[i]) + res = cheb.cheb2poly(chebint) + assert_almost_equal(trim(res), trim(tgt)) + + # check single integration with integration constant and lbnd + for i in range(5): + scl = i + 1 + pol = [0]*i + [1] + chebpol = cheb.poly2cheb(pol) + chebint = cheb.chebint(chebpol, m=1, k=[i], lbnd=-1) + assert_almost_equal(cheb.chebval(-1, chebint), i) + + # check single integration with integration constant and scaling + for i in range(5): + scl = i + 1 + pol = [0]*i + [1] + tgt = [i] + [0]*i + [2/scl] + chebpol = cheb.poly2cheb(pol) + chebint = cheb.chebint(chebpol, m=1, k=[i], scl=2) + res = cheb.cheb2poly(chebint) + assert_almost_equal(trim(res), trim(tgt)) + + # check multiple integrations with default k + for i in range(5): + for j in range(2, 5): + pol = [0]*i + [1] + tgt = pol[:] + for k in range(j): + tgt = cheb.chebint(tgt, m=1) + res = cheb.chebint(pol, m=j) + assert_almost_equal(trim(res), trim(tgt)) + + # check multiple integrations with defined k + for i in range(5): + for j in range(2, 5): + pol = [0]*i + [1] + tgt = pol[:] + for k in range(j): + tgt = cheb.chebint(tgt, m=1, k=[k]) + res = cheb.chebint(pol, m=j, k=list(range(j))) + assert_almost_equal(trim(res), trim(tgt)) + + # check multiple integrations with lbnd + for i in range(5): + for j in range(2, 5): + pol = [0]*i + [1] + tgt = pol[:] + for k in range(j): + tgt = cheb.chebint(tgt, m=1, k=[k], lbnd=-1) + res = cheb.chebint(pol, m=j, k=list(range(j)), lbnd=-1) + assert_almost_equal(trim(res), trim(tgt)) + + # check multiple integrations with scaling + for i in range(5): + for j in range(2, 5): + pol = [0]*i + [1] + tgt = pol[:] + for k in range(j): + tgt = cheb.chebint(tgt, m=1, k=[k], scl=2) + res = cheb.chebint(pol, m=j, k=list(range(j)), scl=2) + assert_almost_equal(trim(res), trim(tgt)) + + def test_chebint_axis(self): + # check that axis keyword works + c2d = np.random.random((3, 4)) + + tgt = np.vstack([cheb.chebint(c) for c in c2d.T]).T + res = cheb.chebint(c2d, axis=0) + assert_almost_equal(res, tgt) + + tgt = np.vstack([cheb.chebint(c) for c in c2d]) + res = cheb.chebint(c2d, axis=1) + assert_almost_equal(res, tgt) + + tgt = np.vstack([cheb.chebint(c, k=3) for c in c2d]) + res = cheb.chebint(c2d, k=3, axis=1) + assert_almost_equal(res, tgt) + + +class TestDerivative: + + def test_chebder(self): + # check exceptions + assert_raises(TypeError, cheb.chebder, [0], .5) + assert_raises(ValueError, cheb.chebder, [0], -1) + + # check that zeroth derivative does nothing + for i in range(5): + tgt = [0]*i + [1] + res = cheb.chebder(tgt, m=0) + assert_equal(trim(res), trim(tgt)) + + # check that derivation is the inverse of integration + for i in range(5): + for j in range(2, 5): + tgt = [0]*i + [1] + res = cheb.chebder(cheb.chebint(tgt, m=j), m=j) + assert_almost_equal(trim(res), trim(tgt)) + + # check derivation with scaling + for i in range(5): + for j in range(2, 5): + tgt = [0]*i + [1] + res = cheb.chebder(cheb.chebint(tgt, m=j, scl=2), m=j, scl=.5) + assert_almost_equal(trim(res), trim(tgt)) + + def test_chebder_axis(self): + # check that axis keyword works + c2d = np.random.random((3, 4)) + + tgt = np.vstack([cheb.chebder(c) for c in c2d.T]).T + res = cheb.chebder(c2d, axis=0) + assert_almost_equal(res, tgt) + + tgt = np.vstack([cheb.chebder(c) for c in c2d]) + res = cheb.chebder(c2d, axis=1) + assert_almost_equal(res, tgt) + + +class TestVander: + # some random values in [-1, 1) + x = np.random.random((3, 5))*2 - 1 + + def test_chebvander(self): + # check for 1d x + x = np.arange(3) + v = cheb.chebvander(x, 3) + assert_(v.shape == (3, 4)) + for i in range(4): + coef = [0]*i + [1] + assert_almost_equal(v[..., i], cheb.chebval(x, coef)) + + # check for 2d x + x = np.array([[1, 2], [3, 4], [5, 6]]) + v = cheb.chebvander(x, 3) + assert_(v.shape == (3, 2, 4)) + for i in range(4): + coef = [0]*i + [1] + assert_almost_equal(v[..., i], cheb.chebval(x, coef)) + + def test_chebvander2d(self): + # also tests chebval2d for non-square coefficient array + x1, x2, x3 = self.x + c = np.random.random((2, 3)) + van = cheb.chebvander2d(x1, x2, [1, 2]) + tgt = cheb.chebval2d(x1, x2, c) + res = np.dot(van, c.flat) + assert_almost_equal(res, tgt) + + # check shape + van = cheb.chebvander2d([x1], [x2], [1, 2]) + assert_(van.shape == (1, 5, 6)) + + def test_chebvander3d(self): + # also tests chebval3d for non-square coefficient array + x1, x2, x3 = self.x + c = np.random.random((2, 3, 4)) + van = cheb.chebvander3d(x1, x2, x3, [1, 2, 3]) + tgt = cheb.chebval3d(x1, x2, x3, c) + res = np.dot(van, c.flat) + assert_almost_equal(res, tgt) + + # check shape + van = cheb.chebvander3d([x1], [x2], [x3], [1, 2, 3]) + assert_(van.shape == (1, 5, 24)) + + +class TestFitting: + + def test_chebfit(self): + def f(x): + return x*(x - 1)*(x - 2) + + def f2(x): + return x**4 + x**2 + 1 + + # Test exceptions + assert_raises(ValueError, cheb.chebfit, [1], [1], -1) + assert_raises(TypeError, cheb.chebfit, [[1]], [1], 0) + assert_raises(TypeError, cheb.chebfit, [], [1], 0) + assert_raises(TypeError, cheb.chebfit, [1], [[[1]]], 0) + assert_raises(TypeError, cheb.chebfit, [1, 2], [1], 0) + assert_raises(TypeError, cheb.chebfit, [1], [1, 2], 0) + assert_raises(TypeError, cheb.chebfit, [1], [1], 0, w=[[1]]) + assert_raises(TypeError, cheb.chebfit, [1], [1], 0, w=[1, 1]) + assert_raises(ValueError, cheb.chebfit, [1], [1], [-1,]) + assert_raises(ValueError, cheb.chebfit, [1], [1], [2, -1, 6]) + assert_raises(TypeError, cheb.chebfit, [1], [1], []) + + # Test fit + x = np.linspace(0, 2) + y = f(x) + # + coef3 = cheb.chebfit(x, y, 3) + assert_equal(len(coef3), 4) + assert_almost_equal(cheb.chebval(x, coef3), y) + coef3 = cheb.chebfit(x, y, [0, 1, 2, 3]) + assert_equal(len(coef3), 4) + assert_almost_equal(cheb.chebval(x, coef3), y) + # + coef4 = cheb.chebfit(x, y, 4) + assert_equal(len(coef4), 5) + assert_almost_equal(cheb.chebval(x, coef4), y) + coef4 = cheb.chebfit(x, y, [0, 1, 2, 3, 4]) + assert_equal(len(coef4), 5) + assert_almost_equal(cheb.chebval(x, coef4), y) + # check things still work if deg is not in strict increasing + coef4 = cheb.chebfit(x, y, [2, 3, 4, 1, 0]) + assert_equal(len(coef4), 5) + assert_almost_equal(cheb.chebval(x, coef4), y) + # + coef2d = cheb.chebfit(x, np.array([y, y]).T, 3) + assert_almost_equal(coef2d, np.array([coef3, coef3]).T) + coef2d = cheb.chebfit(x, np.array([y, y]).T, [0, 1, 2, 3]) + assert_almost_equal(coef2d, np.array([coef3, coef3]).T) + # test weighting + w = np.zeros_like(x) + yw = y.copy() + w[1::2] = 1 + y[0::2] = 0 + wcoef3 = cheb.chebfit(x, yw, 3, w=w) + assert_almost_equal(wcoef3, coef3) + wcoef3 = cheb.chebfit(x, yw, [0, 1, 2, 3], w=w) + assert_almost_equal(wcoef3, coef3) + # + wcoef2d = cheb.chebfit(x, np.array([yw, yw]).T, 3, w=w) + assert_almost_equal(wcoef2d, np.array([coef3, coef3]).T) + wcoef2d = cheb.chebfit(x, np.array([yw, yw]).T, [0, 1, 2, 3], w=w) + assert_almost_equal(wcoef2d, np.array([coef3, coef3]).T) + # test scaling with complex values x points whose square + # is zero when summed. + x = [1, 1j, -1, -1j] + assert_almost_equal(cheb.chebfit(x, x, 1), [0, 1]) + assert_almost_equal(cheb.chebfit(x, x, [0, 1]), [0, 1]) + # test fitting only even polynomials + x = np.linspace(-1, 1) + y = f2(x) + coef1 = cheb.chebfit(x, y, 4) + assert_almost_equal(cheb.chebval(x, coef1), y) + coef2 = cheb.chebfit(x, y, [0, 2, 4]) + assert_almost_equal(cheb.chebval(x, coef2), y) + assert_almost_equal(coef1, coef2) + + +class TestInterpolate: + + def f(self, x): + return x * (x - 1) * (x - 2) + + def test_raises(self): + assert_raises(ValueError, cheb.chebinterpolate, self.f, -1) + assert_raises(TypeError, cheb.chebinterpolate, self.f, 10.) + + def test_dimensions(self): + for deg in range(1, 5): + assert_(cheb.chebinterpolate(self.f, deg).shape == (deg + 1,)) + + def test_approximation(self): + + def powx(x, p): + return x**p + + x = np.linspace(-1, 1, 10) + for deg in range(0, 10): + for p in range(0, deg + 1): + c = cheb.chebinterpolate(powx, deg, (p,)) + assert_almost_equal(cheb.chebval(x, c), powx(x, p), decimal=12) + + +class TestCompanion: + + def test_raises(self): + assert_raises(ValueError, cheb.chebcompanion, []) + assert_raises(ValueError, cheb.chebcompanion, [1]) + + def test_dimensions(self): + for i in range(1, 5): + coef = [0]*i + [1] + assert_(cheb.chebcompanion(coef).shape == (i, i)) + + def test_linear_root(self): + assert_(cheb.chebcompanion([1, 2])[0, 0] == -.5) + + +class TestGauss: + + def test_100(self): + x, w = cheb.chebgauss(100) + + # test orthogonality. Note that the results need to be normalized, + # otherwise the huge values that can arise from fast growing + # functions like Laguerre can be very confusing. + v = cheb.chebvander(x, 99) + vv = np.dot(v.T * w, v) + vd = 1/np.sqrt(vv.diagonal()) + vv = vd[:, None] * vv * vd + assert_almost_equal(vv, np.eye(100)) + + # check that the integral of 1 is correct + tgt = np.pi + assert_almost_equal(w.sum(), tgt) + + +class TestMisc: + + def test_chebfromroots(self): + res = cheb.chebfromroots([]) + assert_almost_equal(trim(res), [1]) + for i in range(1, 5): + roots = np.cos(np.linspace(-np.pi, 0, 2*i + 1)[1::2]) + tgt = [0]*i + [1] + res = cheb.chebfromroots(roots)*2**(i-1) + assert_almost_equal(trim(res), trim(tgt)) + + def test_chebroots(self): + assert_almost_equal(cheb.chebroots([1]), []) + assert_almost_equal(cheb.chebroots([1, 2]), [-.5]) + for i in range(2, 5): + tgt = np.linspace(-1, 1, i) + res = cheb.chebroots(cheb.chebfromroots(tgt)) + assert_almost_equal(trim(res), trim(tgt)) + + def test_chebtrim(self): + coef = [2, -1, 1, 0] + + # Test exceptions + assert_raises(ValueError, cheb.chebtrim, coef, -1) + + # Test results + assert_equal(cheb.chebtrim(coef), coef[:-1]) + assert_equal(cheb.chebtrim(coef, 1), coef[:-3]) + assert_equal(cheb.chebtrim(coef, 2), [0]) + + def test_chebline(self): + assert_equal(cheb.chebline(3, 4), [3, 4]) + + def test_cheb2poly(self): + for i in range(10): + assert_almost_equal(cheb.cheb2poly([0]*i + [1]), Tlist[i]) + + def test_poly2cheb(self): + for i in range(10): + assert_almost_equal(cheb.poly2cheb(Tlist[i]), [0]*i + [1]) + + def test_weight(self): + x = np.linspace(-1, 1, 11)[1:-1] + tgt = 1./(np.sqrt(1 + x) * np.sqrt(1 - x)) + res = cheb.chebweight(x) + assert_almost_equal(res, tgt) + + def test_chebpts1(self): + #test exceptions + assert_raises(ValueError, cheb.chebpts1, 1.5) + assert_raises(ValueError, cheb.chebpts1, 0) + + #test points + tgt = [0] + assert_almost_equal(cheb.chebpts1(1), tgt) + tgt = [-0.70710678118654746, 0.70710678118654746] + assert_almost_equal(cheb.chebpts1(2), tgt) + tgt = [-0.86602540378443871, 0, 0.86602540378443871] + assert_almost_equal(cheb.chebpts1(3), tgt) + tgt = [-0.9238795325, -0.3826834323, 0.3826834323, 0.9238795325] + assert_almost_equal(cheb.chebpts1(4), tgt) + + def test_chebpts2(self): + #test exceptions + assert_raises(ValueError, cheb.chebpts2, 1.5) + assert_raises(ValueError, cheb.chebpts2, 1) + + #test points + tgt = [-1, 1] + assert_almost_equal(cheb.chebpts2(2), tgt) + tgt = [-1, 0, 1] + assert_almost_equal(cheb.chebpts2(3), tgt) + tgt = [-1, -0.5, .5, 1] + assert_almost_equal(cheb.chebpts2(4), tgt) + tgt = [-1.0, -0.707106781187, 0, 0.707106781187, 1.0] + assert_almost_equal(cheb.chebpts2(5), tgt) diff --git a/deepseekvl2/lib/python3.10/site-packages/numpy/polynomial/tests/test_classes.py b/deepseekvl2/lib/python3.10/site-packages/numpy/polynomial/tests/test_classes.py new file mode 100644 index 0000000000000000000000000000000000000000..6322062f29ece2f52754ac7aedf2591b3a983709 --- /dev/null +++ b/deepseekvl2/lib/python3.10/site-packages/numpy/polynomial/tests/test_classes.py @@ -0,0 +1,600 @@ +"""Test inter-conversion of different polynomial classes. + +This tests the convert and cast methods of all the polynomial classes. + +""" +import operator as op +from numbers import Number + +import pytest +import numpy as np +from numpy.polynomial import ( + Polynomial, Legendre, Chebyshev, Laguerre, Hermite, HermiteE) +from numpy.testing import ( + assert_almost_equal, assert_raises, assert_equal, assert_, + ) +from numpy.polynomial.polyutils import RankWarning + +# +# fixtures +# + +classes = ( + Polynomial, Legendre, Chebyshev, Laguerre, + Hermite, HermiteE + ) +classids = tuple(cls.__name__ for cls in classes) + +@pytest.fixture(params=classes, ids=classids) +def Poly(request): + return request.param + +# +# helper functions +# +random = np.random.random + + +def assert_poly_almost_equal(p1, p2, msg=""): + try: + assert_(np.all(p1.domain == p2.domain)) + assert_(np.all(p1.window == p2.window)) + assert_almost_equal(p1.coef, p2.coef) + except AssertionError: + msg = f"Result: {p1}\nTarget: {p2}" + raise AssertionError(msg) + + +# +# Test conversion methods that depend on combinations of two classes. +# + +Poly1 = Poly +Poly2 = Poly + + +def test_conversion(Poly1, Poly2): + x = np.linspace(0, 1, 10) + coef = random((3,)) + + d1 = Poly1.domain + random((2,))*.25 + w1 = Poly1.window + random((2,))*.25 + p1 = Poly1(coef, domain=d1, window=w1) + + d2 = Poly2.domain + random((2,))*.25 + w2 = Poly2.window + random((2,))*.25 + p2 = p1.convert(kind=Poly2, domain=d2, window=w2) + + assert_almost_equal(p2.domain, d2) + assert_almost_equal(p2.window, w2) + assert_almost_equal(p2(x), p1(x)) + + +def test_cast(Poly1, Poly2): + x = np.linspace(0, 1, 10) + coef = random((3,)) + + d1 = Poly1.domain + random((2,))*.25 + w1 = Poly1.window + random((2,))*.25 + p1 = Poly1(coef, domain=d1, window=w1) + + d2 = Poly2.domain + random((2,))*.25 + w2 = Poly2.window + random((2,))*.25 + p2 = Poly2.cast(p1, domain=d2, window=w2) + + assert_almost_equal(p2.domain, d2) + assert_almost_equal(p2.window, w2) + assert_almost_equal(p2(x), p1(x)) + + +# +# test methods that depend on one class +# + + +def test_identity(Poly): + d = Poly.domain + random((2,))*.25 + w = Poly.window + random((2,))*.25 + x = np.linspace(d[0], d[1], 11) + p = Poly.identity(domain=d, window=w) + assert_equal(p.domain, d) + assert_equal(p.window, w) + assert_almost_equal(p(x), x) + + +def test_basis(Poly): + d = Poly.domain + random((2,))*.25 + w = Poly.window + random((2,))*.25 + p = Poly.basis(5, domain=d, window=w) + assert_equal(p.domain, d) + assert_equal(p.window, w) + assert_equal(p.coef, [0]*5 + [1]) + + +def test_fromroots(Poly): + # check that requested roots are zeros of a polynomial + # of correct degree, domain, and window. + d = Poly.domain + random((2,))*.25 + w = Poly.window + random((2,))*.25 + r = random((5,)) + p1 = Poly.fromroots(r, domain=d, window=w) + assert_equal(p1.degree(), len(r)) + assert_equal(p1.domain, d) + assert_equal(p1.window, w) + assert_almost_equal(p1(r), 0) + + # check that polynomial is monic + pdom = Polynomial.domain + pwin = Polynomial.window + p2 = Polynomial.cast(p1, domain=pdom, window=pwin) + assert_almost_equal(p2.coef[-1], 1) + + +def test_bad_conditioned_fit(Poly): + + x = [0., 0., 1.] + y = [1., 2., 3.] + + # check RankWarning is raised + with pytest.warns(RankWarning) as record: + Poly.fit(x, y, 2) + assert record[0].message.args[0] == "The fit may be poorly conditioned" + + +def test_fit(Poly): + + def f(x): + return x*(x - 1)*(x - 2) + x = np.linspace(0, 3) + y = f(x) + + # check default value of domain and window + p = Poly.fit(x, y, 3) + assert_almost_equal(p.domain, [0, 3]) + assert_almost_equal(p(x), y) + assert_equal(p.degree(), 3) + + # check with given domains and window + d = Poly.domain + random((2,))*.25 + w = Poly.window + random((2,))*.25 + p = Poly.fit(x, y, 3, domain=d, window=w) + assert_almost_equal(p(x), y) + assert_almost_equal(p.domain, d) + assert_almost_equal(p.window, w) + p = Poly.fit(x, y, [0, 1, 2, 3], domain=d, window=w) + assert_almost_equal(p(x), y) + assert_almost_equal(p.domain, d) + assert_almost_equal(p.window, w) + + # check with class domain default + p = Poly.fit(x, y, 3, []) + assert_equal(p.domain, Poly.domain) + assert_equal(p.window, Poly.window) + p = Poly.fit(x, y, [0, 1, 2, 3], []) + assert_equal(p.domain, Poly.domain) + assert_equal(p.window, Poly.window) + + # check that fit accepts weights. + w = np.zeros_like(x) + z = y + random(y.shape)*.25 + w[::2] = 1 + p1 = Poly.fit(x[::2], z[::2], 3) + p2 = Poly.fit(x, z, 3, w=w) + p3 = Poly.fit(x, z, [0, 1, 2, 3], w=w) + assert_almost_equal(p1(x), p2(x)) + assert_almost_equal(p2(x), p3(x)) + + +def test_equal(Poly): + p1 = Poly([1, 2, 3], domain=[0, 1], window=[2, 3]) + p2 = Poly([1, 1, 1], domain=[0, 1], window=[2, 3]) + p3 = Poly([1, 2, 3], domain=[1, 2], window=[2, 3]) + p4 = Poly([1, 2, 3], domain=[0, 1], window=[1, 2]) + assert_(p1 == p1) + assert_(not p1 == p2) + assert_(not p1 == p3) + assert_(not p1 == p4) + + +def test_not_equal(Poly): + p1 = Poly([1, 2, 3], domain=[0, 1], window=[2, 3]) + p2 = Poly([1, 1, 1], domain=[0, 1], window=[2, 3]) + p3 = Poly([1, 2, 3], domain=[1, 2], window=[2, 3]) + p4 = Poly([1, 2, 3], domain=[0, 1], window=[1, 2]) + assert_(not p1 != p1) + assert_(p1 != p2) + assert_(p1 != p3) + assert_(p1 != p4) + + +def test_add(Poly): + # This checks commutation, not numerical correctness + c1 = list(random((4,)) + .5) + c2 = list(random((3,)) + .5) + p1 = Poly(c1) + p2 = Poly(c2) + p3 = p1 + p2 + assert_poly_almost_equal(p2 + p1, p3) + assert_poly_almost_equal(p1 + c2, p3) + assert_poly_almost_equal(c2 + p1, p3) + assert_poly_almost_equal(p1 + tuple(c2), p3) + assert_poly_almost_equal(tuple(c2) + p1, p3) + assert_poly_almost_equal(p1 + np.array(c2), p3) + assert_poly_almost_equal(np.array(c2) + p1, p3) + assert_raises(TypeError, op.add, p1, Poly([0], domain=Poly.domain + 1)) + assert_raises(TypeError, op.add, p1, Poly([0], window=Poly.window + 1)) + if Poly is Polynomial: + assert_raises(TypeError, op.add, p1, Chebyshev([0])) + else: + assert_raises(TypeError, op.add, p1, Polynomial([0])) + + +def test_sub(Poly): + # This checks commutation, not numerical correctness + c1 = list(random((4,)) + .5) + c2 = list(random((3,)) + .5) + p1 = Poly(c1) + p2 = Poly(c2) + p3 = p1 - p2 + assert_poly_almost_equal(p2 - p1, -p3) + assert_poly_almost_equal(p1 - c2, p3) + assert_poly_almost_equal(c2 - p1, -p3) + assert_poly_almost_equal(p1 - tuple(c2), p3) + assert_poly_almost_equal(tuple(c2) - p1, -p3) + assert_poly_almost_equal(p1 - np.array(c2), p3) + assert_poly_almost_equal(np.array(c2) - p1, -p3) + assert_raises(TypeError, op.sub, p1, Poly([0], domain=Poly.domain + 1)) + assert_raises(TypeError, op.sub, p1, Poly([0], window=Poly.window + 1)) + if Poly is Polynomial: + assert_raises(TypeError, op.sub, p1, Chebyshev([0])) + else: + assert_raises(TypeError, op.sub, p1, Polynomial([0])) + + +def test_mul(Poly): + c1 = list(random((4,)) + .5) + c2 = list(random((3,)) + .5) + p1 = Poly(c1) + p2 = Poly(c2) + p3 = p1 * p2 + assert_poly_almost_equal(p2 * p1, p3) + assert_poly_almost_equal(p1 * c2, p3) + assert_poly_almost_equal(c2 * p1, p3) + assert_poly_almost_equal(p1 * tuple(c2), p3) + assert_poly_almost_equal(tuple(c2) * p1, p3) + assert_poly_almost_equal(p1 * np.array(c2), p3) + assert_poly_almost_equal(np.array(c2) * p1, p3) + assert_poly_almost_equal(p1 * 2, p1 * Poly([2])) + assert_poly_almost_equal(2 * p1, p1 * Poly([2])) + assert_raises(TypeError, op.mul, p1, Poly([0], domain=Poly.domain + 1)) + assert_raises(TypeError, op.mul, p1, Poly([0], window=Poly.window + 1)) + if Poly is Polynomial: + assert_raises(TypeError, op.mul, p1, Chebyshev([0])) + else: + assert_raises(TypeError, op.mul, p1, Polynomial([0])) + + +def test_floordiv(Poly): + c1 = list(random((4,)) + .5) + c2 = list(random((3,)) + .5) + c3 = list(random((2,)) + .5) + p1 = Poly(c1) + p2 = Poly(c2) + p3 = Poly(c3) + p4 = p1 * p2 + p3 + c4 = list(p4.coef) + assert_poly_almost_equal(p4 // p2, p1) + assert_poly_almost_equal(p4 // c2, p1) + assert_poly_almost_equal(c4 // p2, p1) + assert_poly_almost_equal(p4 // tuple(c2), p1) + assert_poly_almost_equal(tuple(c4) // p2, p1) + assert_poly_almost_equal(p4 // np.array(c2), p1) + assert_poly_almost_equal(np.array(c4) // p2, p1) + assert_poly_almost_equal(2 // p2, Poly([0])) + assert_poly_almost_equal(p2 // 2, 0.5*p2) + assert_raises( + TypeError, op.floordiv, p1, Poly([0], domain=Poly.domain + 1)) + assert_raises( + TypeError, op.floordiv, p1, Poly([0], window=Poly.window + 1)) + if Poly is Polynomial: + assert_raises(TypeError, op.floordiv, p1, Chebyshev([0])) + else: + assert_raises(TypeError, op.floordiv, p1, Polynomial([0])) + + +def test_truediv(Poly): + # true division is valid only if the denominator is a Number and + # not a python bool. + p1 = Poly([1,2,3]) + p2 = p1 * 5 + + for stype in np.ScalarType: + if not issubclass(stype, Number) or issubclass(stype, bool): + continue + s = stype(5) + assert_poly_almost_equal(op.truediv(p2, s), p1) + assert_raises(TypeError, op.truediv, s, p2) + for stype in (int, float): + s = stype(5) + assert_poly_almost_equal(op.truediv(p2, s), p1) + assert_raises(TypeError, op.truediv, s, p2) + for stype in [complex]: + s = stype(5, 0) + assert_poly_almost_equal(op.truediv(p2, s), p1) + assert_raises(TypeError, op.truediv, s, p2) + for s in [tuple(), list(), dict(), bool(), np.array([1])]: + assert_raises(TypeError, op.truediv, p2, s) + assert_raises(TypeError, op.truediv, s, p2) + for ptype in classes: + assert_raises(TypeError, op.truediv, p2, ptype(1)) + + +def test_mod(Poly): + # This checks commutation, not numerical correctness + c1 = list(random((4,)) + .5) + c2 = list(random((3,)) + .5) + c3 = list(random((2,)) + .5) + p1 = Poly(c1) + p2 = Poly(c2) + p3 = Poly(c3) + p4 = p1 * p2 + p3 + c4 = list(p4.coef) + assert_poly_almost_equal(p4 % p2, p3) + assert_poly_almost_equal(p4 % c2, p3) + assert_poly_almost_equal(c4 % p2, p3) + assert_poly_almost_equal(p4 % tuple(c2), p3) + assert_poly_almost_equal(tuple(c4) % p2, p3) + assert_poly_almost_equal(p4 % np.array(c2), p3) + assert_poly_almost_equal(np.array(c4) % p2, p3) + assert_poly_almost_equal(2 % p2, Poly([2])) + assert_poly_almost_equal(p2 % 2, Poly([0])) + assert_raises(TypeError, op.mod, p1, Poly([0], domain=Poly.domain + 1)) + assert_raises(TypeError, op.mod, p1, Poly([0], window=Poly.window + 1)) + if Poly is Polynomial: + assert_raises(TypeError, op.mod, p1, Chebyshev([0])) + else: + assert_raises(TypeError, op.mod, p1, Polynomial([0])) + + +def test_divmod(Poly): + # This checks commutation, not numerical correctness + c1 = list(random((4,)) + .5) + c2 = list(random((3,)) + .5) + c3 = list(random((2,)) + .5) + p1 = Poly(c1) + p2 = Poly(c2) + p3 = Poly(c3) + p4 = p1 * p2 + p3 + c4 = list(p4.coef) + quo, rem = divmod(p4, p2) + assert_poly_almost_equal(quo, p1) + assert_poly_almost_equal(rem, p3) + quo, rem = divmod(p4, c2) + assert_poly_almost_equal(quo, p1) + assert_poly_almost_equal(rem, p3) + quo, rem = divmod(c4, p2) + assert_poly_almost_equal(quo, p1) + assert_poly_almost_equal(rem, p3) + quo, rem = divmod(p4, tuple(c2)) + assert_poly_almost_equal(quo, p1) + assert_poly_almost_equal(rem, p3) + quo, rem = divmod(tuple(c4), p2) + assert_poly_almost_equal(quo, p1) + assert_poly_almost_equal(rem, p3) + quo, rem = divmod(p4, np.array(c2)) + assert_poly_almost_equal(quo, p1) + assert_poly_almost_equal(rem, p3) + quo, rem = divmod(np.array(c4), p2) + assert_poly_almost_equal(quo, p1) + assert_poly_almost_equal(rem, p3) + quo, rem = divmod(p2, 2) + assert_poly_almost_equal(quo, 0.5*p2) + assert_poly_almost_equal(rem, Poly([0])) + quo, rem = divmod(2, p2) + assert_poly_almost_equal(quo, Poly([0])) + assert_poly_almost_equal(rem, Poly([2])) + assert_raises(TypeError, divmod, p1, Poly([0], domain=Poly.domain + 1)) + assert_raises(TypeError, divmod, p1, Poly([0], window=Poly.window + 1)) + if Poly is Polynomial: + assert_raises(TypeError, divmod, p1, Chebyshev([0])) + else: + assert_raises(TypeError, divmod, p1, Polynomial([0])) + + +def test_roots(Poly): + d = Poly.domain * 1.25 + .25 + w = Poly.window + tgt = np.linspace(d[0], d[1], 5) + res = np.sort(Poly.fromroots(tgt, domain=d, window=w).roots()) + assert_almost_equal(res, tgt) + # default domain and window + res = np.sort(Poly.fromroots(tgt).roots()) + assert_almost_equal(res, tgt) + + +def test_degree(Poly): + p = Poly.basis(5) + assert_equal(p.degree(), 5) + + +def test_copy(Poly): + p1 = Poly.basis(5) + p2 = p1.copy() + assert_(p1 == p2) + assert_(p1 is not p2) + assert_(p1.coef is not p2.coef) + assert_(p1.domain is not p2.domain) + assert_(p1.window is not p2.window) + + +def test_integ(Poly): + P = Polynomial + # Check defaults + p0 = Poly.cast(P([1*2, 2*3, 3*4])) + p1 = P.cast(p0.integ()) + p2 = P.cast(p0.integ(2)) + assert_poly_almost_equal(p1, P([0, 2, 3, 4])) + assert_poly_almost_equal(p2, P([0, 0, 1, 1, 1])) + # Check with k + p0 = Poly.cast(P([1*2, 2*3, 3*4])) + p1 = P.cast(p0.integ(k=1)) + p2 = P.cast(p0.integ(2, k=[1, 1])) + assert_poly_almost_equal(p1, P([1, 2, 3, 4])) + assert_poly_almost_equal(p2, P([1, 1, 1, 1, 1])) + # Check with lbnd + p0 = Poly.cast(P([1*2, 2*3, 3*4])) + p1 = P.cast(p0.integ(lbnd=1)) + p2 = P.cast(p0.integ(2, lbnd=1)) + assert_poly_almost_equal(p1, P([-9, 2, 3, 4])) + assert_poly_almost_equal(p2, P([6, -9, 1, 1, 1])) + # Check scaling + d = 2*Poly.domain + p0 = Poly.cast(P([1*2, 2*3, 3*4]), domain=d) + p1 = P.cast(p0.integ()) + p2 = P.cast(p0.integ(2)) + assert_poly_almost_equal(p1, P([0, 2, 3, 4])) + assert_poly_almost_equal(p2, P([0, 0, 1, 1, 1])) + + +def test_deriv(Poly): + # Check that the derivative is the inverse of integration. It is + # assumes that the integration has been checked elsewhere. + d = Poly.domain + random((2,))*.25 + w = Poly.window + random((2,))*.25 + p1 = Poly([1, 2, 3], domain=d, window=w) + p2 = p1.integ(2, k=[1, 2]) + p3 = p1.integ(1, k=[1]) + assert_almost_equal(p2.deriv(1).coef, p3.coef) + assert_almost_equal(p2.deriv(2).coef, p1.coef) + # default domain and window + p1 = Poly([1, 2, 3]) + p2 = p1.integ(2, k=[1, 2]) + p3 = p1.integ(1, k=[1]) + assert_almost_equal(p2.deriv(1).coef, p3.coef) + assert_almost_equal(p2.deriv(2).coef, p1.coef) + + +def test_linspace(Poly): + d = Poly.domain + random((2,))*.25 + w = Poly.window + random((2,))*.25 + p = Poly([1, 2, 3], domain=d, window=w) + # check default domain + xtgt = np.linspace(d[0], d[1], 20) + ytgt = p(xtgt) + xres, yres = p.linspace(20) + assert_almost_equal(xres, xtgt) + assert_almost_equal(yres, ytgt) + # check specified domain + xtgt = np.linspace(0, 2, 20) + ytgt = p(xtgt) + xres, yres = p.linspace(20, domain=[0, 2]) + assert_almost_equal(xres, xtgt) + assert_almost_equal(yres, ytgt) + + +def test_pow(Poly): + d = Poly.domain + random((2,))*.25 + w = Poly.window + random((2,))*.25 + tgt = Poly([1], domain=d, window=w) + tst = Poly([1, 2, 3], domain=d, window=w) + for i in range(5): + assert_poly_almost_equal(tst**i, tgt) + tgt = tgt * tst + # default domain and window + tgt = Poly([1]) + tst = Poly([1, 2, 3]) + for i in range(5): + assert_poly_almost_equal(tst**i, tgt) + tgt = tgt * tst + # check error for invalid powers + assert_raises(ValueError, op.pow, tgt, 1.5) + assert_raises(ValueError, op.pow, tgt, -1) + + +def test_call(Poly): + P = Polynomial + d = Poly.domain + x = np.linspace(d[0], d[1], 11) + + # Check defaults + p = Poly.cast(P([1, 2, 3])) + tgt = 1 + x*(2 + 3*x) + res = p(x) + assert_almost_equal(res, tgt) + + +def test_cutdeg(Poly): + p = Poly([1, 2, 3]) + assert_raises(ValueError, p.cutdeg, .5) + assert_raises(ValueError, p.cutdeg, -1) + assert_equal(len(p.cutdeg(3)), 3) + assert_equal(len(p.cutdeg(2)), 3) + assert_equal(len(p.cutdeg(1)), 2) + assert_equal(len(p.cutdeg(0)), 1) + + +def test_truncate(Poly): + p = Poly([1, 2, 3]) + assert_raises(ValueError, p.truncate, .5) + assert_raises(ValueError, p.truncate, 0) + assert_equal(len(p.truncate(4)), 3) + assert_equal(len(p.truncate(3)), 3) + assert_equal(len(p.truncate(2)), 2) + assert_equal(len(p.truncate(1)), 1) + + +def test_trim(Poly): + c = [1, 1e-6, 1e-12, 0] + p = Poly(c) + assert_equal(p.trim().coef, c[:3]) + assert_equal(p.trim(1e-10).coef, c[:2]) + assert_equal(p.trim(1e-5).coef, c[:1]) + + +def test_mapparms(Poly): + # check with defaults. Should be identity. + d = Poly.domain + w = Poly.window + p = Poly([1], domain=d, window=w) + assert_almost_equal([0, 1], p.mapparms()) + # + w = 2*d + 1 + p = Poly([1], domain=d, window=w) + assert_almost_equal([1, 2], p.mapparms()) + + +def test_ufunc_override(Poly): + p = Poly([1, 2, 3]) + x = np.ones(3) + assert_raises(TypeError, np.add, p, x) + assert_raises(TypeError, np.add, x, p) + + +# +# Test class method that only exists for some classes +# + + +class TestInterpolate: + + def f(self, x): + return x * (x - 1) * (x - 2) + + def test_raises(self): + assert_raises(ValueError, Chebyshev.interpolate, self.f, -1) + assert_raises(TypeError, Chebyshev.interpolate, self.f, 10.) + + def test_dimensions(self): + for deg in range(1, 5): + assert_(Chebyshev.interpolate(self.f, deg).degree() == deg) + + def test_approximation(self): + + def powx(x, p): + return x**p + + x = np.linspace(0, 2, 10) + for deg in range(0, 10): + for t in range(0, deg + 1): + p = Chebyshev.interpolate(powx, deg, domain=[0, 2], args=(t,)) + assert_almost_equal(p(x), powx(x, t), decimal=11) diff --git a/deepseekvl2/lib/python3.10/site-packages/numpy/polynomial/tests/test_hermite.py b/deepseekvl2/lib/python3.10/site-packages/numpy/polynomial/tests/test_hermite.py new file mode 100644 index 0000000000000000000000000000000000000000..53ee0844e3c58456807bfd7828bdb9cf58f8ed76 --- /dev/null +++ b/deepseekvl2/lib/python3.10/site-packages/numpy/polynomial/tests/test_hermite.py @@ -0,0 +1,555 @@ +"""Tests for hermite module. + +""" +from functools import reduce + +import numpy as np +import numpy.polynomial.hermite as herm +from numpy.polynomial.polynomial import polyval +from numpy.testing import ( + assert_almost_equal, assert_raises, assert_equal, assert_, + ) + +H0 = np.array([1]) +H1 = np.array([0, 2]) +H2 = np.array([-2, 0, 4]) +H3 = np.array([0, -12, 0, 8]) +H4 = np.array([12, 0, -48, 0, 16]) +H5 = np.array([0, 120, 0, -160, 0, 32]) +H6 = np.array([-120, 0, 720, 0, -480, 0, 64]) +H7 = np.array([0, -1680, 0, 3360, 0, -1344, 0, 128]) +H8 = np.array([1680, 0, -13440, 0, 13440, 0, -3584, 0, 256]) +H9 = np.array([0, 30240, 0, -80640, 0, 48384, 0, -9216, 0, 512]) + +Hlist = [H0, H1, H2, H3, H4, H5, H6, H7, H8, H9] + + +def trim(x): + return herm.hermtrim(x, tol=1e-6) + + +class TestConstants: + + def test_hermdomain(self): + assert_equal(herm.hermdomain, [-1, 1]) + + def test_hermzero(self): + assert_equal(herm.hermzero, [0]) + + def test_hermone(self): + assert_equal(herm.hermone, [1]) + + def test_hermx(self): + assert_equal(herm.hermx, [0, .5]) + + +class TestArithmetic: + x = np.linspace(-3, 3, 100) + + def test_hermadd(self): + for i in range(5): + for j in range(5): + msg = f"At i={i}, j={j}" + tgt = np.zeros(max(i, j) + 1) + tgt[i] += 1 + tgt[j] += 1 + res = herm.hermadd([0]*i + [1], [0]*j + [1]) + assert_equal(trim(res), trim(tgt), err_msg=msg) + + def test_hermsub(self): + for i in range(5): + for j in range(5): + msg = f"At i={i}, j={j}" + tgt = np.zeros(max(i, j) + 1) + tgt[i] += 1 + tgt[j] -= 1 + res = herm.hermsub([0]*i + [1], [0]*j + [1]) + assert_equal(trim(res), trim(tgt), err_msg=msg) + + def test_hermmulx(self): + assert_equal(herm.hermmulx([0]), [0]) + assert_equal(herm.hermmulx([1]), [0, .5]) + for i in range(1, 5): + ser = [0]*i + [1] + tgt = [0]*(i - 1) + [i, 0, .5] + assert_equal(herm.hermmulx(ser), tgt) + + def test_hermmul(self): + # check values of result + for i in range(5): + pol1 = [0]*i + [1] + val1 = herm.hermval(self.x, pol1) + for j in range(5): + msg = f"At i={i}, j={j}" + pol2 = [0]*j + [1] + val2 = herm.hermval(self.x, pol2) + pol3 = herm.hermmul(pol1, pol2) + val3 = herm.hermval(self.x, pol3) + assert_(len(pol3) == i + j + 1, msg) + assert_almost_equal(val3, val1*val2, err_msg=msg) + + def test_hermdiv(self): + for i in range(5): + for j in range(5): + msg = f"At i={i}, j={j}" + ci = [0]*i + [1] + cj = [0]*j + [1] + tgt = herm.hermadd(ci, cj) + quo, rem = herm.hermdiv(tgt, ci) + res = herm.hermadd(herm.hermmul(quo, ci), rem) + assert_equal(trim(res), trim(tgt), err_msg=msg) + + def test_hermpow(self): + for i in range(5): + for j in range(5): + msg = f"At i={i}, j={j}" + c = np.arange(i + 1) + tgt = reduce(herm.hermmul, [c]*j, np.array([1])) + res = herm.hermpow(c, j) + assert_equal(trim(res), trim(tgt), err_msg=msg) + + +class TestEvaluation: + # coefficients of 1 + 2*x + 3*x**2 + c1d = np.array([2.5, 1., .75]) + c2d = np.einsum('i,j->ij', c1d, c1d) + c3d = np.einsum('i,j,k->ijk', c1d, c1d, c1d) + + # some random values in [-1, 1) + x = np.random.random((3, 5))*2 - 1 + y = polyval(x, [1., 2., 3.]) + + def test_hermval(self): + #check empty input + assert_equal(herm.hermval([], [1]).size, 0) + + #check normal input) + x = np.linspace(-1, 1) + y = [polyval(x, c) for c in Hlist] + for i in range(10): + msg = f"At i={i}" + tgt = y[i] + res = herm.hermval(x, [0]*i + [1]) + assert_almost_equal(res, tgt, err_msg=msg) + + #check that shape is preserved + for i in range(3): + dims = [2]*i + x = np.zeros(dims) + assert_equal(herm.hermval(x, [1]).shape, dims) + assert_equal(herm.hermval(x, [1, 0]).shape, dims) + assert_equal(herm.hermval(x, [1, 0, 0]).shape, dims) + + def test_hermval2d(self): + x1, x2, x3 = self.x + y1, y2, y3 = self.y + + #test exceptions + assert_raises(ValueError, herm.hermval2d, x1, x2[:2], self.c2d) + + #test values + tgt = y1*y2 + res = herm.hermval2d(x1, x2, self.c2d) + assert_almost_equal(res, tgt) + + #test shape + z = np.ones((2, 3)) + res = herm.hermval2d(z, z, self.c2d) + assert_(res.shape == (2, 3)) + + def test_hermval3d(self): + x1, x2, x3 = self.x + y1, y2, y3 = self.y + + #test exceptions + assert_raises(ValueError, herm.hermval3d, x1, x2, x3[:2], self.c3d) + + #test values + tgt = y1*y2*y3 + res = herm.hermval3d(x1, x2, x3, self.c3d) + assert_almost_equal(res, tgt) + + #test shape + z = np.ones((2, 3)) + res = herm.hermval3d(z, z, z, self.c3d) + assert_(res.shape == (2, 3)) + + def test_hermgrid2d(self): + x1, x2, x3 = self.x + y1, y2, y3 = self.y + + #test values + tgt = np.einsum('i,j->ij', y1, y2) + res = herm.hermgrid2d(x1, x2, self.c2d) + assert_almost_equal(res, tgt) + + #test shape + z = np.ones((2, 3)) + res = herm.hermgrid2d(z, z, self.c2d) + assert_(res.shape == (2, 3)*2) + + def test_hermgrid3d(self): + x1, x2, x3 = self.x + y1, y2, y3 = self.y + + #test values + tgt = np.einsum('i,j,k->ijk', y1, y2, y3) + res = herm.hermgrid3d(x1, x2, x3, self.c3d) + assert_almost_equal(res, tgt) + + #test shape + z = np.ones((2, 3)) + res = herm.hermgrid3d(z, z, z, self.c3d) + assert_(res.shape == (2, 3)*3) + + +class TestIntegral: + + def test_hermint(self): + # check exceptions + assert_raises(TypeError, herm.hermint, [0], .5) + assert_raises(ValueError, herm.hermint, [0], -1) + assert_raises(ValueError, herm.hermint, [0], 1, [0, 0]) + assert_raises(ValueError, herm.hermint, [0], lbnd=[0]) + assert_raises(ValueError, herm.hermint, [0], scl=[0]) + assert_raises(TypeError, herm.hermint, [0], axis=.5) + + # test integration of zero polynomial + for i in range(2, 5): + k = [0]*(i - 2) + [1] + res = herm.hermint([0], m=i, k=k) + assert_almost_equal(res, [0, .5]) + + # check single integration with integration constant + for i in range(5): + scl = i + 1 + pol = [0]*i + [1] + tgt = [i] + [0]*i + [1/scl] + hermpol = herm.poly2herm(pol) + hermint = herm.hermint(hermpol, m=1, k=[i]) + res = herm.herm2poly(hermint) + assert_almost_equal(trim(res), trim(tgt)) + + # check single integration with integration constant and lbnd + for i in range(5): + scl = i + 1 + pol = [0]*i + [1] + hermpol = herm.poly2herm(pol) + hermint = herm.hermint(hermpol, m=1, k=[i], lbnd=-1) + assert_almost_equal(herm.hermval(-1, hermint), i) + + # check single integration with integration constant and scaling + for i in range(5): + scl = i + 1 + pol = [0]*i + [1] + tgt = [i] + [0]*i + [2/scl] + hermpol = herm.poly2herm(pol) + hermint = herm.hermint(hermpol, m=1, k=[i], scl=2) + res = herm.herm2poly(hermint) + assert_almost_equal(trim(res), trim(tgt)) + + # check multiple integrations with default k + for i in range(5): + for j in range(2, 5): + pol = [0]*i + [1] + tgt = pol[:] + for k in range(j): + tgt = herm.hermint(tgt, m=1) + res = herm.hermint(pol, m=j) + assert_almost_equal(trim(res), trim(tgt)) + + # check multiple integrations with defined k + for i in range(5): + for j in range(2, 5): + pol = [0]*i + [1] + tgt = pol[:] + for k in range(j): + tgt = herm.hermint(tgt, m=1, k=[k]) + res = herm.hermint(pol, m=j, k=list(range(j))) + assert_almost_equal(trim(res), trim(tgt)) + + # check multiple integrations with lbnd + for i in range(5): + for j in range(2, 5): + pol = [0]*i + [1] + tgt = pol[:] + for k in range(j): + tgt = herm.hermint(tgt, m=1, k=[k], lbnd=-1) + res = herm.hermint(pol, m=j, k=list(range(j)), lbnd=-1) + assert_almost_equal(trim(res), trim(tgt)) + + # check multiple integrations with scaling + for i in range(5): + for j in range(2, 5): + pol = [0]*i + [1] + tgt = pol[:] + for k in range(j): + tgt = herm.hermint(tgt, m=1, k=[k], scl=2) + res = herm.hermint(pol, m=j, k=list(range(j)), scl=2) + assert_almost_equal(trim(res), trim(tgt)) + + def test_hermint_axis(self): + # check that axis keyword works + c2d = np.random.random((3, 4)) + + tgt = np.vstack([herm.hermint(c) for c in c2d.T]).T + res = herm.hermint(c2d, axis=0) + assert_almost_equal(res, tgt) + + tgt = np.vstack([herm.hermint(c) for c in c2d]) + res = herm.hermint(c2d, axis=1) + assert_almost_equal(res, tgt) + + tgt = np.vstack([herm.hermint(c, k=3) for c in c2d]) + res = herm.hermint(c2d, k=3, axis=1) + assert_almost_equal(res, tgt) + + +class TestDerivative: + + def test_hermder(self): + # check exceptions + assert_raises(TypeError, herm.hermder, [0], .5) + assert_raises(ValueError, herm.hermder, [0], -1) + + # check that zeroth derivative does nothing + for i in range(5): + tgt = [0]*i + [1] + res = herm.hermder(tgt, m=0) + assert_equal(trim(res), trim(tgt)) + + # check that derivation is the inverse of integration + for i in range(5): + for j in range(2, 5): + tgt = [0]*i + [1] + res = herm.hermder(herm.hermint(tgt, m=j), m=j) + assert_almost_equal(trim(res), trim(tgt)) + + # check derivation with scaling + for i in range(5): + for j in range(2, 5): + tgt = [0]*i + [1] + res = herm.hermder(herm.hermint(tgt, m=j, scl=2), m=j, scl=.5) + assert_almost_equal(trim(res), trim(tgt)) + + def test_hermder_axis(self): + # check that axis keyword works + c2d = np.random.random((3, 4)) + + tgt = np.vstack([herm.hermder(c) for c in c2d.T]).T + res = herm.hermder(c2d, axis=0) + assert_almost_equal(res, tgt) + + tgt = np.vstack([herm.hermder(c) for c in c2d]) + res = herm.hermder(c2d, axis=1) + assert_almost_equal(res, tgt) + + +class TestVander: + # some random values in [-1, 1) + x = np.random.random((3, 5))*2 - 1 + + def test_hermvander(self): + # check for 1d x + x = np.arange(3) + v = herm.hermvander(x, 3) + assert_(v.shape == (3, 4)) + for i in range(4): + coef = [0]*i + [1] + assert_almost_equal(v[..., i], herm.hermval(x, coef)) + + # check for 2d x + x = np.array([[1, 2], [3, 4], [5, 6]]) + v = herm.hermvander(x, 3) + assert_(v.shape == (3, 2, 4)) + for i in range(4): + coef = [0]*i + [1] + assert_almost_equal(v[..., i], herm.hermval(x, coef)) + + def test_hermvander2d(self): + # also tests hermval2d for non-square coefficient array + x1, x2, x3 = self.x + c = np.random.random((2, 3)) + van = herm.hermvander2d(x1, x2, [1, 2]) + tgt = herm.hermval2d(x1, x2, c) + res = np.dot(van, c.flat) + assert_almost_equal(res, tgt) + + # check shape + van = herm.hermvander2d([x1], [x2], [1, 2]) + assert_(van.shape == (1, 5, 6)) + + def test_hermvander3d(self): + # also tests hermval3d for non-square coefficient array + x1, x2, x3 = self.x + c = np.random.random((2, 3, 4)) + van = herm.hermvander3d(x1, x2, x3, [1, 2, 3]) + tgt = herm.hermval3d(x1, x2, x3, c) + res = np.dot(van, c.flat) + assert_almost_equal(res, tgt) + + # check shape + van = herm.hermvander3d([x1], [x2], [x3], [1, 2, 3]) + assert_(van.shape == (1, 5, 24)) + + +class TestFitting: + + def test_hermfit(self): + def f(x): + return x*(x - 1)*(x - 2) + + def f2(x): + return x**4 + x**2 + 1 + + # Test exceptions + assert_raises(ValueError, herm.hermfit, [1], [1], -1) + assert_raises(TypeError, herm.hermfit, [[1]], [1], 0) + assert_raises(TypeError, herm.hermfit, [], [1], 0) + assert_raises(TypeError, herm.hermfit, [1], [[[1]]], 0) + assert_raises(TypeError, herm.hermfit, [1, 2], [1], 0) + assert_raises(TypeError, herm.hermfit, [1], [1, 2], 0) + assert_raises(TypeError, herm.hermfit, [1], [1], 0, w=[[1]]) + assert_raises(TypeError, herm.hermfit, [1], [1], 0, w=[1, 1]) + assert_raises(ValueError, herm.hermfit, [1], [1], [-1,]) + assert_raises(ValueError, herm.hermfit, [1], [1], [2, -1, 6]) + assert_raises(TypeError, herm.hermfit, [1], [1], []) + + # Test fit + x = np.linspace(0, 2) + y = f(x) + # + coef3 = herm.hermfit(x, y, 3) + assert_equal(len(coef3), 4) + assert_almost_equal(herm.hermval(x, coef3), y) + coef3 = herm.hermfit(x, y, [0, 1, 2, 3]) + assert_equal(len(coef3), 4) + assert_almost_equal(herm.hermval(x, coef3), y) + # + coef4 = herm.hermfit(x, y, 4) + assert_equal(len(coef4), 5) + assert_almost_equal(herm.hermval(x, coef4), y) + coef4 = herm.hermfit(x, y, [0, 1, 2, 3, 4]) + assert_equal(len(coef4), 5) + assert_almost_equal(herm.hermval(x, coef4), y) + # check things still work if deg is not in strict increasing + coef4 = herm.hermfit(x, y, [2, 3, 4, 1, 0]) + assert_equal(len(coef4), 5) + assert_almost_equal(herm.hermval(x, coef4), y) + # + coef2d = herm.hermfit(x, np.array([y, y]).T, 3) + assert_almost_equal(coef2d, np.array([coef3, coef3]).T) + coef2d = herm.hermfit(x, np.array([y, y]).T, [0, 1, 2, 3]) + assert_almost_equal(coef2d, np.array([coef3, coef3]).T) + # test weighting + w = np.zeros_like(x) + yw = y.copy() + w[1::2] = 1 + y[0::2] = 0 + wcoef3 = herm.hermfit(x, yw, 3, w=w) + assert_almost_equal(wcoef3, coef3) + wcoef3 = herm.hermfit(x, yw, [0, 1, 2, 3], w=w) + assert_almost_equal(wcoef3, coef3) + # + wcoef2d = herm.hermfit(x, np.array([yw, yw]).T, 3, w=w) + assert_almost_equal(wcoef2d, np.array([coef3, coef3]).T) + wcoef2d = herm.hermfit(x, np.array([yw, yw]).T, [0, 1, 2, 3], w=w) + assert_almost_equal(wcoef2d, np.array([coef3, coef3]).T) + # test scaling with complex values x points whose square + # is zero when summed. + x = [1, 1j, -1, -1j] + assert_almost_equal(herm.hermfit(x, x, 1), [0, .5]) + assert_almost_equal(herm.hermfit(x, x, [0, 1]), [0, .5]) + # test fitting only even Legendre polynomials + x = np.linspace(-1, 1) + y = f2(x) + coef1 = herm.hermfit(x, y, 4) + assert_almost_equal(herm.hermval(x, coef1), y) + coef2 = herm.hermfit(x, y, [0, 2, 4]) + assert_almost_equal(herm.hermval(x, coef2), y) + assert_almost_equal(coef1, coef2) + + +class TestCompanion: + + def test_raises(self): + assert_raises(ValueError, herm.hermcompanion, []) + assert_raises(ValueError, herm.hermcompanion, [1]) + + def test_dimensions(self): + for i in range(1, 5): + coef = [0]*i + [1] + assert_(herm.hermcompanion(coef).shape == (i, i)) + + def test_linear_root(self): + assert_(herm.hermcompanion([1, 2])[0, 0] == -.25) + + +class TestGauss: + + def test_100(self): + x, w = herm.hermgauss(100) + + # test orthogonality. Note that the results need to be normalized, + # otherwise the huge values that can arise from fast growing + # functions like Laguerre can be very confusing. + v = herm.hermvander(x, 99) + vv = np.dot(v.T * w, v) + vd = 1/np.sqrt(vv.diagonal()) + vv = vd[:, None] * vv * vd + assert_almost_equal(vv, np.eye(100)) + + # check that the integral of 1 is correct + tgt = np.sqrt(np.pi) + assert_almost_equal(w.sum(), tgt) + + +class TestMisc: + + def test_hermfromroots(self): + res = herm.hermfromroots([]) + assert_almost_equal(trim(res), [1]) + for i in range(1, 5): + roots = np.cos(np.linspace(-np.pi, 0, 2*i + 1)[1::2]) + pol = herm.hermfromroots(roots) + res = herm.hermval(roots, pol) + tgt = 0 + assert_(len(pol) == i + 1) + assert_almost_equal(herm.herm2poly(pol)[-1], 1) + assert_almost_equal(res, tgt) + + def test_hermroots(self): + assert_almost_equal(herm.hermroots([1]), []) + assert_almost_equal(herm.hermroots([1, 1]), [-.5]) + for i in range(2, 5): + tgt = np.linspace(-1, 1, i) + res = herm.hermroots(herm.hermfromroots(tgt)) + assert_almost_equal(trim(res), trim(tgt)) + + def test_hermtrim(self): + coef = [2, -1, 1, 0] + + # Test exceptions + assert_raises(ValueError, herm.hermtrim, coef, -1) + + # Test results + assert_equal(herm.hermtrim(coef), coef[:-1]) + assert_equal(herm.hermtrim(coef, 1), coef[:-3]) + assert_equal(herm.hermtrim(coef, 2), [0]) + + def test_hermline(self): + assert_equal(herm.hermline(3, 4), [3, 2]) + + def test_herm2poly(self): + for i in range(10): + assert_almost_equal(herm.herm2poly([0]*i + [1]), Hlist[i]) + + def test_poly2herm(self): + for i in range(10): + assert_almost_equal(herm.poly2herm(Hlist[i]), [0]*i + [1]) + + def test_weight(self): + x = np.linspace(-5, 5, 11) + tgt = np.exp(-x**2) + res = herm.hermweight(x) + assert_almost_equal(res, tgt) diff --git a/deepseekvl2/lib/python3.10/site-packages/numpy/polynomial/tests/test_hermite_e.py b/deepseekvl2/lib/python3.10/site-packages/numpy/polynomial/tests/test_hermite_e.py new file mode 100644 index 0000000000000000000000000000000000000000..2d262a3306222bd79f682b09763b0bd2b90ba8fe --- /dev/null +++ b/deepseekvl2/lib/python3.10/site-packages/numpy/polynomial/tests/test_hermite_e.py @@ -0,0 +1,556 @@ +"""Tests for hermite_e module. + +""" +from functools import reduce + +import numpy as np +import numpy.polynomial.hermite_e as herme +from numpy.polynomial.polynomial import polyval +from numpy.testing import ( + assert_almost_equal, assert_raises, assert_equal, assert_, + ) + +He0 = np.array([1]) +He1 = np.array([0, 1]) +He2 = np.array([-1, 0, 1]) +He3 = np.array([0, -3, 0, 1]) +He4 = np.array([3, 0, -6, 0, 1]) +He5 = np.array([0, 15, 0, -10, 0, 1]) +He6 = np.array([-15, 0, 45, 0, -15, 0, 1]) +He7 = np.array([0, -105, 0, 105, 0, -21, 0, 1]) +He8 = np.array([105, 0, -420, 0, 210, 0, -28, 0, 1]) +He9 = np.array([0, 945, 0, -1260, 0, 378, 0, -36, 0, 1]) + +Helist = [He0, He1, He2, He3, He4, He5, He6, He7, He8, He9] + + +def trim(x): + return herme.hermetrim(x, tol=1e-6) + + +class TestConstants: + + def test_hermedomain(self): + assert_equal(herme.hermedomain, [-1, 1]) + + def test_hermezero(self): + assert_equal(herme.hermezero, [0]) + + def test_hermeone(self): + assert_equal(herme.hermeone, [1]) + + def test_hermex(self): + assert_equal(herme.hermex, [0, 1]) + + +class TestArithmetic: + x = np.linspace(-3, 3, 100) + + def test_hermeadd(self): + for i in range(5): + for j in range(5): + msg = f"At i={i}, j={j}" + tgt = np.zeros(max(i, j) + 1) + tgt[i] += 1 + tgt[j] += 1 + res = herme.hermeadd([0]*i + [1], [0]*j + [1]) + assert_equal(trim(res), trim(tgt), err_msg=msg) + + def test_hermesub(self): + for i in range(5): + for j in range(5): + msg = f"At i={i}, j={j}" + tgt = np.zeros(max(i, j) + 1) + tgt[i] += 1 + tgt[j] -= 1 + res = herme.hermesub([0]*i + [1], [0]*j + [1]) + assert_equal(trim(res), trim(tgt), err_msg=msg) + + def test_hermemulx(self): + assert_equal(herme.hermemulx([0]), [0]) + assert_equal(herme.hermemulx([1]), [0, 1]) + for i in range(1, 5): + ser = [0]*i + [1] + tgt = [0]*(i - 1) + [i, 0, 1] + assert_equal(herme.hermemulx(ser), tgt) + + def test_hermemul(self): + # check values of result + for i in range(5): + pol1 = [0]*i + [1] + val1 = herme.hermeval(self.x, pol1) + for j in range(5): + msg = f"At i={i}, j={j}" + pol2 = [0]*j + [1] + val2 = herme.hermeval(self.x, pol2) + pol3 = herme.hermemul(pol1, pol2) + val3 = herme.hermeval(self.x, pol3) + assert_(len(pol3) == i + j + 1, msg) + assert_almost_equal(val3, val1*val2, err_msg=msg) + + def test_hermediv(self): + for i in range(5): + for j in range(5): + msg = f"At i={i}, j={j}" + ci = [0]*i + [1] + cj = [0]*j + [1] + tgt = herme.hermeadd(ci, cj) + quo, rem = herme.hermediv(tgt, ci) + res = herme.hermeadd(herme.hermemul(quo, ci), rem) + assert_equal(trim(res), trim(tgt), err_msg=msg) + + def test_hermepow(self): + for i in range(5): + for j in range(5): + msg = f"At i={i}, j={j}" + c = np.arange(i + 1) + tgt = reduce(herme.hermemul, [c]*j, np.array([1])) + res = herme.hermepow(c, j) + assert_equal(trim(res), trim(tgt), err_msg=msg) + + +class TestEvaluation: + # coefficients of 1 + 2*x + 3*x**2 + c1d = np.array([4., 2., 3.]) + c2d = np.einsum('i,j->ij', c1d, c1d) + c3d = np.einsum('i,j,k->ijk', c1d, c1d, c1d) + + # some random values in [-1, 1) + x = np.random.random((3, 5))*2 - 1 + y = polyval(x, [1., 2., 3.]) + + def test_hermeval(self): + #check empty input + assert_equal(herme.hermeval([], [1]).size, 0) + + #check normal input) + x = np.linspace(-1, 1) + y = [polyval(x, c) for c in Helist] + for i in range(10): + msg = f"At i={i}" + tgt = y[i] + res = herme.hermeval(x, [0]*i + [1]) + assert_almost_equal(res, tgt, err_msg=msg) + + #check that shape is preserved + for i in range(3): + dims = [2]*i + x = np.zeros(dims) + assert_equal(herme.hermeval(x, [1]).shape, dims) + assert_equal(herme.hermeval(x, [1, 0]).shape, dims) + assert_equal(herme.hermeval(x, [1, 0, 0]).shape, dims) + + def test_hermeval2d(self): + x1, x2, x3 = self.x + y1, y2, y3 = self.y + + #test exceptions + assert_raises(ValueError, herme.hermeval2d, x1, x2[:2], self.c2d) + + #test values + tgt = y1*y2 + res = herme.hermeval2d(x1, x2, self.c2d) + assert_almost_equal(res, tgt) + + #test shape + z = np.ones((2, 3)) + res = herme.hermeval2d(z, z, self.c2d) + assert_(res.shape == (2, 3)) + + def test_hermeval3d(self): + x1, x2, x3 = self.x + y1, y2, y3 = self.y + + #test exceptions + assert_raises(ValueError, herme.hermeval3d, x1, x2, x3[:2], self.c3d) + + #test values + tgt = y1*y2*y3 + res = herme.hermeval3d(x1, x2, x3, self.c3d) + assert_almost_equal(res, tgt) + + #test shape + z = np.ones((2, 3)) + res = herme.hermeval3d(z, z, z, self.c3d) + assert_(res.shape == (2, 3)) + + def test_hermegrid2d(self): + x1, x2, x3 = self.x + y1, y2, y3 = self.y + + #test values + tgt = np.einsum('i,j->ij', y1, y2) + res = herme.hermegrid2d(x1, x2, self.c2d) + assert_almost_equal(res, tgt) + + #test shape + z = np.ones((2, 3)) + res = herme.hermegrid2d(z, z, self.c2d) + assert_(res.shape == (2, 3)*2) + + def test_hermegrid3d(self): + x1, x2, x3 = self.x + y1, y2, y3 = self.y + + #test values + tgt = np.einsum('i,j,k->ijk', y1, y2, y3) + res = herme.hermegrid3d(x1, x2, x3, self.c3d) + assert_almost_equal(res, tgt) + + #test shape + z = np.ones((2, 3)) + res = herme.hermegrid3d(z, z, z, self.c3d) + assert_(res.shape == (2, 3)*3) + + +class TestIntegral: + + def test_hermeint(self): + # check exceptions + assert_raises(TypeError, herme.hermeint, [0], .5) + assert_raises(ValueError, herme.hermeint, [0], -1) + assert_raises(ValueError, herme.hermeint, [0], 1, [0, 0]) + assert_raises(ValueError, herme.hermeint, [0], lbnd=[0]) + assert_raises(ValueError, herme.hermeint, [0], scl=[0]) + assert_raises(TypeError, herme.hermeint, [0], axis=.5) + + # test integration of zero polynomial + for i in range(2, 5): + k = [0]*(i - 2) + [1] + res = herme.hermeint([0], m=i, k=k) + assert_almost_equal(res, [0, 1]) + + # check single integration with integration constant + for i in range(5): + scl = i + 1 + pol = [0]*i + [1] + tgt = [i] + [0]*i + [1/scl] + hermepol = herme.poly2herme(pol) + hermeint = herme.hermeint(hermepol, m=1, k=[i]) + res = herme.herme2poly(hermeint) + assert_almost_equal(trim(res), trim(tgt)) + + # check single integration with integration constant and lbnd + for i in range(5): + scl = i + 1 + pol = [0]*i + [1] + hermepol = herme.poly2herme(pol) + hermeint = herme.hermeint(hermepol, m=1, k=[i], lbnd=-1) + assert_almost_equal(herme.hermeval(-1, hermeint), i) + + # check single integration with integration constant and scaling + for i in range(5): + scl = i + 1 + pol = [0]*i + [1] + tgt = [i] + [0]*i + [2/scl] + hermepol = herme.poly2herme(pol) + hermeint = herme.hermeint(hermepol, m=1, k=[i], scl=2) + res = herme.herme2poly(hermeint) + assert_almost_equal(trim(res), trim(tgt)) + + # check multiple integrations with default k + for i in range(5): + for j in range(2, 5): + pol = [0]*i + [1] + tgt = pol[:] + for k in range(j): + tgt = herme.hermeint(tgt, m=1) + res = herme.hermeint(pol, m=j) + assert_almost_equal(trim(res), trim(tgt)) + + # check multiple integrations with defined k + for i in range(5): + for j in range(2, 5): + pol = [0]*i + [1] + tgt = pol[:] + for k in range(j): + tgt = herme.hermeint(tgt, m=1, k=[k]) + res = herme.hermeint(pol, m=j, k=list(range(j))) + assert_almost_equal(trim(res), trim(tgt)) + + # check multiple integrations with lbnd + for i in range(5): + for j in range(2, 5): + pol = [0]*i + [1] + tgt = pol[:] + for k in range(j): + tgt = herme.hermeint(tgt, m=1, k=[k], lbnd=-1) + res = herme.hermeint(pol, m=j, k=list(range(j)), lbnd=-1) + assert_almost_equal(trim(res), trim(tgt)) + + # check multiple integrations with scaling + for i in range(5): + for j in range(2, 5): + pol = [0]*i + [1] + tgt = pol[:] + for k in range(j): + tgt = herme.hermeint(tgt, m=1, k=[k], scl=2) + res = herme.hermeint(pol, m=j, k=list(range(j)), scl=2) + assert_almost_equal(trim(res), trim(tgt)) + + def test_hermeint_axis(self): + # check that axis keyword works + c2d = np.random.random((3, 4)) + + tgt = np.vstack([herme.hermeint(c) for c in c2d.T]).T + res = herme.hermeint(c2d, axis=0) + assert_almost_equal(res, tgt) + + tgt = np.vstack([herme.hermeint(c) for c in c2d]) + res = herme.hermeint(c2d, axis=1) + assert_almost_equal(res, tgt) + + tgt = np.vstack([herme.hermeint(c, k=3) for c in c2d]) + res = herme.hermeint(c2d, k=3, axis=1) + assert_almost_equal(res, tgt) + + +class TestDerivative: + + def test_hermeder(self): + # check exceptions + assert_raises(TypeError, herme.hermeder, [0], .5) + assert_raises(ValueError, herme.hermeder, [0], -1) + + # check that zeroth derivative does nothing + for i in range(5): + tgt = [0]*i + [1] + res = herme.hermeder(tgt, m=0) + assert_equal(trim(res), trim(tgt)) + + # check that derivation is the inverse of integration + for i in range(5): + for j in range(2, 5): + tgt = [0]*i + [1] + res = herme.hermeder(herme.hermeint(tgt, m=j), m=j) + assert_almost_equal(trim(res), trim(tgt)) + + # check derivation with scaling + for i in range(5): + for j in range(2, 5): + tgt = [0]*i + [1] + res = herme.hermeder( + herme.hermeint(tgt, m=j, scl=2), m=j, scl=.5) + assert_almost_equal(trim(res), trim(tgt)) + + def test_hermeder_axis(self): + # check that axis keyword works + c2d = np.random.random((3, 4)) + + tgt = np.vstack([herme.hermeder(c) for c in c2d.T]).T + res = herme.hermeder(c2d, axis=0) + assert_almost_equal(res, tgt) + + tgt = np.vstack([herme.hermeder(c) for c in c2d]) + res = herme.hermeder(c2d, axis=1) + assert_almost_equal(res, tgt) + + +class TestVander: + # some random values in [-1, 1) + x = np.random.random((3, 5))*2 - 1 + + def test_hermevander(self): + # check for 1d x + x = np.arange(3) + v = herme.hermevander(x, 3) + assert_(v.shape == (3, 4)) + for i in range(4): + coef = [0]*i + [1] + assert_almost_equal(v[..., i], herme.hermeval(x, coef)) + + # check for 2d x + x = np.array([[1, 2], [3, 4], [5, 6]]) + v = herme.hermevander(x, 3) + assert_(v.shape == (3, 2, 4)) + for i in range(4): + coef = [0]*i + [1] + assert_almost_equal(v[..., i], herme.hermeval(x, coef)) + + def test_hermevander2d(self): + # also tests hermeval2d for non-square coefficient array + x1, x2, x3 = self.x + c = np.random.random((2, 3)) + van = herme.hermevander2d(x1, x2, [1, 2]) + tgt = herme.hermeval2d(x1, x2, c) + res = np.dot(van, c.flat) + assert_almost_equal(res, tgt) + + # check shape + van = herme.hermevander2d([x1], [x2], [1, 2]) + assert_(van.shape == (1, 5, 6)) + + def test_hermevander3d(self): + # also tests hermeval3d for non-square coefficient array + x1, x2, x3 = self.x + c = np.random.random((2, 3, 4)) + van = herme.hermevander3d(x1, x2, x3, [1, 2, 3]) + tgt = herme.hermeval3d(x1, x2, x3, c) + res = np.dot(van, c.flat) + assert_almost_equal(res, tgt) + + # check shape + van = herme.hermevander3d([x1], [x2], [x3], [1, 2, 3]) + assert_(van.shape == (1, 5, 24)) + + +class TestFitting: + + def test_hermefit(self): + def f(x): + return x*(x - 1)*(x - 2) + + def f2(x): + return x**4 + x**2 + 1 + + # Test exceptions + assert_raises(ValueError, herme.hermefit, [1], [1], -1) + assert_raises(TypeError, herme.hermefit, [[1]], [1], 0) + assert_raises(TypeError, herme.hermefit, [], [1], 0) + assert_raises(TypeError, herme.hermefit, [1], [[[1]]], 0) + assert_raises(TypeError, herme.hermefit, [1, 2], [1], 0) + assert_raises(TypeError, herme.hermefit, [1], [1, 2], 0) + assert_raises(TypeError, herme.hermefit, [1], [1], 0, w=[[1]]) + assert_raises(TypeError, herme.hermefit, [1], [1], 0, w=[1, 1]) + assert_raises(ValueError, herme.hermefit, [1], [1], [-1,]) + assert_raises(ValueError, herme.hermefit, [1], [1], [2, -1, 6]) + assert_raises(TypeError, herme.hermefit, [1], [1], []) + + # Test fit + x = np.linspace(0, 2) + y = f(x) + # + coef3 = herme.hermefit(x, y, 3) + assert_equal(len(coef3), 4) + assert_almost_equal(herme.hermeval(x, coef3), y) + coef3 = herme.hermefit(x, y, [0, 1, 2, 3]) + assert_equal(len(coef3), 4) + assert_almost_equal(herme.hermeval(x, coef3), y) + # + coef4 = herme.hermefit(x, y, 4) + assert_equal(len(coef4), 5) + assert_almost_equal(herme.hermeval(x, coef4), y) + coef4 = herme.hermefit(x, y, [0, 1, 2, 3, 4]) + assert_equal(len(coef4), 5) + assert_almost_equal(herme.hermeval(x, coef4), y) + # check things still work if deg is not in strict increasing + coef4 = herme.hermefit(x, y, [2, 3, 4, 1, 0]) + assert_equal(len(coef4), 5) + assert_almost_equal(herme.hermeval(x, coef4), y) + # + coef2d = herme.hermefit(x, np.array([y, y]).T, 3) + assert_almost_equal(coef2d, np.array([coef3, coef3]).T) + coef2d = herme.hermefit(x, np.array([y, y]).T, [0, 1, 2, 3]) + assert_almost_equal(coef2d, np.array([coef3, coef3]).T) + # test weighting + w = np.zeros_like(x) + yw = y.copy() + w[1::2] = 1 + y[0::2] = 0 + wcoef3 = herme.hermefit(x, yw, 3, w=w) + assert_almost_equal(wcoef3, coef3) + wcoef3 = herme.hermefit(x, yw, [0, 1, 2, 3], w=w) + assert_almost_equal(wcoef3, coef3) + # + wcoef2d = herme.hermefit(x, np.array([yw, yw]).T, 3, w=w) + assert_almost_equal(wcoef2d, np.array([coef3, coef3]).T) + wcoef2d = herme.hermefit(x, np.array([yw, yw]).T, [0, 1, 2, 3], w=w) + assert_almost_equal(wcoef2d, np.array([coef3, coef3]).T) + # test scaling with complex values x points whose square + # is zero when summed. + x = [1, 1j, -1, -1j] + assert_almost_equal(herme.hermefit(x, x, 1), [0, 1]) + assert_almost_equal(herme.hermefit(x, x, [0, 1]), [0, 1]) + # test fitting only even Legendre polynomials + x = np.linspace(-1, 1) + y = f2(x) + coef1 = herme.hermefit(x, y, 4) + assert_almost_equal(herme.hermeval(x, coef1), y) + coef2 = herme.hermefit(x, y, [0, 2, 4]) + assert_almost_equal(herme.hermeval(x, coef2), y) + assert_almost_equal(coef1, coef2) + + +class TestCompanion: + + def test_raises(self): + assert_raises(ValueError, herme.hermecompanion, []) + assert_raises(ValueError, herme.hermecompanion, [1]) + + def test_dimensions(self): + for i in range(1, 5): + coef = [0]*i + [1] + assert_(herme.hermecompanion(coef).shape == (i, i)) + + def test_linear_root(self): + assert_(herme.hermecompanion([1, 2])[0, 0] == -.5) + + +class TestGauss: + + def test_100(self): + x, w = herme.hermegauss(100) + + # test orthogonality. Note that the results need to be normalized, + # otherwise the huge values that can arise from fast growing + # functions like Laguerre can be very confusing. + v = herme.hermevander(x, 99) + vv = np.dot(v.T * w, v) + vd = 1/np.sqrt(vv.diagonal()) + vv = vd[:, None] * vv * vd + assert_almost_equal(vv, np.eye(100)) + + # check that the integral of 1 is correct + tgt = np.sqrt(2*np.pi) + assert_almost_equal(w.sum(), tgt) + + +class TestMisc: + + def test_hermefromroots(self): + res = herme.hermefromroots([]) + assert_almost_equal(trim(res), [1]) + for i in range(1, 5): + roots = np.cos(np.linspace(-np.pi, 0, 2*i + 1)[1::2]) + pol = herme.hermefromroots(roots) + res = herme.hermeval(roots, pol) + tgt = 0 + assert_(len(pol) == i + 1) + assert_almost_equal(herme.herme2poly(pol)[-1], 1) + assert_almost_equal(res, tgt) + + def test_hermeroots(self): + assert_almost_equal(herme.hermeroots([1]), []) + assert_almost_equal(herme.hermeroots([1, 1]), [-1]) + for i in range(2, 5): + tgt = np.linspace(-1, 1, i) + res = herme.hermeroots(herme.hermefromroots(tgt)) + assert_almost_equal(trim(res), trim(tgt)) + + def test_hermetrim(self): + coef = [2, -1, 1, 0] + + # Test exceptions + assert_raises(ValueError, herme.hermetrim, coef, -1) + + # Test results + assert_equal(herme.hermetrim(coef), coef[:-1]) + assert_equal(herme.hermetrim(coef, 1), coef[:-3]) + assert_equal(herme.hermetrim(coef, 2), [0]) + + def test_hermeline(self): + assert_equal(herme.hermeline(3, 4), [3, 4]) + + def test_herme2poly(self): + for i in range(10): + assert_almost_equal(herme.herme2poly([0]*i + [1]), Helist[i]) + + def test_poly2herme(self): + for i in range(10): + assert_almost_equal(herme.poly2herme(Helist[i]), [0]*i + [1]) + + def test_weight(self): + x = np.linspace(-5, 5, 11) + tgt = np.exp(-.5*x**2) + res = herme.hermeweight(x) + assert_almost_equal(res, tgt) diff --git a/deepseekvl2/lib/python3.10/site-packages/numpy/polynomial/tests/test_laguerre.py b/deepseekvl2/lib/python3.10/site-packages/numpy/polynomial/tests/test_laguerre.py new file mode 100644 index 0000000000000000000000000000000000000000..227ef3c5576dd666e2eb76576eb260d5ba48cb0e --- /dev/null +++ b/deepseekvl2/lib/python3.10/site-packages/numpy/polynomial/tests/test_laguerre.py @@ -0,0 +1,537 @@ +"""Tests for laguerre module. + +""" +from functools import reduce + +import numpy as np +import numpy.polynomial.laguerre as lag +from numpy.polynomial.polynomial import polyval +from numpy.testing import ( + assert_almost_equal, assert_raises, assert_equal, assert_, + ) + +L0 = np.array([1])/1 +L1 = np.array([1, -1])/1 +L2 = np.array([2, -4, 1])/2 +L3 = np.array([6, -18, 9, -1])/6 +L4 = np.array([24, -96, 72, -16, 1])/24 +L5 = np.array([120, -600, 600, -200, 25, -1])/120 +L6 = np.array([720, -4320, 5400, -2400, 450, -36, 1])/720 + +Llist = [L0, L1, L2, L3, L4, L5, L6] + + +def trim(x): + return lag.lagtrim(x, tol=1e-6) + + +class TestConstants: + + def test_lagdomain(self): + assert_equal(lag.lagdomain, [0, 1]) + + def test_lagzero(self): + assert_equal(lag.lagzero, [0]) + + def test_lagone(self): + assert_equal(lag.lagone, [1]) + + def test_lagx(self): + assert_equal(lag.lagx, [1, -1]) + + +class TestArithmetic: + x = np.linspace(-3, 3, 100) + + def test_lagadd(self): + for i in range(5): + for j in range(5): + msg = f"At i={i}, j={j}" + tgt = np.zeros(max(i, j) + 1) + tgt[i] += 1 + tgt[j] += 1 + res = lag.lagadd([0]*i + [1], [0]*j + [1]) + assert_equal(trim(res), trim(tgt), err_msg=msg) + + def test_lagsub(self): + for i in range(5): + for j in range(5): + msg = f"At i={i}, j={j}" + tgt = np.zeros(max(i, j) + 1) + tgt[i] += 1 + tgt[j] -= 1 + res = lag.lagsub([0]*i + [1], [0]*j + [1]) + assert_equal(trim(res), trim(tgt), err_msg=msg) + + def test_lagmulx(self): + assert_equal(lag.lagmulx([0]), [0]) + assert_equal(lag.lagmulx([1]), [1, -1]) + for i in range(1, 5): + ser = [0]*i + [1] + tgt = [0]*(i - 1) + [-i, 2*i + 1, -(i + 1)] + assert_almost_equal(lag.lagmulx(ser), tgt) + + def test_lagmul(self): + # check values of result + for i in range(5): + pol1 = [0]*i + [1] + val1 = lag.lagval(self.x, pol1) + for j in range(5): + msg = f"At i={i}, j={j}" + pol2 = [0]*j + [1] + val2 = lag.lagval(self.x, pol2) + pol3 = lag.lagmul(pol1, pol2) + val3 = lag.lagval(self.x, pol3) + assert_(len(pol3) == i + j + 1, msg) + assert_almost_equal(val3, val1*val2, err_msg=msg) + + def test_lagdiv(self): + for i in range(5): + for j in range(5): + msg = f"At i={i}, j={j}" + ci = [0]*i + [1] + cj = [0]*j + [1] + tgt = lag.lagadd(ci, cj) + quo, rem = lag.lagdiv(tgt, ci) + res = lag.lagadd(lag.lagmul(quo, ci), rem) + assert_almost_equal(trim(res), trim(tgt), err_msg=msg) + + def test_lagpow(self): + for i in range(5): + for j in range(5): + msg = f"At i={i}, j={j}" + c = np.arange(i + 1) + tgt = reduce(lag.lagmul, [c]*j, np.array([1])) + res = lag.lagpow(c, j) + assert_equal(trim(res), trim(tgt), err_msg=msg) + + +class TestEvaluation: + # coefficients of 1 + 2*x + 3*x**2 + c1d = np.array([9., -14., 6.]) + c2d = np.einsum('i,j->ij', c1d, c1d) + c3d = np.einsum('i,j,k->ijk', c1d, c1d, c1d) + + # some random values in [-1, 1) + x = np.random.random((3, 5))*2 - 1 + y = polyval(x, [1., 2., 3.]) + + def test_lagval(self): + #check empty input + assert_equal(lag.lagval([], [1]).size, 0) + + #check normal input) + x = np.linspace(-1, 1) + y = [polyval(x, c) for c in Llist] + for i in range(7): + msg = f"At i={i}" + tgt = y[i] + res = lag.lagval(x, [0]*i + [1]) + assert_almost_equal(res, tgt, err_msg=msg) + + #check that shape is preserved + for i in range(3): + dims = [2]*i + x = np.zeros(dims) + assert_equal(lag.lagval(x, [1]).shape, dims) + assert_equal(lag.lagval(x, [1, 0]).shape, dims) + assert_equal(lag.lagval(x, [1, 0, 0]).shape, dims) + + def test_lagval2d(self): + x1, x2, x3 = self.x + y1, y2, y3 = self.y + + #test exceptions + assert_raises(ValueError, lag.lagval2d, x1, x2[:2], self.c2d) + + #test values + tgt = y1*y2 + res = lag.lagval2d(x1, x2, self.c2d) + assert_almost_equal(res, tgt) + + #test shape + z = np.ones((2, 3)) + res = lag.lagval2d(z, z, self.c2d) + assert_(res.shape == (2, 3)) + + def test_lagval3d(self): + x1, x2, x3 = self.x + y1, y2, y3 = self.y + + #test exceptions + assert_raises(ValueError, lag.lagval3d, x1, x2, x3[:2], self.c3d) + + #test values + tgt = y1*y2*y3 + res = lag.lagval3d(x1, x2, x3, self.c3d) + assert_almost_equal(res, tgt) + + #test shape + z = np.ones((2, 3)) + res = lag.lagval3d(z, z, z, self.c3d) + assert_(res.shape == (2, 3)) + + def test_laggrid2d(self): + x1, x2, x3 = self.x + y1, y2, y3 = self.y + + #test values + tgt = np.einsum('i,j->ij', y1, y2) + res = lag.laggrid2d(x1, x2, self.c2d) + assert_almost_equal(res, tgt) + + #test shape + z = np.ones((2, 3)) + res = lag.laggrid2d(z, z, self.c2d) + assert_(res.shape == (2, 3)*2) + + def test_laggrid3d(self): + x1, x2, x3 = self.x + y1, y2, y3 = self.y + + #test values + tgt = np.einsum('i,j,k->ijk', y1, y2, y3) + res = lag.laggrid3d(x1, x2, x3, self.c3d) + assert_almost_equal(res, tgt) + + #test shape + z = np.ones((2, 3)) + res = lag.laggrid3d(z, z, z, self.c3d) + assert_(res.shape == (2, 3)*3) + + +class TestIntegral: + + def test_lagint(self): + # check exceptions + assert_raises(TypeError, lag.lagint, [0], .5) + assert_raises(ValueError, lag.lagint, [0], -1) + assert_raises(ValueError, lag.lagint, [0], 1, [0, 0]) + assert_raises(ValueError, lag.lagint, [0], lbnd=[0]) + assert_raises(ValueError, lag.lagint, [0], scl=[0]) + assert_raises(TypeError, lag.lagint, [0], axis=.5) + + # test integration of zero polynomial + for i in range(2, 5): + k = [0]*(i - 2) + [1] + res = lag.lagint([0], m=i, k=k) + assert_almost_equal(res, [1, -1]) + + # check single integration with integration constant + for i in range(5): + scl = i + 1 + pol = [0]*i + [1] + tgt = [i] + [0]*i + [1/scl] + lagpol = lag.poly2lag(pol) + lagint = lag.lagint(lagpol, m=1, k=[i]) + res = lag.lag2poly(lagint) + assert_almost_equal(trim(res), trim(tgt)) + + # check single integration with integration constant and lbnd + for i in range(5): + scl = i + 1 + pol = [0]*i + [1] + lagpol = lag.poly2lag(pol) + lagint = lag.lagint(lagpol, m=1, k=[i], lbnd=-1) + assert_almost_equal(lag.lagval(-1, lagint), i) + + # check single integration with integration constant and scaling + for i in range(5): + scl = i + 1 + pol = [0]*i + [1] + tgt = [i] + [0]*i + [2/scl] + lagpol = lag.poly2lag(pol) + lagint = lag.lagint(lagpol, m=1, k=[i], scl=2) + res = lag.lag2poly(lagint) + assert_almost_equal(trim(res), trim(tgt)) + + # check multiple integrations with default k + for i in range(5): + for j in range(2, 5): + pol = [0]*i + [1] + tgt = pol[:] + for k in range(j): + tgt = lag.lagint(tgt, m=1) + res = lag.lagint(pol, m=j) + assert_almost_equal(trim(res), trim(tgt)) + + # check multiple integrations with defined k + for i in range(5): + for j in range(2, 5): + pol = [0]*i + [1] + tgt = pol[:] + for k in range(j): + tgt = lag.lagint(tgt, m=1, k=[k]) + res = lag.lagint(pol, m=j, k=list(range(j))) + assert_almost_equal(trim(res), trim(tgt)) + + # check multiple integrations with lbnd + for i in range(5): + for j in range(2, 5): + pol = [0]*i + [1] + tgt = pol[:] + for k in range(j): + tgt = lag.lagint(tgt, m=1, k=[k], lbnd=-1) + res = lag.lagint(pol, m=j, k=list(range(j)), lbnd=-1) + assert_almost_equal(trim(res), trim(tgt)) + + # check multiple integrations with scaling + for i in range(5): + for j in range(2, 5): + pol = [0]*i + [1] + tgt = pol[:] + for k in range(j): + tgt = lag.lagint(tgt, m=1, k=[k], scl=2) + res = lag.lagint(pol, m=j, k=list(range(j)), scl=2) + assert_almost_equal(trim(res), trim(tgt)) + + def test_lagint_axis(self): + # check that axis keyword works + c2d = np.random.random((3, 4)) + + tgt = np.vstack([lag.lagint(c) for c in c2d.T]).T + res = lag.lagint(c2d, axis=0) + assert_almost_equal(res, tgt) + + tgt = np.vstack([lag.lagint(c) for c in c2d]) + res = lag.lagint(c2d, axis=1) + assert_almost_equal(res, tgt) + + tgt = np.vstack([lag.lagint(c, k=3) for c in c2d]) + res = lag.lagint(c2d, k=3, axis=1) + assert_almost_equal(res, tgt) + + +class TestDerivative: + + def test_lagder(self): + # check exceptions + assert_raises(TypeError, lag.lagder, [0], .5) + assert_raises(ValueError, lag.lagder, [0], -1) + + # check that zeroth derivative does nothing + for i in range(5): + tgt = [0]*i + [1] + res = lag.lagder(tgt, m=0) + assert_equal(trim(res), trim(tgt)) + + # check that derivation is the inverse of integration + for i in range(5): + for j in range(2, 5): + tgt = [0]*i + [1] + res = lag.lagder(lag.lagint(tgt, m=j), m=j) + assert_almost_equal(trim(res), trim(tgt)) + + # check derivation with scaling + for i in range(5): + for j in range(2, 5): + tgt = [0]*i + [1] + res = lag.lagder(lag.lagint(tgt, m=j, scl=2), m=j, scl=.5) + assert_almost_equal(trim(res), trim(tgt)) + + def test_lagder_axis(self): + # check that axis keyword works + c2d = np.random.random((3, 4)) + + tgt = np.vstack([lag.lagder(c) for c in c2d.T]).T + res = lag.lagder(c2d, axis=0) + assert_almost_equal(res, tgt) + + tgt = np.vstack([lag.lagder(c) for c in c2d]) + res = lag.lagder(c2d, axis=1) + assert_almost_equal(res, tgt) + + +class TestVander: + # some random values in [-1, 1) + x = np.random.random((3, 5))*2 - 1 + + def test_lagvander(self): + # check for 1d x + x = np.arange(3) + v = lag.lagvander(x, 3) + assert_(v.shape == (3, 4)) + for i in range(4): + coef = [0]*i + [1] + assert_almost_equal(v[..., i], lag.lagval(x, coef)) + + # check for 2d x + x = np.array([[1, 2], [3, 4], [5, 6]]) + v = lag.lagvander(x, 3) + assert_(v.shape == (3, 2, 4)) + for i in range(4): + coef = [0]*i + [1] + assert_almost_equal(v[..., i], lag.lagval(x, coef)) + + def test_lagvander2d(self): + # also tests lagval2d for non-square coefficient array + x1, x2, x3 = self.x + c = np.random.random((2, 3)) + van = lag.lagvander2d(x1, x2, [1, 2]) + tgt = lag.lagval2d(x1, x2, c) + res = np.dot(van, c.flat) + assert_almost_equal(res, tgt) + + # check shape + van = lag.lagvander2d([x1], [x2], [1, 2]) + assert_(van.shape == (1, 5, 6)) + + def test_lagvander3d(self): + # also tests lagval3d for non-square coefficient array + x1, x2, x3 = self.x + c = np.random.random((2, 3, 4)) + van = lag.lagvander3d(x1, x2, x3, [1, 2, 3]) + tgt = lag.lagval3d(x1, x2, x3, c) + res = np.dot(van, c.flat) + assert_almost_equal(res, tgt) + + # check shape + van = lag.lagvander3d([x1], [x2], [x3], [1, 2, 3]) + assert_(van.shape == (1, 5, 24)) + + +class TestFitting: + + def test_lagfit(self): + def f(x): + return x*(x - 1)*(x - 2) + + # Test exceptions + assert_raises(ValueError, lag.lagfit, [1], [1], -1) + assert_raises(TypeError, lag.lagfit, [[1]], [1], 0) + assert_raises(TypeError, lag.lagfit, [], [1], 0) + assert_raises(TypeError, lag.lagfit, [1], [[[1]]], 0) + assert_raises(TypeError, lag.lagfit, [1, 2], [1], 0) + assert_raises(TypeError, lag.lagfit, [1], [1, 2], 0) + assert_raises(TypeError, lag.lagfit, [1], [1], 0, w=[[1]]) + assert_raises(TypeError, lag.lagfit, [1], [1], 0, w=[1, 1]) + assert_raises(ValueError, lag.lagfit, [1], [1], [-1,]) + assert_raises(ValueError, lag.lagfit, [1], [1], [2, -1, 6]) + assert_raises(TypeError, lag.lagfit, [1], [1], []) + + # Test fit + x = np.linspace(0, 2) + y = f(x) + # + coef3 = lag.lagfit(x, y, 3) + assert_equal(len(coef3), 4) + assert_almost_equal(lag.lagval(x, coef3), y) + coef3 = lag.lagfit(x, y, [0, 1, 2, 3]) + assert_equal(len(coef3), 4) + assert_almost_equal(lag.lagval(x, coef3), y) + # + coef4 = lag.lagfit(x, y, 4) + assert_equal(len(coef4), 5) + assert_almost_equal(lag.lagval(x, coef4), y) + coef4 = lag.lagfit(x, y, [0, 1, 2, 3, 4]) + assert_equal(len(coef4), 5) + assert_almost_equal(lag.lagval(x, coef4), y) + # + coef2d = lag.lagfit(x, np.array([y, y]).T, 3) + assert_almost_equal(coef2d, np.array([coef3, coef3]).T) + coef2d = lag.lagfit(x, np.array([y, y]).T, [0, 1, 2, 3]) + assert_almost_equal(coef2d, np.array([coef3, coef3]).T) + # test weighting + w = np.zeros_like(x) + yw = y.copy() + w[1::2] = 1 + y[0::2] = 0 + wcoef3 = lag.lagfit(x, yw, 3, w=w) + assert_almost_equal(wcoef3, coef3) + wcoef3 = lag.lagfit(x, yw, [0, 1, 2, 3], w=w) + assert_almost_equal(wcoef3, coef3) + # + wcoef2d = lag.lagfit(x, np.array([yw, yw]).T, 3, w=w) + assert_almost_equal(wcoef2d, np.array([coef3, coef3]).T) + wcoef2d = lag.lagfit(x, np.array([yw, yw]).T, [0, 1, 2, 3], w=w) + assert_almost_equal(wcoef2d, np.array([coef3, coef3]).T) + # test scaling with complex values x points whose square + # is zero when summed. + x = [1, 1j, -1, -1j] + assert_almost_equal(lag.lagfit(x, x, 1), [1, -1]) + assert_almost_equal(lag.lagfit(x, x, [0, 1]), [1, -1]) + + +class TestCompanion: + + def test_raises(self): + assert_raises(ValueError, lag.lagcompanion, []) + assert_raises(ValueError, lag.lagcompanion, [1]) + + def test_dimensions(self): + for i in range(1, 5): + coef = [0]*i + [1] + assert_(lag.lagcompanion(coef).shape == (i, i)) + + def test_linear_root(self): + assert_(lag.lagcompanion([1, 2])[0, 0] == 1.5) + + +class TestGauss: + + def test_100(self): + x, w = lag.laggauss(100) + + # test orthogonality. Note that the results need to be normalized, + # otherwise the huge values that can arise from fast growing + # functions like Laguerre can be very confusing. + v = lag.lagvander(x, 99) + vv = np.dot(v.T * w, v) + vd = 1/np.sqrt(vv.diagonal()) + vv = vd[:, None] * vv * vd + assert_almost_equal(vv, np.eye(100)) + + # check that the integral of 1 is correct + tgt = 1.0 + assert_almost_equal(w.sum(), tgt) + + +class TestMisc: + + def test_lagfromroots(self): + res = lag.lagfromroots([]) + assert_almost_equal(trim(res), [1]) + for i in range(1, 5): + roots = np.cos(np.linspace(-np.pi, 0, 2*i + 1)[1::2]) + pol = lag.lagfromroots(roots) + res = lag.lagval(roots, pol) + tgt = 0 + assert_(len(pol) == i + 1) + assert_almost_equal(lag.lag2poly(pol)[-1], 1) + assert_almost_equal(res, tgt) + + def test_lagroots(self): + assert_almost_equal(lag.lagroots([1]), []) + assert_almost_equal(lag.lagroots([0, 1]), [1]) + for i in range(2, 5): + tgt = np.linspace(0, 3, i) + res = lag.lagroots(lag.lagfromroots(tgt)) + assert_almost_equal(trim(res), trim(tgt)) + + def test_lagtrim(self): + coef = [2, -1, 1, 0] + + # Test exceptions + assert_raises(ValueError, lag.lagtrim, coef, -1) + + # Test results + assert_equal(lag.lagtrim(coef), coef[:-1]) + assert_equal(lag.lagtrim(coef, 1), coef[:-3]) + assert_equal(lag.lagtrim(coef, 2), [0]) + + def test_lagline(self): + assert_equal(lag.lagline(3, 4), [7, -4]) + + def test_lag2poly(self): + for i in range(7): + assert_almost_equal(lag.lag2poly([0]*i + [1]), Llist[i]) + + def test_poly2lag(self): + for i in range(7): + assert_almost_equal(lag.poly2lag(Llist[i]), [0]*i + [1]) + + def test_weight(self): + x = np.linspace(0, 10, 11) + tgt = np.exp(-x) + res = lag.lagweight(x) + assert_almost_equal(res, tgt) diff --git a/deepseekvl2/lib/python3.10/site-packages/numpy/polynomial/tests/test_legendre.py b/deepseekvl2/lib/python3.10/site-packages/numpy/polynomial/tests/test_legendre.py new file mode 100644 index 0000000000000000000000000000000000000000..92399c160ecb75fbb1f9a5a7f2bba0fe90d84a54 --- /dev/null +++ b/deepseekvl2/lib/python3.10/site-packages/numpy/polynomial/tests/test_legendre.py @@ -0,0 +1,568 @@ +"""Tests for legendre module. + +""" +from functools import reduce + +import numpy as np +import numpy.polynomial.legendre as leg +from numpy.polynomial.polynomial import polyval +from numpy.testing import ( + assert_almost_equal, assert_raises, assert_equal, assert_, + ) + +L0 = np.array([1]) +L1 = np.array([0, 1]) +L2 = np.array([-1, 0, 3])/2 +L3 = np.array([0, -3, 0, 5])/2 +L4 = np.array([3, 0, -30, 0, 35])/8 +L5 = np.array([0, 15, 0, -70, 0, 63])/8 +L6 = np.array([-5, 0, 105, 0, -315, 0, 231])/16 +L7 = np.array([0, -35, 0, 315, 0, -693, 0, 429])/16 +L8 = np.array([35, 0, -1260, 0, 6930, 0, -12012, 0, 6435])/128 +L9 = np.array([0, 315, 0, -4620, 0, 18018, 0, -25740, 0, 12155])/128 + +Llist = [L0, L1, L2, L3, L4, L5, L6, L7, L8, L9] + + +def trim(x): + return leg.legtrim(x, tol=1e-6) + + +class TestConstants: + + def test_legdomain(self): + assert_equal(leg.legdomain, [-1, 1]) + + def test_legzero(self): + assert_equal(leg.legzero, [0]) + + def test_legone(self): + assert_equal(leg.legone, [1]) + + def test_legx(self): + assert_equal(leg.legx, [0, 1]) + + +class TestArithmetic: + x = np.linspace(-1, 1, 100) + + def test_legadd(self): + for i in range(5): + for j in range(5): + msg = f"At i={i}, j={j}" + tgt = np.zeros(max(i, j) + 1) + tgt[i] += 1 + tgt[j] += 1 + res = leg.legadd([0]*i + [1], [0]*j + [1]) + assert_equal(trim(res), trim(tgt), err_msg=msg) + + def test_legsub(self): + for i in range(5): + for j in range(5): + msg = f"At i={i}, j={j}" + tgt = np.zeros(max(i, j) + 1) + tgt[i] += 1 + tgt[j] -= 1 + res = leg.legsub([0]*i + [1], [0]*j + [1]) + assert_equal(trim(res), trim(tgt), err_msg=msg) + + def test_legmulx(self): + assert_equal(leg.legmulx([0]), [0]) + assert_equal(leg.legmulx([1]), [0, 1]) + for i in range(1, 5): + tmp = 2*i + 1 + ser = [0]*i + [1] + tgt = [0]*(i - 1) + [i/tmp, 0, (i + 1)/tmp] + assert_equal(leg.legmulx(ser), tgt) + + def test_legmul(self): + # check values of result + for i in range(5): + pol1 = [0]*i + [1] + val1 = leg.legval(self.x, pol1) + for j in range(5): + msg = f"At i={i}, j={j}" + pol2 = [0]*j + [1] + val2 = leg.legval(self.x, pol2) + pol3 = leg.legmul(pol1, pol2) + val3 = leg.legval(self.x, pol3) + assert_(len(pol3) == i + j + 1, msg) + assert_almost_equal(val3, val1*val2, err_msg=msg) + + def test_legdiv(self): + for i in range(5): + for j in range(5): + msg = f"At i={i}, j={j}" + ci = [0]*i + [1] + cj = [0]*j + [1] + tgt = leg.legadd(ci, cj) + quo, rem = leg.legdiv(tgt, ci) + res = leg.legadd(leg.legmul(quo, ci), rem) + assert_equal(trim(res), trim(tgt), err_msg=msg) + + def test_legpow(self): + for i in range(5): + for j in range(5): + msg = f"At i={i}, j={j}" + c = np.arange(i + 1) + tgt = reduce(leg.legmul, [c]*j, np.array([1])) + res = leg.legpow(c, j) + assert_equal(trim(res), trim(tgt), err_msg=msg) + + +class TestEvaluation: + # coefficients of 1 + 2*x + 3*x**2 + c1d = np.array([2., 2., 2.]) + c2d = np.einsum('i,j->ij', c1d, c1d) + c3d = np.einsum('i,j,k->ijk', c1d, c1d, c1d) + + # some random values in [-1, 1) + x = np.random.random((3, 5))*2 - 1 + y = polyval(x, [1., 2., 3.]) + + def test_legval(self): + #check empty input + assert_equal(leg.legval([], [1]).size, 0) + + #check normal input) + x = np.linspace(-1, 1) + y = [polyval(x, c) for c in Llist] + for i in range(10): + msg = f"At i={i}" + tgt = y[i] + res = leg.legval(x, [0]*i + [1]) + assert_almost_equal(res, tgt, err_msg=msg) + + #check that shape is preserved + for i in range(3): + dims = [2]*i + x = np.zeros(dims) + assert_equal(leg.legval(x, [1]).shape, dims) + assert_equal(leg.legval(x, [1, 0]).shape, dims) + assert_equal(leg.legval(x, [1, 0, 0]).shape, dims) + + def test_legval2d(self): + x1, x2, x3 = self.x + y1, y2, y3 = self.y + + #test exceptions + assert_raises(ValueError, leg.legval2d, x1, x2[:2], self.c2d) + + #test values + tgt = y1*y2 + res = leg.legval2d(x1, x2, self.c2d) + assert_almost_equal(res, tgt) + + #test shape + z = np.ones((2, 3)) + res = leg.legval2d(z, z, self.c2d) + assert_(res.shape == (2, 3)) + + def test_legval3d(self): + x1, x2, x3 = self.x + y1, y2, y3 = self.y + + #test exceptions + assert_raises(ValueError, leg.legval3d, x1, x2, x3[:2], self.c3d) + + #test values + tgt = y1*y2*y3 + res = leg.legval3d(x1, x2, x3, self.c3d) + assert_almost_equal(res, tgt) + + #test shape + z = np.ones((2, 3)) + res = leg.legval3d(z, z, z, self.c3d) + assert_(res.shape == (2, 3)) + + def test_leggrid2d(self): + x1, x2, x3 = self.x + y1, y2, y3 = self.y + + #test values + tgt = np.einsum('i,j->ij', y1, y2) + res = leg.leggrid2d(x1, x2, self.c2d) + assert_almost_equal(res, tgt) + + #test shape + z = np.ones((2, 3)) + res = leg.leggrid2d(z, z, self.c2d) + assert_(res.shape == (2, 3)*2) + + def test_leggrid3d(self): + x1, x2, x3 = self.x + y1, y2, y3 = self.y + + #test values + tgt = np.einsum('i,j,k->ijk', y1, y2, y3) + res = leg.leggrid3d(x1, x2, x3, self.c3d) + assert_almost_equal(res, tgt) + + #test shape + z = np.ones((2, 3)) + res = leg.leggrid3d(z, z, z, self.c3d) + assert_(res.shape == (2, 3)*3) + + +class TestIntegral: + + def test_legint(self): + # check exceptions + assert_raises(TypeError, leg.legint, [0], .5) + assert_raises(ValueError, leg.legint, [0], -1) + assert_raises(ValueError, leg.legint, [0], 1, [0, 0]) + assert_raises(ValueError, leg.legint, [0], lbnd=[0]) + assert_raises(ValueError, leg.legint, [0], scl=[0]) + assert_raises(TypeError, leg.legint, [0], axis=.5) + + # test integration of zero polynomial + for i in range(2, 5): + k = [0]*(i - 2) + [1] + res = leg.legint([0], m=i, k=k) + assert_almost_equal(res, [0, 1]) + + # check single integration with integration constant + for i in range(5): + scl = i + 1 + pol = [0]*i + [1] + tgt = [i] + [0]*i + [1/scl] + legpol = leg.poly2leg(pol) + legint = leg.legint(legpol, m=1, k=[i]) + res = leg.leg2poly(legint) + assert_almost_equal(trim(res), trim(tgt)) + + # check single integration with integration constant and lbnd + for i in range(5): + scl = i + 1 + pol = [0]*i + [1] + legpol = leg.poly2leg(pol) + legint = leg.legint(legpol, m=1, k=[i], lbnd=-1) + assert_almost_equal(leg.legval(-1, legint), i) + + # check single integration with integration constant and scaling + for i in range(5): + scl = i + 1 + pol = [0]*i + [1] + tgt = [i] + [0]*i + [2/scl] + legpol = leg.poly2leg(pol) + legint = leg.legint(legpol, m=1, k=[i], scl=2) + res = leg.leg2poly(legint) + assert_almost_equal(trim(res), trim(tgt)) + + # check multiple integrations with default k + for i in range(5): + for j in range(2, 5): + pol = [0]*i + [1] + tgt = pol[:] + for k in range(j): + tgt = leg.legint(tgt, m=1) + res = leg.legint(pol, m=j) + assert_almost_equal(trim(res), trim(tgt)) + + # check multiple integrations with defined k + for i in range(5): + for j in range(2, 5): + pol = [0]*i + [1] + tgt = pol[:] + for k in range(j): + tgt = leg.legint(tgt, m=1, k=[k]) + res = leg.legint(pol, m=j, k=list(range(j))) + assert_almost_equal(trim(res), trim(tgt)) + + # check multiple integrations with lbnd + for i in range(5): + for j in range(2, 5): + pol = [0]*i + [1] + tgt = pol[:] + for k in range(j): + tgt = leg.legint(tgt, m=1, k=[k], lbnd=-1) + res = leg.legint(pol, m=j, k=list(range(j)), lbnd=-1) + assert_almost_equal(trim(res), trim(tgt)) + + # check multiple integrations with scaling + for i in range(5): + for j in range(2, 5): + pol = [0]*i + [1] + tgt = pol[:] + for k in range(j): + tgt = leg.legint(tgt, m=1, k=[k], scl=2) + res = leg.legint(pol, m=j, k=list(range(j)), scl=2) + assert_almost_equal(trim(res), trim(tgt)) + + def test_legint_axis(self): + # check that axis keyword works + c2d = np.random.random((3, 4)) + + tgt = np.vstack([leg.legint(c) for c in c2d.T]).T + res = leg.legint(c2d, axis=0) + assert_almost_equal(res, tgt) + + tgt = np.vstack([leg.legint(c) for c in c2d]) + res = leg.legint(c2d, axis=1) + assert_almost_equal(res, tgt) + + tgt = np.vstack([leg.legint(c, k=3) for c in c2d]) + res = leg.legint(c2d, k=3, axis=1) + assert_almost_equal(res, tgt) + + def test_legint_zerointord(self): + assert_equal(leg.legint((1, 2, 3), 0), (1, 2, 3)) + + +class TestDerivative: + + def test_legder(self): + # check exceptions + assert_raises(TypeError, leg.legder, [0], .5) + assert_raises(ValueError, leg.legder, [0], -1) + + # check that zeroth derivative does nothing + for i in range(5): + tgt = [0]*i + [1] + res = leg.legder(tgt, m=0) + assert_equal(trim(res), trim(tgt)) + + # check that derivation is the inverse of integration + for i in range(5): + for j in range(2, 5): + tgt = [0]*i + [1] + res = leg.legder(leg.legint(tgt, m=j), m=j) + assert_almost_equal(trim(res), trim(tgt)) + + # check derivation with scaling + for i in range(5): + for j in range(2, 5): + tgt = [0]*i + [1] + res = leg.legder(leg.legint(tgt, m=j, scl=2), m=j, scl=.5) + assert_almost_equal(trim(res), trim(tgt)) + + def test_legder_axis(self): + # check that axis keyword works + c2d = np.random.random((3, 4)) + + tgt = np.vstack([leg.legder(c) for c in c2d.T]).T + res = leg.legder(c2d, axis=0) + assert_almost_equal(res, tgt) + + tgt = np.vstack([leg.legder(c) for c in c2d]) + res = leg.legder(c2d, axis=1) + assert_almost_equal(res, tgt) + + def test_legder_orderhigherthancoeff(self): + c = (1, 2, 3, 4) + assert_equal(leg.legder(c, 4), [0]) + +class TestVander: + # some random values in [-1, 1) + x = np.random.random((3, 5))*2 - 1 + + def test_legvander(self): + # check for 1d x + x = np.arange(3) + v = leg.legvander(x, 3) + assert_(v.shape == (3, 4)) + for i in range(4): + coef = [0]*i + [1] + assert_almost_equal(v[..., i], leg.legval(x, coef)) + + # check for 2d x + x = np.array([[1, 2], [3, 4], [5, 6]]) + v = leg.legvander(x, 3) + assert_(v.shape == (3, 2, 4)) + for i in range(4): + coef = [0]*i + [1] + assert_almost_equal(v[..., i], leg.legval(x, coef)) + + def test_legvander2d(self): + # also tests polyval2d for non-square coefficient array + x1, x2, x3 = self.x + c = np.random.random((2, 3)) + van = leg.legvander2d(x1, x2, [1, 2]) + tgt = leg.legval2d(x1, x2, c) + res = np.dot(van, c.flat) + assert_almost_equal(res, tgt) + + # check shape + van = leg.legvander2d([x1], [x2], [1, 2]) + assert_(van.shape == (1, 5, 6)) + + def test_legvander3d(self): + # also tests polyval3d for non-square coefficient array + x1, x2, x3 = self.x + c = np.random.random((2, 3, 4)) + van = leg.legvander3d(x1, x2, x3, [1, 2, 3]) + tgt = leg.legval3d(x1, x2, x3, c) + res = np.dot(van, c.flat) + assert_almost_equal(res, tgt) + + # check shape + van = leg.legvander3d([x1], [x2], [x3], [1, 2, 3]) + assert_(van.shape == (1, 5, 24)) + + def test_legvander_negdeg(self): + assert_raises(ValueError, leg.legvander, (1, 2, 3), -1) + + +class TestFitting: + + def test_legfit(self): + def f(x): + return x*(x - 1)*(x - 2) + + def f2(x): + return x**4 + x**2 + 1 + + # Test exceptions + assert_raises(ValueError, leg.legfit, [1], [1], -1) + assert_raises(TypeError, leg.legfit, [[1]], [1], 0) + assert_raises(TypeError, leg.legfit, [], [1], 0) + assert_raises(TypeError, leg.legfit, [1], [[[1]]], 0) + assert_raises(TypeError, leg.legfit, [1, 2], [1], 0) + assert_raises(TypeError, leg.legfit, [1], [1, 2], 0) + assert_raises(TypeError, leg.legfit, [1], [1], 0, w=[[1]]) + assert_raises(TypeError, leg.legfit, [1], [1], 0, w=[1, 1]) + assert_raises(ValueError, leg.legfit, [1], [1], [-1,]) + assert_raises(ValueError, leg.legfit, [1], [1], [2, -1, 6]) + assert_raises(TypeError, leg.legfit, [1], [1], []) + + # Test fit + x = np.linspace(0, 2) + y = f(x) + # + coef3 = leg.legfit(x, y, 3) + assert_equal(len(coef3), 4) + assert_almost_equal(leg.legval(x, coef3), y) + coef3 = leg.legfit(x, y, [0, 1, 2, 3]) + assert_equal(len(coef3), 4) + assert_almost_equal(leg.legval(x, coef3), y) + # + coef4 = leg.legfit(x, y, 4) + assert_equal(len(coef4), 5) + assert_almost_equal(leg.legval(x, coef4), y) + coef4 = leg.legfit(x, y, [0, 1, 2, 3, 4]) + assert_equal(len(coef4), 5) + assert_almost_equal(leg.legval(x, coef4), y) + # check things still work if deg is not in strict increasing + coef4 = leg.legfit(x, y, [2, 3, 4, 1, 0]) + assert_equal(len(coef4), 5) + assert_almost_equal(leg.legval(x, coef4), y) + # + coef2d = leg.legfit(x, np.array([y, y]).T, 3) + assert_almost_equal(coef2d, np.array([coef3, coef3]).T) + coef2d = leg.legfit(x, np.array([y, y]).T, [0, 1, 2, 3]) + assert_almost_equal(coef2d, np.array([coef3, coef3]).T) + # test weighting + w = np.zeros_like(x) + yw = y.copy() + w[1::2] = 1 + y[0::2] = 0 + wcoef3 = leg.legfit(x, yw, 3, w=w) + assert_almost_equal(wcoef3, coef3) + wcoef3 = leg.legfit(x, yw, [0, 1, 2, 3], w=w) + assert_almost_equal(wcoef3, coef3) + # + wcoef2d = leg.legfit(x, np.array([yw, yw]).T, 3, w=w) + assert_almost_equal(wcoef2d, np.array([coef3, coef3]).T) + wcoef2d = leg.legfit(x, np.array([yw, yw]).T, [0, 1, 2, 3], w=w) + assert_almost_equal(wcoef2d, np.array([coef3, coef3]).T) + # test scaling with complex values x points whose square + # is zero when summed. + x = [1, 1j, -1, -1j] + assert_almost_equal(leg.legfit(x, x, 1), [0, 1]) + assert_almost_equal(leg.legfit(x, x, [0, 1]), [0, 1]) + # test fitting only even Legendre polynomials + x = np.linspace(-1, 1) + y = f2(x) + coef1 = leg.legfit(x, y, 4) + assert_almost_equal(leg.legval(x, coef1), y) + coef2 = leg.legfit(x, y, [0, 2, 4]) + assert_almost_equal(leg.legval(x, coef2), y) + assert_almost_equal(coef1, coef2) + + +class TestCompanion: + + def test_raises(self): + assert_raises(ValueError, leg.legcompanion, []) + assert_raises(ValueError, leg.legcompanion, [1]) + + def test_dimensions(self): + for i in range(1, 5): + coef = [0]*i + [1] + assert_(leg.legcompanion(coef).shape == (i, i)) + + def test_linear_root(self): + assert_(leg.legcompanion([1, 2])[0, 0] == -.5) + + +class TestGauss: + + def test_100(self): + x, w = leg.leggauss(100) + + # test orthogonality. Note that the results need to be normalized, + # otherwise the huge values that can arise from fast growing + # functions like Laguerre can be very confusing. + v = leg.legvander(x, 99) + vv = np.dot(v.T * w, v) + vd = 1/np.sqrt(vv.diagonal()) + vv = vd[:, None] * vv * vd + assert_almost_equal(vv, np.eye(100)) + + # check that the integral of 1 is correct + tgt = 2.0 + assert_almost_equal(w.sum(), tgt) + + +class TestMisc: + + def test_legfromroots(self): + res = leg.legfromroots([]) + assert_almost_equal(trim(res), [1]) + for i in range(1, 5): + roots = np.cos(np.linspace(-np.pi, 0, 2*i + 1)[1::2]) + pol = leg.legfromroots(roots) + res = leg.legval(roots, pol) + tgt = 0 + assert_(len(pol) == i + 1) + assert_almost_equal(leg.leg2poly(pol)[-1], 1) + assert_almost_equal(res, tgt) + + def test_legroots(self): + assert_almost_equal(leg.legroots([1]), []) + assert_almost_equal(leg.legroots([1, 2]), [-.5]) + for i in range(2, 5): + tgt = np.linspace(-1, 1, i) + res = leg.legroots(leg.legfromroots(tgt)) + assert_almost_equal(trim(res), trim(tgt)) + + def test_legtrim(self): + coef = [2, -1, 1, 0] + + # Test exceptions + assert_raises(ValueError, leg.legtrim, coef, -1) + + # Test results + assert_equal(leg.legtrim(coef), coef[:-1]) + assert_equal(leg.legtrim(coef, 1), coef[:-3]) + assert_equal(leg.legtrim(coef, 2), [0]) + + def test_legline(self): + assert_equal(leg.legline(3, 4), [3, 4]) + + def test_legline_zeroscl(self): + assert_equal(leg.legline(3, 0), [3]) + + def test_leg2poly(self): + for i in range(10): + assert_almost_equal(leg.leg2poly([0]*i + [1]), Llist[i]) + + def test_poly2leg(self): + for i in range(10): + assert_almost_equal(leg.poly2leg(Llist[i]), [0]*i + [1]) + + def test_weight(self): + x = np.linspace(-1, 1, 11) + tgt = 1. + res = leg.legweight(x) + assert_almost_equal(res, tgt) diff --git a/deepseekvl2/lib/python3.10/site-packages/numpy/polynomial/tests/test_polynomial.py b/deepseekvl2/lib/python3.10/site-packages/numpy/polynomial/tests/test_polynomial.py new file mode 100644 index 0000000000000000000000000000000000000000..6b3ef2388f630f0233c79f31a9a1f4039f4e4f4a --- /dev/null +++ b/deepseekvl2/lib/python3.10/site-packages/numpy/polynomial/tests/test_polynomial.py @@ -0,0 +1,611 @@ +"""Tests for polynomial module. + +""" +from functools import reduce + +import numpy as np +import numpy.polynomial.polynomial as poly +import pickle +from copy import deepcopy +from numpy.testing import ( + assert_almost_equal, assert_raises, assert_equal, assert_, + assert_warns, assert_array_equal, assert_raises_regex) + + +def trim(x): + return poly.polytrim(x, tol=1e-6) + +T0 = [1] +T1 = [0, 1] +T2 = [-1, 0, 2] +T3 = [0, -3, 0, 4] +T4 = [1, 0, -8, 0, 8] +T5 = [0, 5, 0, -20, 0, 16] +T6 = [-1, 0, 18, 0, -48, 0, 32] +T7 = [0, -7, 0, 56, 0, -112, 0, 64] +T8 = [1, 0, -32, 0, 160, 0, -256, 0, 128] +T9 = [0, 9, 0, -120, 0, 432, 0, -576, 0, 256] + +Tlist = [T0, T1, T2, T3, T4, T5, T6, T7, T8, T9] + + +class TestConstants: + + def test_polydomain(self): + assert_equal(poly.polydomain, [-1, 1]) + + def test_polyzero(self): + assert_equal(poly.polyzero, [0]) + + def test_polyone(self): + assert_equal(poly.polyone, [1]) + + def test_polyx(self): + assert_equal(poly.polyx, [0, 1]) + + def test_copy(self): + x = poly.Polynomial([1, 2, 3]) + y = deepcopy(x) + assert_equal(x, y) + + def test_pickle(self): + x = poly.Polynomial([1, 2, 3]) + y = pickle.loads(pickle.dumps(x)) + assert_equal(x, y) + +class TestArithmetic: + + def test_polyadd(self): + for i in range(5): + for j in range(5): + msg = f"At i={i}, j={j}" + tgt = np.zeros(max(i, j) + 1) + tgt[i] += 1 + tgt[j] += 1 + res = poly.polyadd([0]*i + [1], [0]*j + [1]) + assert_equal(trim(res), trim(tgt), err_msg=msg) + + def test_polysub(self): + for i in range(5): + for j in range(5): + msg = f"At i={i}, j={j}" + tgt = np.zeros(max(i, j) + 1) + tgt[i] += 1 + tgt[j] -= 1 + res = poly.polysub([0]*i + [1], [0]*j + [1]) + assert_equal(trim(res), trim(tgt), err_msg=msg) + + def test_polymulx(self): + assert_equal(poly.polymulx([0]), [0]) + assert_equal(poly.polymulx([1]), [0, 1]) + for i in range(1, 5): + ser = [0]*i + [1] + tgt = [0]*(i + 1) + [1] + assert_equal(poly.polymulx(ser), tgt) + + def test_polymul(self): + for i in range(5): + for j in range(5): + msg = f"At i={i}, j={j}" + tgt = np.zeros(i + j + 1) + tgt[i + j] += 1 + res = poly.polymul([0]*i + [1], [0]*j + [1]) + assert_equal(trim(res), trim(tgt), err_msg=msg) + + def test_polydiv(self): + # check zero division + assert_raises(ZeroDivisionError, poly.polydiv, [1], [0]) + + # check scalar division + quo, rem = poly.polydiv([2], [2]) + assert_equal((quo, rem), (1, 0)) + quo, rem = poly.polydiv([2, 2], [2]) + assert_equal((quo, rem), ((1, 1), 0)) + + # check rest. + for i in range(5): + for j in range(5): + msg = f"At i={i}, j={j}" + ci = [0]*i + [1, 2] + cj = [0]*j + [1, 2] + tgt = poly.polyadd(ci, cj) + quo, rem = poly.polydiv(tgt, ci) + res = poly.polyadd(poly.polymul(quo, ci), rem) + assert_equal(res, tgt, err_msg=msg) + + def test_polypow(self): + for i in range(5): + for j in range(5): + msg = f"At i={i}, j={j}" + c = np.arange(i + 1) + tgt = reduce(poly.polymul, [c]*j, np.array([1])) + res = poly.polypow(c, j) + assert_equal(trim(res), trim(tgt), err_msg=msg) + + +class TestEvaluation: + # coefficients of 1 + 2*x + 3*x**2 + c1d = np.array([1., 2., 3.]) + c2d = np.einsum('i,j->ij', c1d, c1d) + c3d = np.einsum('i,j,k->ijk', c1d, c1d, c1d) + + # some random values in [-1, 1) + x = np.random.random((3, 5))*2 - 1 + y = poly.polyval(x, [1., 2., 3.]) + + def test_polyval(self): + #check empty input + assert_equal(poly.polyval([], [1]).size, 0) + + #check normal input) + x = np.linspace(-1, 1) + y = [x**i for i in range(5)] + for i in range(5): + tgt = y[i] + res = poly.polyval(x, [0]*i + [1]) + assert_almost_equal(res, tgt) + tgt = x*(x**2 - 1) + res = poly.polyval(x, [0, -1, 0, 1]) + assert_almost_equal(res, tgt) + + #check that shape is preserved + for i in range(3): + dims = [2]*i + x = np.zeros(dims) + assert_equal(poly.polyval(x, [1]).shape, dims) + assert_equal(poly.polyval(x, [1, 0]).shape, dims) + assert_equal(poly.polyval(x, [1, 0, 0]).shape, dims) + + #check masked arrays are processed correctly + mask = [False, True, False] + mx = np.ma.array([1, 2, 3], mask=mask) + res = np.polyval([7, 5, 3], mx) + assert_array_equal(res.mask, mask) + + #check subtypes of ndarray are preserved + class C(np.ndarray): + pass + + cx = np.array([1, 2, 3]).view(C) + assert_equal(type(np.polyval([2, 3, 4], cx)), C) + + def test_polyvalfromroots(self): + # check exception for broadcasting x values over root array with + # too few dimensions + assert_raises(ValueError, poly.polyvalfromroots, + [1], [1], tensor=False) + + # check empty input + assert_equal(poly.polyvalfromroots([], [1]).size, 0) + assert_(poly.polyvalfromroots([], [1]).shape == (0,)) + + # check empty input + multidimensional roots + assert_equal(poly.polyvalfromroots([], [[1] * 5]).size, 0) + assert_(poly.polyvalfromroots([], [[1] * 5]).shape == (5, 0)) + + # check scalar input + assert_equal(poly.polyvalfromroots(1, 1), 0) + assert_(poly.polyvalfromroots(1, np.ones((3, 3))).shape == (3,)) + + # check normal input) + x = np.linspace(-1, 1) + y = [x**i for i in range(5)] + for i in range(1, 5): + tgt = y[i] + res = poly.polyvalfromroots(x, [0]*i) + assert_almost_equal(res, tgt) + tgt = x*(x - 1)*(x + 1) + res = poly.polyvalfromroots(x, [-1, 0, 1]) + assert_almost_equal(res, tgt) + + # check that shape is preserved + for i in range(3): + dims = [2]*i + x = np.zeros(dims) + assert_equal(poly.polyvalfromroots(x, [1]).shape, dims) + assert_equal(poly.polyvalfromroots(x, [1, 0]).shape, dims) + assert_equal(poly.polyvalfromroots(x, [1, 0, 0]).shape, dims) + + # check compatibility with factorization + ptest = [15, 2, -16, -2, 1] + r = poly.polyroots(ptest) + x = np.linspace(-1, 1) + assert_almost_equal(poly.polyval(x, ptest), + poly.polyvalfromroots(x, r)) + + # check multidimensional arrays of roots and values + # check tensor=False + rshape = (3, 5) + x = np.arange(-3, 2) + r = np.random.randint(-5, 5, size=rshape) + res = poly.polyvalfromroots(x, r, tensor=False) + tgt = np.empty(r.shape[1:]) + for ii in range(tgt.size): + tgt[ii] = poly.polyvalfromroots(x[ii], r[:, ii]) + assert_equal(res, tgt) + + # check tensor=True + x = np.vstack([x, 2*x]) + res = poly.polyvalfromroots(x, r, tensor=True) + tgt = np.empty(r.shape[1:] + x.shape) + for ii in range(r.shape[1]): + for jj in range(x.shape[0]): + tgt[ii, jj, :] = poly.polyvalfromroots(x[jj], r[:, ii]) + assert_equal(res, tgt) + + def test_polyval2d(self): + x1, x2, x3 = self.x + y1, y2, y3 = self.y + + #test exceptions + assert_raises_regex(ValueError, 'incompatible', + poly.polyval2d, x1, x2[:2], self.c2d) + + #test values + tgt = y1*y2 + res = poly.polyval2d(x1, x2, self.c2d) + assert_almost_equal(res, tgt) + + #test shape + z = np.ones((2, 3)) + res = poly.polyval2d(z, z, self.c2d) + assert_(res.shape == (2, 3)) + + def test_polyval3d(self): + x1, x2, x3 = self.x + y1, y2, y3 = self.y + + #test exceptions + assert_raises_regex(ValueError, 'incompatible', + poly.polyval3d, x1, x2, x3[:2], self.c3d) + + #test values + tgt = y1*y2*y3 + res = poly.polyval3d(x1, x2, x3, self.c3d) + assert_almost_equal(res, tgt) + + #test shape + z = np.ones((2, 3)) + res = poly.polyval3d(z, z, z, self.c3d) + assert_(res.shape == (2, 3)) + + def test_polygrid2d(self): + x1, x2, x3 = self.x + y1, y2, y3 = self.y + + #test values + tgt = np.einsum('i,j->ij', y1, y2) + res = poly.polygrid2d(x1, x2, self.c2d) + assert_almost_equal(res, tgt) + + #test shape + z = np.ones((2, 3)) + res = poly.polygrid2d(z, z, self.c2d) + assert_(res.shape == (2, 3)*2) + + def test_polygrid3d(self): + x1, x2, x3 = self.x + y1, y2, y3 = self.y + + #test values + tgt = np.einsum('i,j,k->ijk', y1, y2, y3) + res = poly.polygrid3d(x1, x2, x3, self.c3d) + assert_almost_equal(res, tgt) + + #test shape + z = np.ones((2, 3)) + res = poly.polygrid3d(z, z, z, self.c3d) + assert_(res.shape == (2, 3)*3) + + +class TestIntegral: + + def test_polyint(self): + # check exceptions + assert_raises(TypeError, poly.polyint, [0], .5) + assert_raises(ValueError, poly.polyint, [0], -1) + assert_raises(ValueError, poly.polyint, [0], 1, [0, 0]) + assert_raises(ValueError, poly.polyint, [0], lbnd=[0]) + assert_raises(ValueError, poly.polyint, [0], scl=[0]) + assert_raises(TypeError, poly.polyint, [0], axis=.5) + with assert_warns(DeprecationWarning): + poly.polyint([1, 1], 1.) + + # test integration of zero polynomial + for i in range(2, 5): + k = [0]*(i - 2) + [1] + res = poly.polyint([0], m=i, k=k) + assert_almost_equal(res, [0, 1]) + + # check single integration with integration constant + for i in range(5): + scl = i + 1 + pol = [0]*i + [1] + tgt = [i] + [0]*i + [1/scl] + res = poly.polyint(pol, m=1, k=[i]) + assert_almost_equal(trim(res), trim(tgt)) + + # check single integration with integration constant and lbnd + for i in range(5): + scl = i + 1 + pol = [0]*i + [1] + res = poly.polyint(pol, m=1, k=[i], lbnd=-1) + assert_almost_equal(poly.polyval(-1, res), i) + + # check single integration with integration constant and scaling + for i in range(5): + scl = i + 1 + pol = [0]*i + [1] + tgt = [i] + [0]*i + [2/scl] + res = poly.polyint(pol, m=1, k=[i], scl=2) + assert_almost_equal(trim(res), trim(tgt)) + + # check multiple integrations with default k + for i in range(5): + for j in range(2, 5): + pol = [0]*i + [1] + tgt = pol[:] + for k in range(j): + tgt = poly.polyint(tgt, m=1) + res = poly.polyint(pol, m=j) + assert_almost_equal(trim(res), trim(tgt)) + + # check multiple integrations with defined k + for i in range(5): + for j in range(2, 5): + pol = [0]*i + [1] + tgt = pol[:] + for k in range(j): + tgt = poly.polyint(tgt, m=1, k=[k]) + res = poly.polyint(pol, m=j, k=list(range(j))) + assert_almost_equal(trim(res), trim(tgt)) + + # check multiple integrations with lbnd + for i in range(5): + for j in range(2, 5): + pol = [0]*i + [1] + tgt = pol[:] + for k in range(j): + tgt = poly.polyint(tgt, m=1, k=[k], lbnd=-1) + res = poly.polyint(pol, m=j, k=list(range(j)), lbnd=-1) + assert_almost_equal(trim(res), trim(tgt)) + + # check multiple integrations with scaling + for i in range(5): + for j in range(2, 5): + pol = [0]*i + [1] + tgt = pol[:] + for k in range(j): + tgt = poly.polyint(tgt, m=1, k=[k], scl=2) + res = poly.polyint(pol, m=j, k=list(range(j)), scl=2) + assert_almost_equal(trim(res), trim(tgt)) + + def test_polyint_axis(self): + # check that axis keyword works + c2d = np.random.random((3, 4)) + + tgt = np.vstack([poly.polyint(c) for c in c2d.T]).T + res = poly.polyint(c2d, axis=0) + assert_almost_equal(res, tgt) + + tgt = np.vstack([poly.polyint(c) for c in c2d]) + res = poly.polyint(c2d, axis=1) + assert_almost_equal(res, tgt) + + tgt = np.vstack([poly.polyint(c, k=3) for c in c2d]) + res = poly.polyint(c2d, k=3, axis=1) + assert_almost_equal(res, tgt) + + +class TestDerivative: + + def test_polyder(self): + # check exceptions + assert_raises(TypeError, poly.polyder, [0], .5) + assert_raises(ValueError, poly.polyder, [0], -1) + + # check that zeroth derivative does nothing + for i in range(5): + tgt = [0]*i + [1] + res = poly.polyder(tgt, m=0) + assert_equal(trim(res), trim(tgt)) + + # check that derivation is the inverse of integration + for i in range(5): + for j in range(2, 5): + tgt = [0]*i + [1] + res = poly.polyder(poly.polyint(tgt, m=j), m=j) + assert_almost_equal(trim(res), trim(tgt)) + + # check derivation with scaling + for i in range(5): + for j in range(2, 5): + tgt = [0]*i + [1] + res = poly.polyder(poly.polyint(tgt, m=j, scl=2), m=j, scl=.5) + assert_almost_equal(trim(res), trim(tgt)) + + def test_polyder_axis(self): + # check that axis keyword works + c2d = np.random.random((3, 4)) + + tgt = np.vstack([poly.polyder(c) for c in c2d.T]).T + res = poly.polyder(c2d, axis=0) + assert_almost_equal(res, tgt) + + tgt = np.vstack([poly.polyder(c) for c in c2d]) + res = poly.polyder(c2d, axis=1) + assert_almost_equal(res, tgt) + + +class TestVander: + # some random values in [-1, 1) + x = np.random.random((3, 5))*2 - 1 + + def test_polyvander(self): + # check for 1d x + x = np.arange(3) + v = poly.polyvander(x, 3) + assert_(v.shape == (3, 4)) + for i in range(4): + coef = [0]*i + [1] + assert_almost_equal(v[..., i], poly.polyval(x, coef)) + + # check for 2d x + x = np.array([[1, 2], [3, 4], [5, 6]]) + v = poly.polyvander(x, 3) + assert_(v.shape == (3, 2, 4)) + for i in range(4): + coef = [0]*i + [1] + assert_almost_equal(v[..., i], poly.polyval(x, coef)) + + def test_polyvander2d(self): + # also tests polyval2d for non-square coefficient array + x1, x2, x3 = self.x + c = np.random.random((2, 3)) + van = poly.polyvander2d(x1, x2, [1, 2]) + tgt = poly.polyval2d(x1, x2, c) + res = np.dot(van, c.flat) + assert_almost_equal(res, tgt) + + # check shape + van = poly.polyvander2d([x1], [x2], [1, 2]) + assert_(van.shape == (1, 5, 6)) + + def test_polyvander3d(self): + # also tests polyval3d for non-square coefficient array + x1, x2, x3 = self.x + c = np.random.random((2, 3, 4)) + van = poly.polyvander3d(x1, x2, x3, [1, 2, 3]) + tgt = poly.polyval3d(x1, x2, x3, c) + res = np.dot(van, c.flat) + assert_almost_equal(res, tgt) + + # check shape + van = poly.polyvander3d([x1], [x2], [x3], [1, 2, 3]) + assert_(van.shape == (1, 5, 24)) + + def test_polyvandernegdeg(self): + x = np.arange(3) + assert_raises(ValueError, poly.polyvander, x, -1) + + +class TestCompanion: + + def test_raises(self): + assert_raises(ValueError, poly.polycompanion, []) + assert_raises(ValueError, poly.polycompanion, [1]) + + def test_dimensions(self): + for i in range(1, 5): + coef = [0]*i + [1] + assert_(poly.polycompanion(coef).shape == (i, i)) + + def test_linear_root(self): + assert_(poly.polycompanion([1, 2])[0, 0] == -.5) + + +class TestMisc: + + def test_polyfromroots(self): + res = poly.polyfromroots([]) + assert_almost_equal(trim(res), [1]) + for i in range(1, 5): + roots = np.cos(np.linspace(-np.pi, 0, 2*i + 1)[1::2]) + tgt = Tlist[i] + res = poly.polyfromroots(roots)*2**(i-1) + assert_almost_equal(trim(res), trim(tgt)) + + def test_polyroots(self): + assert_almost_equal(poly.polyroots([1]), []) + assert_almost_equal(poly.polyroots([1, 2]), [-.5]) + for i in range(2, 5): + tgt = np.linspace(-1, 1, i) + res = poly.polyroots(poly.polyfromroots(tgt)) + assert_almost_equal(trim(res), trim(tgt)) + + def test_polyfit(self): + def f(x): + return x*(x - 1)*(x - 2) + + def f2(x): + return x**4 + x**2 + 1 + + # Test exceptions + assert_raises(ValueError, poly.polyfit, [1], [1], -1) + assert_raises(TypeError, poly.polyfit, [[1]], [1], 0) + assert_raises(TypeError, poly.polyfit, [], [1], 0) + assert_raises(TypeError, poly.polyfit, [1], [[[1]]], 0) + assert_raises(TypeError, poly.polyfit, [1, 2], [1], 0) + assert_raises(TypeError, poly.polyfit, [1], [1, 2], 0) + assert_raises(TypeError, poly.polyfit, [1], [1], 0, w=[[1]]) + assert_raises(TypeError, poly.polyfit, [1], [1], 0, w=[1, 1]) + assert_raises(ValueError, poly.polyfit, [1], [1], [-1,]) + assert_raises(ValueError, poly.polyfit, [1], [1], [2, -1, 6]) + assert_raises(TypeError, poly.polyfit, [1], [1], []) + + # Test fit + x = np.linspace(0, 2) + y = f(x) + # + coef3 = poly.polyfit(x, y, 3) + assert_equal(len(coef3), 4) + assert_almost_equal(poly.polyval(x, coef3), y) + coef3 = poly.polyfit(x, y, [0, 1, 2, 3]) + assert_equal(len(coef3), 4) + assert_almost_equal(poly.polyval(x, coef3), y) + # + coef4 = poly.polyfit(x, y, 4) + assert_equal(len(coef4), 5) + assert_almost_equal(poly.polyval(x, coef4), y) + coef4 = poly.polyfit(x, y, [0, 1, 2, 3, 4]) + assert_equal(len(coef4), 5) + assert_almost_equal(poly.polyval(x, coef4), y) + # + coef2d = poly.polyfit(x, np.array([y, y]).T, 3) + assert_almost_equal(coef2d, np.array([coef3, coef3]).T) + coef2d = poly.polyfit(x, np.array([y, y]).T, [0, 1, 2, 3]) + assert_almost_equal(coef2d, np.array([coef3, coef3]).T) + # test weighting + w = np.zeros_like(x) + yw = y.copy() + w[1::2] = 1 + yw[0::2] = 0 + wcoef3 = poly.polyfit(x, yw, 3, w=w) + assert_almost_equal(wcoef3, coef3) + wcoef3 = poly.polyfit(x, yw, [0, 1, 2, 3], w=w) + assert_almost_equal(wcoef3, coef3) + # + wcoef2d = poly.polyfit(x, np.array([yw, yw]).T, 3, w=w) + assert_almost_equal(wcoef2d, np.array([coef3, coef3]).T) + wcoef2d = poly.polyfit(x, np.array([yw, yw]).T, [0, 1, 2, 3], w=w) + assert_almost_equal(wcoef2d, np.array([coef3, coef3]).T) + # test scaling with complex values x points whose square + # is zero when summed. + x = [1, 1j, -1, -1j] + assert_almost_equal(poly.polyfit(x, x, 1), [0, 1]) + assert_almost_equal(poly.polyfit(x, x, [0, 1]), [0, 1]) + # test fitting only even Polyendre polynomials + x = np.linspace(-1, 1) + y = f2(x) + coef1 = poly.polyfit(x, y, 4) + assert_almost_equal(poly.polyval(x, coef1), y) + coef2 = poly.polyfit(x, y, [0, 2, 4]) + assert_almost_equal(poly.polyval(x, coef2), y) + assert_almost_equal(coef1, coef2) + + def test_polytrim(self): + coef = [2, -1, 1, 0] + + # Test exceptions + assert_raises(ValueError, poly.polytrim, coef, -1) + + # Test results + assert_equal(poly.polytrim(coef), coef[:-1]) + assert_equal(poly.polytrim(coef, 1), coef[:-3]) + assert_equal(poly.polytrim(coef, 2), [0]) + + def test_polyline(self): + assert_equal(poly.polyline(3, 4), [3, 4]) + + def test_polyline_zero(self): + assert_equal(poly.polyline(3, 0), [3]) diff --git a/deepseekvl2/lib/python3.10/site-packages/numpy/polynomial/tests/test_polyutils.py b/deepseekvl2/lib/python3.10/site-packages/numpy/polynomial/tests/test_polyutils.py new file mode 100644 index 0000000000000000000000000000000000000000..cc630790da1ce8fd1ca413cd530ae5636cce5aa8 --- /dev/null +++ b/deepseekvl2/lib/python3.10/site-packages/numpy/polynomial/tests/test_polyutils.py @@ -0,0 +1,121 @@ +"""Tests for polyutils module. + +""" +import numpy as np +import numpy.polynomial.polyutils as pu +from numpy.testing import ( + assert_almost_equal, assert_raises, assert_equal, assert_, + ) + + +class TestMisc: + + def test_trimseq(self): + for i in range(5): + tgt = [1] + res = pu.trimseq([1] + [0]*5) + assert_equal(res, tgt) + + def test_as_series(self): + # check exceptions + assert_raises(ValueError, pu.as_series, [[]]) + assert_raises(ValueError, pu.as_series, [[[1, 2]]]) + assert_raises(ValueError, pu.as_series, [[1], ['a']]) + # check common types + types = ['i', 'd', 'O'] + for i in range(len(types)): + for j in range(i): + ci = np.ones(1, types[i]) + cj = np.ones(1, types[j]) + [resi, resj] = pu.as_series([ci, cj]) + assert_(resi.dtype.char == resj.dtype.char) + assert_(resj.dtype.char == types[i]) + + def test_trimcoef(self): + coef = [2, -1, 1, 0] + # Test exceptions + assert_raises(ValueError, pu.trimcoef, coef, -1) + # Test results + assert_equal(pu.trimcoef(coef), coef[:-1]) + assert_equal(pu.trimcoef(coef, 1), coef[:-3]) + assert_equal(pu.trimcoef(coef, 2), [0]) + + def test_vander_nd_exception(self): + # n_dims != len(points) + assert_raises(ValueError, pu._vander_nd, (), (1, 2, 3), [90]) + # n_dims != len(degrees) + assert_raises(ValueError, pu._vander_nd, (), (), [90.65]) + # n_dims == 0 + assert_raises(ValueError, pu._vander_nd, (), (), []) + + def test_div_zerodiv(self): + # c2[-1] == 0 + assert_raises(ZeroDivisionError, pu._div, pu._div, (1, 2, 3), [0]) + + def test_pow_too_large(self): + # power > maxpower + assert_raises(ValueError, pu._pow, (), [1, 2, 3], 5, 4) + +class TestDomain: + + def test_getdomain(self): + # test for real values + x = [1, 10, 3, -1] + tgt = [-1, 10] + res = pu.getdomain(x) + assert_almost_equal(res, tgt) + + # test for complex values + x = [1 + 1j, 1 - 1j, 0, 2] + tgt = [-1j, 2 + 1j] + res = pu.getdomain(x) + assert_almost_equal(res, tgt) + + def test_mapdomain(self): + # test for real values + dom1 = [0, 4] + dom2 = [1, 3] + tgt = dom2 + res = pu.mapdomain(dom1, dom1, dom2) + assert_almost_equal(res, tgt) + + # test for complex values + dom1 = [0 - 1j, 2 + 1j] + dom2 = [-2, 2] + tgt = dom2 + x = dom1 + res = pu.mapdomain(x, dom1, dom2) + assert_almost_equal(res, tgt) + + # test for multidimensional arrays + dom1 = [0, 4] + dom2 = [1, 3] + tgt = np.array([dom2, dom2]) + x = np.array([dom1, dom1]) + res = pu.mapdomain(x, dom1, dom2) + assert_almost_equal(res, tgt) + + # test that subtypes are preserved. + class MyNDArray(np.ndarray): + pass + + dom1 = [0, 4] + dom2 = [1, 3] + x = np.array([dom1, dom1]).view(MyNDArray) + res = pu.mapdomain(x, dom1, dom2) + assert_(isinstance(res, MyNDArray)) + + def test_mapparms(self): + # test for real values + dom1 = [0, 4] + dom2 = [1, 3] + tgt = [1, .5] + res = pu. mapparms(dom1, dom2) + assert_almost_equal(res, tgt) + + # test for complex values + dom1 = [0 - 1j, 2 + 1j] + dom2 = [-2, 2] + tgt = [-1 + 1j, 1 - 1j] + res = pu.mapparms(dom1, dom2) + assert_almost_equal(res, tgt) diff --git a/deepseekvl2/lib/python3.10/site-packages/numpy/polynomial/tests/test_printing.py b/deepseekvl2/lib/python3.10/site-packages/numpy/polynomial/tests/test_printing.py new file mode 100644 index 0000000000000000000000000000000000000000..6f2a5092d7225c797b60fd8f2602f2f9276cdd74 --- /dev/null +++ b/deepseekvl2/lib/python3.10/site-packages/numpy/polynomial/tests/test_printing.py @@ -0,0 +1,530 @@ +from math import nan, inf +import pytest +from numpy.core import array, arange, printoptions +import numpy.polynomial as poly +from numpy.testing import assert_equal, assert_ + +# For testing polynomial printing with object arrays +from fractions import Fraction +from decimal import Decimal + + +class TestStrUnicodeSuperSubscripts: + + @pytest.fixture(scope='class', autouse=True) + def use_unicode(self): + poly.set_default_printstyle('unicode') + + @pytest.mark.parametrize(('inp', 'tgt'), ( + ([1, 2, 3], "1.0 + 2.0·x + 3.0·x²"), + ([-1, 0, 3, -1], "-1.0 + 0.0·x + 3.0·x² - 1.0·x³"), + (arange(12), ("0.0 + 1.0·x + 2.0·x² + 3.0·x³ + 4.0·x⁴ + 5.0·x⁵ + " + "6.0·x⁶ + 7.0·x⁷ +\n8.0·x⁸ + 9.0·x⁹ + 10.0·x¹⁰ + " + "11.0·x¹¹")), + )) + def test_polynomial_str(self, inp, tgt): + res = str(poly.Polynomial(inp)) + assert_equal(res, tgt) + + @pytest.mark.parametrize(('inp', 'tgt'), ( + ([1, 2, 3], "1.0 + 2.0·T₁(x) + 3.0·T₂(x)"), + ([-1, 0, 3, -1], "-1.0 + 0.0·T₁(x) + 3.0·T₂(x) - 1.0·T₃(x)"), + (arange(12), ("0.0 + 1.0·T₁(x) + 2.0·T₂(x) + 3.0·T₃(x) + 4.0·T₄(x) + " + "5.0·T₅(x) +\n6.0·T₆(x) + 7.0·T₇(x) + 8.0·T₈(x) + " + "9.0·T₉(x) + 10.0·T₁₀(x) + 11.0·T₁₁(x)")), + )) + def test_chebyshev_str(self, inp, tgt): + res = str(poly.Chebyshev(inp)) + assert_equal(res, tgt) + + @pytest.mark.parametrize(('inp', 'tgt'), ( + ([1, 2, 3], "1.0 + 2.0·P₁(x) + 3.0·P₂(x)"), + ([-1, 0, 3, -1], "-1.0 + 0.0·P₁(x) + 3.0·P₂(x) - 1.0·P₃(x)"), + (arange(12), ("0.0 + 1.0·P₁(x) + 2.0·P₂(x) + 3.0·P₃(x) + 4.0·P₄(x) + " + "5.0·P₅(x) +\n6.0·P₆(x) + 7.0·P₇(x) + 8.0·P₈(x) + " + "9.0·P₉(x) + 10.0·P₁₀(x) + 11.0·P₁₁(x)")), + )) + def test_legendre_str(self, inp, tgt): + res = str(poly.Legendre(inp)) + assert_equal(res, tgt) + + @pytest.mark.parametrize(('inp', 'tgt'), ( + ([1, 2, 3], "1.0 + 2.0·H₁(x) + 3.0·H₂(x)"), + ([-1, 0, 3, -1], "-1.0 + 0.0·H₁(x) + 3.0·H₂(x) - 1.0·H₃(x)"), + (arange(12), ("0.0 + 1.0·H₁(x) + 2.0·H₂(x) + 3.0·H₃(x) + 4.0·H₄(x) + " + "5.0·H₅(x) +\n6.0·H₆(x) + 7.0·H₇(x) + 8.0·H₈(x) + " + "9.0·H₉(x) + 10.0·H₁₀(x) + 11.0·H₁₁(x)")), + )) + def test_hermite_str(self, inp, tgt): + res = str(poly.Hermite(inp)) + assert_equal(res, tgt) + + @pytest.mark.parametrize(('inp', 'tgt'), ( + ([1, 2, 3], "1.0 + 2.0·He₁(x) + 3.0·He₂(x)"), + ([-1, 0, 3, -1], "-1.0 + 0.0·He₁(x) + 3.0·He₂(x) - 1.0·He₃(x)"), + (arange(12), ("0.0 + 1.0·He₁(x) + 2.0·He₂(x) + 3.0·He₃(x) + " + "4.0·He₄(x) + 5.0·He₅(x) +\n6.0·He₆(x) + 7.0·He₇(x) + " + "8.0·He₈(x) + 9.0·He₉(x) + 10.0·He₁₀(x) +\n" + "11.0·He₁₁(x)")), + )) + def test_hermiteE_str(self, inp, tgt): + res = str(poly.HermiteE(inp)) + assert_equal(res, tgt) + + @pytest.mark.parametrize(('inp', 'tgt'), ( + ([1, 2, 3], "1.0 + 2.0·L₁(x) + 3.0·L₂(x)"), + ([-1, 0, 3, -1], "-1.0 + 0.0·L₁(x) + 3.0·L₂(x) - 1.0·L₃(x)"), + (arange(12), ("0.0 + 1.0·L₁(x) + 2.0·L₂(x) + 3.0·L₃(x) + 4.0·L₄(x) + " + "5.0·L₅(x) +\n6.0·L₆(x) + 7.0·L₇(x) + 8.0·L₈(x) + " + "9.0·L₉(x) + 10.0·L₁₀(x) + 11.0·L₁₁(x)")), + )) + def test_laguerre_str(self, inp, tgt): + res = str(poly.Laguerre(inp)) + assert_equal(res, tgt) + + +class TestStrAscii: + + @pytest.fixture(scope='class', autouse=True) + def use_ascii(self): + poly.set_default_printstyle('ascii') + + @pytest.mark.parametrize(('inp', 'tgt'), ( + ([1, 2, 3], "1.0 + 2.0 x + 3.0 x**2"), + ([-1, 0, 3, -1], "-1.0 + 0.0 x + 3.0 x**2 - 1.0 x**3"), + (arange(12), ("0.0 + 1.0 x + 2.0 x**2 + 3.0 x**3 + 4.0 x**4 + " + "5.0 x**5 + 6.0 x**6 +\n7.0 x**7 + 8.0 x**8 + " + "9.0 x**9 + 10.0 x**10 + 11.0 x**11")), + )) + def test_polynomial_str(self, inp, tgt): + res = str(poly.Polynomial(inp)) + assert_equal(res, tgt) + + @pytest.mark.parametrize(('inp', 'tgt'), ( + ([1, 2, 3], "1.0 + 2.0 T_1(x) + 3.0 T_2(x)"), + ([-1, 0, 3, -1], "-1.0 + 0.0 T_1(x) + 3.0 T_2(x) - 1.0 T_3(x)"), + (arange(12), ("0.0 + 1.0 T_1(x) + 2.0 T_2(x) + 3.0 T_3(x) + " + "4.0 T_4(x) + 5.0 T_5(x) +\n6.0 T_6(x) + 7.0 T_7(x) + " + "8.0 T_8(x) + 9.0 T_9(x) + 10.0 T_10(x) +\n" + "11.0 T_11(x)")), + )) + def test_chebyshev_str(self, inp, tgt): + res = str(poly.Chebyshev(inp)) + assert_equal(res, tgt) + + @pytest.mark.parametrize(('inp', 'tgt'), ( + ([1, 2, 3], "1.0 + 2.0 P_1(x) + 3.0 P_2(x)"), + ([-1, 0, 3, -1], "-1.0 + 0.0 P_1(x) + 3.0 P_2(x) - 1.0 P_3(x)"), + (arange(12), ("0.0 + 1.0 P_1(x) + 2.0 P_2(x) + 3.0 P_3(x) + " + "4.0 P_4(x) + 5.0 P_5(x) +\n6.0 P_6(x) + 7.0 P_7(x) + " + "8.0 P_8(x) + 9.0 P_9(x) + 10.0 P_10(x) +\n" + "11.0 P_11(x)")), + )) + def test_legendre_str(self, inp, tgt): + res = str(poly.Legendre(inp)) + assert_equal(res, tgt) + + @pytest.mark.parametrize(('inp', 'tgt'), ( + ([1, 2, 3], "1.0 + 2.0 H_1(x) + 3.0 H_2(x)"), + ([-1, 0, 3, -1], "-1.0 + 0.0 H_1(x) + 3.0 H_2(x) - 1.0 H_3(x)"), + (arange(12), ("0.0 + 1.0 H_1(x) + 2.0 H_2(x) + 3.0 H_3(x) + " + "4.0 H_4(x) + 5.0 H_5(x) +\n6.0 H_6(x) + 7.0 H_7(x) + " + "8.0 H_8(x) + 9.0 H_9(x) + 10.0 H_10(x) +\n" + "11.0 H_11(x)")), + )) + def test_hermite_str(self, inp, tgt): + res = str(poly.Hermite(inp)) + assert_equal(res, tgt) + + @pytest.mark.parametrize(('inp', 'tgt'), ( + ([1, 2, 3], "1.0 + 2.0 He_1(x) + 3.0 He_2(x)"), + ([-1, 0, 3, -1], "-1.0 + 0.0 He_1(x) + 3.0 He_2(x) - 1.0 He_3(x)"), + (arange(12), ("0.0 + 1.0 He_1(x) + 2.0 He_2(x) + 3.0 He_3(x) + " + "4.0 He_4(x) +\n5.0 He_5(x) + 6.0 He_6(x) + " + "7.0 He_7(x) + 8.0 He_8(x) + 9.0 He_9(x) +\n" + "10.0 He_10(x) + 11.0 He_11(x)")), + )) + def test_hermiteE_str(self, inp, tgt): + res = str(poly.HermiteE(inp)) + assert_equal(res, tgt) + + @pytest.mark.parametrize(('inp', 'tgt'), ( + ([1, 2, 3], "1.0 + 2.0 L_1(x) + 3.0 L_2(x)"), + ([-1, 0, 3, -1], "-1.0 + 0.0 L_1(x) + 3.0 L_2(x) - 1.0 L_3(x)"), + (arange(12), ("0.0 + 1.0 L_1(x) + 2.0 L_2(x) + 3.0 L_3(x) + " + "4.0 L_4(x) + 5.0 L_5(x) +\n6.0 L_6(x) + 7.0 L_7(x) + " + "8.0 L_8(x) + 9.0 L_9(x) + 10.0 L_10(x) +\n" + "11.0 L_11(x)")), + )) + def test_laguerre_str(self, inp, tgt): + res = str(poly.Laguerre(inp)) + assert_equal(res, tgt) + + +class TestLinebreaking: + + @pytest.fixture(scope='class', autouse=True) + def use_ascii(self): + poly.set_default_printstyle('ascii') + + def test_single_line_one_less(self): + # With 'ascii' style, len(str(p)) is default linewidth - 1 (i.e. 74) + p = poly.Polynomial([12345678, 12345678, 12345678, 12345678, 123]) + assert_equal(len(str(p)), 74) + assert_equal(str(p), ( + '12345678.0 + 12345678.0 x + 12345678.0 x**2 + ' + '12345678.0 x**3 + 123.0 x**4' + )) + + def test_num_chars_is_linewidth(self): + # len(str(p)) == default linewidth == 75 + p = poly.Polynomial([12345678, 12345678, 12345678, 12345678, 1234]) + assert_equal(len(str(p)), 75) + assert_equal(str(p), ( + '12345678.0 + 12345678.0 x + 12345678.0 x**2 + ' + '12345678.0 x**3 +\n1234.0 x**4' + )) + + def test_first_linebreak_multiline_one_less_than_linewidth(self): + # Multiline str where len(first_line) + len(next_term) == lw - 1 == 74 + p = poly.Polynomial( + [12345678, 12345678, 12345678, 12345678, 1, 12345678] + ) + assert_equal(len(str(p).split('\n')[0]), 74) + assert_equal(str(p), ( + '12345678.0 + 12345678.0 x + 12345678.0 x**2 + ' + '12345678.0 x**3 + 1.0 x**4 +\n12345678.0 x**5' + )) + + def test_first_linebreak_multiline_on_linewidth(self): + # First line is one character longer than previous test + p = poly.Polynomial( + [12345678, 12345678, 12345678, 12345678.12, 1, 12345678] + ) + assert_equal(str(p), ( + '12345678.0 + 12345678.0 x + 12345678.0 x**2 + ' + '12345678.12 x**3 +\n1.0 x**4 + 12345678.0 x**5' + )) + + @pytest.mark.parametrize(('lw', 'tgt'), ( + (75, ('0.0 + 10.0 x + 200.0 x**2 + 3000.0 x**3 + 40000.0 x**4 + ' + '500000.0 x**5 +\n600000.0 x**6 + 70000.0 x**7 + 8000.0 x**8 + ' + '900.0 x**9')), + (45, ('0.0 + 10.0 x + 200.0 x**2 + 3000.0 x**3 +\n40000.0 x**4 + ' + '500000.0 x**5 +\n600000.0 x**6 + 70000.0 x**7 + 8000.0 x**8 +\n' + '900.0 x**9')), + (132, ('0.0 + 10.0 x + 200.0 x**2 + 3000.0 x**3 + 40000.0 x**4 + ' + '500000.0 x**5 + 600000.0 x**6 + 70000.0 x**7 + 8000.0 x**8 + ' + '900.0 x**9')), + )) + def test_linewidth_printoption(self, lw, tgt): + p = poly.Polynomial( + [0, 10, 200, 3000, 40000, 500000, 600000, 70000, 8000, 900] + ) + with printoptions(linewidth=lw): + assert_equal(str(p), tgt) + for line in str(p).split('\n'): + assert_(len(line) < lw) + + +def test_set_default_printoptions(): + p = poly.Polynomial([1, 2, 3]) + c = poly.Chebyshev([1, 2, 3]) + poly.set_default_printstyle('ascii') + assert_equal(str(p), "1.0 + 2.0 x + 3.0 x**2") + assert_equal(str(c), "1.0 + 2.0 T_1(x) + 3.0 T_2(x)") + poly.set_default_printstyle('unicode') + assert_equal(str(p), "1.0 + 2.0·x + 3.0·x²") + assert_equal(str(c), "1.0 + 2.0·T₁(x) + 3.0·T₂(x)") + with pytest.raises(ValueError): + poly.set_default_printstyle('invalid_input') + + +def test_complex_coefficients(): + """Test both numpy and built-in complex.""" + coefs = [0+1j, 1+1j, -2+2j, 3+0j] + # numpy complex + p1 = poly.Polynomial(coefs) + # Python complex + p2 = poly.Polynomial(array(coefs, dtype=object)) + poly.set_default_printstyle('unicode') + assert_equal(str(p1), "1j + (1+1j)·x - (2-2j)·x² + (3+0j)·x³") + assert_equal(str(p2), "1j + (1+1j)·x + (-2+2j)·x² + (3+0j)·x³") + poly.set_default_printstyle('ascii') + assert_equal(str(p1), "1j + (1+1j) x - (2-2j) x**2 + (3+0j) x**3") + assert_equal(str(p2), "1j + (1+1j) x + (-2+2j) x**2 + (3+0j) x**3") + + +@pytest.mark.parametrize(('coefs', 'tgt'), ( + (array([Fraction(1, 2), Fraction(3, 4)], dtype=object), ( + "1/2 + 3/4·x" + )), + (array([1, 2, Fraction(5, 7)], dtype=object), ( + "1 + 2·x + 5/7·x²" + )), + (array([Decimal('1.00'), Decimal('2.2'), 3], dtype=object), ( + "1.00 + 2.2·x + 3·x²" + )), +)) +def test_numeric_object_coefficients(coefs, tgt): + p = poly.Polynomial(coefs) + poly.set_default_printstyle('unicode') + assert_equal(str(p), tgt) + + +@pytest.mark.parametrize(('coefs', 'tgt'), ( + (array([1, 2, 'f'], dtype=object), '1 + 2·x + f·x²'), + (array([1, 2, [3, 4]], dtype=object), '1 + 2·x + [3, 4]·x²'), +)) +def test_nonnumeric_object_coefficients(coefs, tgt): + """ + Test coef fallback for object arrays of non-numeric coefficients. + """ + p = poly.Polynomial(coefs) + poly.set_default_printstyle('unicode') + assert_equal(str(p), tgt) + + +class TestFormat: + def test_format_unicode(self): + poly.set_default_printstyle('ascii') + p = poly.Polynomial([1, 2, 0, -1]) + assert_equal(format(p, 'unicode'), "1.0 + 2.0·x + 0.0·x² - 1.0·x³") + + def test_format_ascii(self): + poly.set_default_printstyle('unicode') + p = poly.Polynomial([1, 2, 0, -1]) + assert_equal( + format(p, 'ascii'), "1.0 + 2.0 x + 0.0 x**2 - 1.0 x**3" + ) + + def test_empty_formatstr(self): + poly.set_default_printstyle('ascii') + p = poly.Polynomial([1, 2, 3]) + assert_equal(format(p), "1.0 + 2.0 x + 3.0 x**2") + assert_equal(f"{p}", "1.0 + 2.0 x + 3.0 x**2") + + def test_bad_formatstr(self): + p = poly.Polynomial([1, 2, 0, -1]) + with pytest.raises(ValueError): + format(p, '.2f') + + +@pytest.mark.parametrize(('poly', 'tgt'), ( + (poly.Polynomial, '1.0 + 2.0·z + 3.0·z²'), + (poly.Chebyshev, '1.0 + 2.0·T₁(z) + 3.0·T₂(z)'), + (poly.Hermite, '1.0 + 2.0·H₁(z) + 3.0·H₂(z)'), + (poly.HermiteE, '1.0 + 2.0·He₁(z) + 3.0·He₂(z)'), + (poly.Laguerre, '1.0 + 2.0·L₁(z) + 3.0·L₂(z)'), + (poly.Legendre, '1.0 + 2.0·P₁(z) + 3.0·P₂(z)'), +)) +def test_symbol(poly, tgt): + p = poly([1, 2, 3], symbol='z') + assert_equal(f"{p:unicode}", tgt) + + +class TestRepr: + def test_polynomial_str(self): + res = repr(poly.Polynomial([0, 1])) + tgt = ( + "Polynomial([0., 1.], domain=[-1, 1], window=[-1, 1], " + "symbol='x')" + ) + assert_equal(res, tgt) + + def test_chebyshev_str(self): + res = repr(poly.Chebyshev([0, 1])) + tgt = ( + "Chebyshev([0., 1.], domain=[-1, 1], window=[-1, 1], " + "symbol='x')" + ) + assert_equal(res, tgt) + + def test_legendre_repr(self): + res = repr(poly.Legendre([0, 1])) + tgt = ( + "Legendre([0., 1.], domain=[-1, 1], window=[-1, 1], " + "symbol='x')" + ) + assert_equal(res, tgt) + + def test_hermite_repr(self): + res = repr(poly.Hermite([0, 1])) + tgt = ( + "Hermite([0., 1.], domain=[-1, 1], window=[-1, 1], " + "symbol='x')" + ) + assert_equal(res, tgt) + + def test_hermiteE_repr(self): + res = repr(poly.HermiteE([0, 1])) + tgt = ( + "HermiteE([0., 1.], domain=[-1, 1], window=[-1, 1], " + "symbol='x')" + ) + assert_equal(res, tgt) + + def test_laguerre_repr(self): + res = repr(poly.Laguerre([0, 1])) + tgt = ( + "Laguerre([0., 1.], domain=[0, 1], window=[0, 1], " + "symbol='x')" + ) + assert_equal(res, tgt) + + +class TestLatexRepr: + """Test the latex repr used by Jupyter""" + + def as_latex(self, obj): + # right now we ignore the formatting of scalars in our tests, since + # it makes them too verbose. Ideally, the formatting of scalars will + # be fixed such that tests below continue to pass + obj._repr_latex_scalar = lambda x, parens=False: str(x) + try: + return obj._repr_latex_() + finally: + del obj._repr_latex_scalar + + def test_simple_polynomial(self): + # default input + p = poly.Polynomial([1, 2, 3]) + assert_equal(self.as_latex(p), + r'$x \mapsto 1.0 + 2.0\,x + 3.0\,x^{2}$') + + # translated input + p = poly.Polynomial([1, 2, 3], domain=[-2, 0]) + assert_equal(self.as_latex(p), + r'$x \mapsto 1.0 + 2.0\,\left(1.0 + x\right) + 3.0\,\left(1.0 + x\right)^{2}$') + + # scaled input + p = poly.Polynomial([1, 2, 3], domain=[-0.5, 0.5]) + assert_equal(self.as_latex(p), + r'$x \mapsto 1.0 + 2.0\,\left(2.0x\right) + 3.0\,\left(2.0x\right)^{2}$') + + # affine input + p = poly.Polynomial([1, 2, 3], domain=[-1, 0]) + assert_equal(self.as_latex(p), + r'$x \mapsto 1.0 + 2.0\,\left(1.0 + 2.0x\right) + 3.0\,\left(1.0 + 2.0x\right)^{2}$') + + def test_basis_func(self): + p = poly.Chebyshev([1, 2, 3]) + assert_equal(self.as_latex(p), + r'$x \mapsto 1.0\,{T}_{0}(x) + 2.0\,{T}_{1}(x) + 3.0\,{T}_{2}(x)$') + # affine input - check no surplus parens are added + p = poly.Chebyshev([1, 2, 3], domain=[-1, 0]) + assert_equal(self.as_latex(p), + r'$x \mapsto 1.0\,{T}_{0}(1.0 + 2.0x) + 2.0\,{T}_{1}(1.0 + 2.0x) + 3.0\,{T}_{2}(1.0 + 2.0x)$') + + def test_multichar_basis_func(self): + p = poly.HermiteE([1, 2, 3]) + assert_equal(self.as_latex(p), + r'$x \mapsto 1.0\,{He}_{0}(x) + 2.0\,{He}_{1}(x) + 3.0\,{He}_{2}(x)$') + + def test_symbol_basic(self): + # default input + p = poly.Polynomial([1, 2, 3], symbol='z') + assert_equal(self.as_latex(p), + r'$z \mapsto 1.0 + 2.0\,z + 3.0\,z^{2}$') + + # translated input + p = poly.Polynomial([1, 2, 3], domain=[-2, 0], symbol='z') + assert_equal( + self.as_latex(p), + ( + r'$z \mapsto 1.0 + 2.0\,\left(1.0 + z\right) + 3.0\,' + r'\left(1.0 + z\right)^{2}$' + ), + ) + + # scaled input + p = poly.Polynomial([1, 2, 3], domain=[-0.5, 0.5], symbol='z') + assert_equal( + self.as_latex(p), + ( + r'$z \mapsto 1.0 + 2.0\,\left(2.0z\right) + 3.0\,' + r'\left(2.0z\right)^{2}$' + ), + ) + + # affine input + p = poly.Polynomial([1, 2, 3], domain=[-1, 0], symbol='z') + assert_equal( + self.as_latex(p), + ( + r'$z \mapsto 1.0 + 2.0\,\left(1.0 + 2.0z\right) + 3.0\,' + r'\left(1.0 + 2.0z\right)^{2}$' + ), + ) + + +SWITCH_TO_EXP = ( + '1.0 + (1.0e-01) x + (1.0e-02) x**2', + '1.2 + (1.2e-01) x + (1.2e-02) x**2', + '1.23 + 0.12 x + (1.23e-02) x**2 + (1.23e-03) x**3', + '1.235 + 0.123 x + (1.235e-02) x**2 + (1.235e-03) x**3', + '1.2346 + 0.1235 x + 0.0123 x**2 + (1.2346e-03) x**3 + (1.2346e-04) x**4', + '1.23457 + 0.12346 x + 0.01235 x**2 + (1.23457e-03) x**3 + ' + '(1.23457e-04) x**4', + '1.234568 + 0.123457 x + 0.012346 x**2 + 0.001235 x**3 + ' + '(1.234568e-04) x**4 + (1.234568e-05) x**5', + '1.2345679 + 0.1234568 x + 0.0123457 x**2 + 0.0012346 x**3 + ' + '(1.2345679e-04) x**4 + (1.2345679e-05) x**5') + +class TestPrintOptions: + """ + Test the output is properly configured via printoptions. + The exponential notation is enabled automatically when the values + are too small or too large. + """ + + @pytest.fixture(scope='class', autouse=True) + def use_ascii(self): + poly.set_default_printstyle('ascii') + + def test_str(self): + p = poly.Polynomial([1/2, 1/7, 1/7*10**8, 1/7*10**9]) + assert_equal(str(p), '0.5 + 0.14285714 x + 14285714.28571429 x**2 ' + '+ (1.42857143e+08) x**3') + + with printoptions(precision=3): + assert_equal(str(p), '0.5 + 0.143 x + 14285714.286 x**2 ' + '+ (1.429e+08) x**3') + + def test_latex(self): + p = poly.Polynomial([1/2, 1/7, 1/7*10**8, 1/7*10**9]) + assert_equal(p._repr_latex_(), + r'$x \mapsto \text{0.5} + \text{0.14285714}\,x + ' + r'\text{14285714.28571429}\,x^{2} + ' + r'\text{(1.42857143e+08)}\,x^{3}$') + + with printoptions(precision=3): + assert_equal(p._repr_latex_(), + r'$x \mapsto \text{0.5} + \text{0.143}\,x + ' + r'\text{14285714.286}\,x^{2} + \text{(1.429e+08)}\,x^{3}$') + + def test_fixed(self): + p = poly.Polynomial([1/2]) + assert_equal(str(p), '0.5') + + with printoptions(floatmode='fixed'): + assert_equal(str(p), '0.50000000') + + with printoptions(floatmode='fixed', precision=4): + assert_equal(str(p), '0.5000') + + def test_switch_to_exp(self): + for i, s in enumerate(SWITCH_TO_EXP): + with printoptions(precision=i): + p = poly.Polynomial([1.23456789*10**-i + for i in range(i//2+3)]) + assert str(p).replace('\n', ' ') == s + + def test_non_finite(self): + p = poly.Polynomial([nan, inf]) + assert str(p) == 'nan + inf x' + assert p._repr_latex_() == r'$x \mapsto \text{nan} + \text{inf}\,x$' + with printoptions(nanstr='NAN', infstr='INF'): + assert str(p) == 'NAN + INF x' + assert p._repr_latex_() == \ + r'$x \mapsto \text{NAN} + \text{INF}\,x$' diff --git a/deepseekvl2/lib/python3.10/site-packages/numpy/polynomial/tests/test_symbol.py b/deepseekvl2/lib/python3.10/site-packages/numpy/polynomial/tests/test_symbol.py new file mode 100644 index 0000000000000000000000000000000000000000..4ea6035ef7a75e6807634ba894e42015c83edb7d --- /dev/null +++ b/deepseekvl2/lib/python3.10/site-packages/numpy/polynomial/tests/test_symbol.py @@ -0,0 +1,216 @@ +""" +Tests related to the ``symbol`` attribute of the ABCPolyBase class. +""" + +import pytest +import numpy.polynomial as poly +from numpy.core import array +from numpy.testing import assert_equal, assert_raises, assert_ + + +class TestInit: + """ + Test polynomial creation with symbol kwarg. + """ + c = [1, 2, 3] + + def test_default_symbol(self): + p = poly.Polynomial(self.c) + assert_equal(p.symbol, 'x') + + @pytest.mark.parametrize(('bad_input', 'exception'), ( + ('', ValueError), + ('3', ValueError), + (None, TypeError), + (1, TypeError), + )) + def test_symbol_bad_input(self, bad_input, exception): + with pytest.raises(exception): + p = poly.Polynomial(self.c, symbol=bad_input) + + @pytest.mark.parametrize('symbol', ( + 'x', + 'x_1', + 'A', + 'xyz', + 'β', + )) + def test_valid_symbols(self, symbol): + """ + Values for symbol that should pass input validation. + """ + p = poly.Polynomial(self.c, symbol=symbol) + assert_equal(p.symbol, symbol) + + def test_property(self): + """ + 'symbol' attribute is read only. + """ + p = poly.Polynomial(self.c, symbol='x') + with pytest.raises(AttributeError): + p.symbol = 'z' + + def test_change_symbol(self): + p = poly.Polynomial(self.c, symbol='y') + # Create new polynomial from p with different symbol + pt = poly.Polynomial(p.coef, symbol='t') + assert_equal(pt.symbol, 't') + + +class TestUnaryOperators: + p = poly.Polynomial([1, 2, 3], symbol='z') + + def test_neg(self): + n = -self.p + assert_equal(n.symbol, 'z') + + def test_scalarmul(self): + out = self.p * 10 + assert_equal(out.symbol, 'z') + + def test_rscalarmul(self): + out = 10 * self.p + assert_equal(out.symbol, 'z') + + def test_pow(self): + out = self.p ** 3 + assert_equal(out.symbol, 'z') + + +@pytest.mark.parametrize( + 'rhs', + ( + poly.Polynomial([4, 5, 6], symbol='z'), + array([4, 5, 6]), + ), +) +class TestBinaryOperatorsSameSymbol: + """ + Ensure symbol is preserved for numeric operations on polynomials with + the same symbol + """ + p = poly.Polynomial([1, 2, 3], symbol='z') + + def test_add(self, rhs): + out = self.p + rhs + assert_equal(out.symbol, 'z') + + def test_sub(self, rhs): + out = self.p - rhs + assert_equal(out.symbol, 'z') + + def test_polymul(self, rhs): + out = self.p * rhs + assert_equal(out.symbol, 'z') + + def test_divmod(self, rhs): + for out in divmod(self.p, rhs): + assert_equal(out.symbol, 'z') + + def test_radd(self, rhs): + out = rhs + self.p + assert_equal(out.symbol, 'z') + + def test_rsub(self, rhs): + out = rhs - self.p + assert_equal(out.symbol, 'z') + + def test_rmul(self, rhs): + out = rhs * self.p + assert_equal(out.symbol, 'z') + + def test_rdivmod(self, rhs): + for out in divmod(rhs, self.p): + assert_equal(out.symbol, 'z') + + +class TestBinaryOperatorsDifferentSymbol: + p = poly.Polynomial([1, 2, 3], symbol='x') + other = poly.Polynomial([4, 5, 6], symbol='y') + ops = (p.__add__, p.__sub__, p.__mul__, p.__floordiv__, p.__mod__) + + @pytest.mark.parametrize('f', ops) + def test_binops_fails(self, f): + assert_raises(ValueError, f, self.other) + + +class TestEquality: + p = poly.Polynomial([1, 2, 3], symbol='x') + + def test_eq(self): + other = poly.Polynomial([1, 2, 3], symbol='x') + assert_(self.p == other) + + def test_neq(self): + other = poly.Polynomial([1, 2, 3], symbol='y') + assert_(not self.p == other) + + +class TestExtraMethods: + """ + Test other methods for manipulating/creating polynomial objects. + """ + p = poly.Polynomial([1, 2, 3, 0], symbol='z') + + def test_copy(self): + other = self.p.copy() + assert_equal(other.symbol, 'z') + + def test_trim(self): + other = self.p.trim() + assert_equal(other.symbol, 'z') + + def test_truncate(self): + other = self.p.truncate(2) + assert_equal(other.symbol, 'z') + + @pytest.mark.parametrize('kwarg', ( + {'domain': [-10, 10]}, + {'window': [-10, 10]}, + {'kind': poly.Chebyshev}, + )) + def test_convert(self, kwarg): + other = self.p.convert(**kwarg) + assert_equal(other.symbol, 'z') + + def test_integ(self): + other = self.p.integ() + assert_equal(other.symbol, 'z') + + def test_deriv(self): + other = self.p.deriv() + assert_equal(other.symbol, 'z') + + +def test_composition(): + p = poly.Polynomial([3, 2, 1], symbol="t") + q = poly.Polynomial([5, 1, 0, -1], symbol="λ_1") + r = p(q) + assert r.symbol == "λ_1" + + +# +# Class methods that result in new polynomial class instances +# + + +def test_fit(): + x, y = (range(10),)*2 + p = poly.Polynomial.fit(x, y, deg=1, symbol='z') + assert_equal(p.symbol, 'z') + + +def test_froomroots(): + roots = [-2, 2] + p = poly.Polynomial.fromroots(roots, symbol='z') + assert_equal(p.symbol, 'z') + + +def test_identity(): + p = poly.Polynomial.identity(domain=[-1, 1], window=[5, 20], symbol='z') + assert_equal(p.symbol, 'z') + + +def test_basis(): + p = poly.Polynomial.basis(3, symbol='z') + assert_equal(p.symbol, 'z') diff --git a/falcon/lib/python3.10/site-packages/nvidia/cusolver/lib/libcusolver.so.11 b/falcon/lib/python3.10/site-packages/nvidia/cusolver/lib/libcusolver.so.11 new file mode 100644 index 0000000000000000000000000000000000000000..2193754f8bc1f26d3a1100ed3527afc6bfc39cfb --- /dev/null +++ b/falcon/lib/python3.10/site-packages/nvidia/cusolver/lib/libcusolver.so.11 @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:10287abc7ce9c5fc7e7c163761566b599eae1b362c47e1000911d198443d6d52 +size 114481816 diff --git a/infer_4_33_0/lib/python3.10/site-packages/wandb/vendor/promise-2.3.0/wandb_promise/__init__.py b/infer_4_33_0/lib/python3.10/site-packages/wandb/vendor/promise-2.3.0/wandb_promise/__init__.py new file mode 100644 index 0000000000000000000000000000000000000000..ed495d0fda9acc8ed0cb6a94ec93485d68781b9e --- /dev/null +++ b/infer_4_33_0/lib/python3.10/site-packages/wandb/vendor/promise-2.3.0/wandb_promise/__init__.py @@ -0,0 +1,38 @@ +from .pyutils.version import get_version + + +try: + # This variable is injected in the __builtins__ by the build + # process. It used to enable importing subpackages when + # the required packages are not installed + __SETUP__ # type: ignore +except NameError: + __SETUP__ = False + + +VERSION = (2, 3, 0, "final", 0) + +__version__ = get_version(VERSION) + +if not __SETUP__: + from .promise import ( + Promise, + promise_for_dict, + promisify, + is_thenable, + async_instance, + get_default_scheduler, + set_default_scheduler, + ) + from .schedulers.immediate import ImmediateScheduler + + __all__ = [ + "Promise", + "promise_for_dict", + "promisify", + "is_thenable", + "async_instance", + "get_default_scheduler", + "set_default_scheduler", + "ImmediateScheduler", + ] diff --git a/infer_4_33_0/lib/python3.10/site-packages/wandb/vendor/promise-2.3.0/wandb_promise/__pycache__/async_.cpython-310.pyc b/infer_4_33_0/lib/python3.10/site-packages/wandb/vendor/promise-2.3.0/wandb_promise/__pycache__/async_.cpython-310.pyc new file mode 100644 index 0000000000000000000000000000000000000000..95633f2f468a1b83a724e8b3cca933704f338181 Binary files /dev/null and b/infer_4_33_0/lib/python3.10/site-packages/wandb/vendor/promise-2.3.0/wandb_promise/__pycache__/async_.cpython-310.pyc differ diff --git a/infer_4_33_0/lib/python3.10/site-packages/wandb/vendor/promise-2.3.0/wandb_promise/compat.py b/infer_4_33_0/lib/python3.10/site-packages/wandb/vendor/promise-2.3.0/wandb_promise/compat.py new file mode 100644 index 0000000000000000000000000000000000000000..21b091989858e0588a4dc5e1ad10baa677cd7e0b --- /dev/null +++ b/infer_4_33_0/lib/python3.10/site-packages/wandb/vendor/promise-2.3.0/wandb_promise/compat.py @@ -0,0 +1,32 @@ +try: + from inspect import iscoroutine +except ImportError: + + def iscoroutine(obj): # type: ignore + return False + + +try: + from asyncio import Future, ensure_future # type: ignore +except ImportError: + + class Future: # type: ignore + def __init__(self): + raise Exception("You need asyncio for using Futures") + + def set_result(self): + raise Exception("You need asyncio for using Futures") + + def set_exception(self): + raise Exception("You need asyncio for using Futures") + + def ensure_future(): # type: ignore + raise Exception("ensure_future needs asyncio for executing") + + +try: + from .iterate_promise import iterate_promise +except (SyntaxError, ImportError): + + def iterate_promise(promise): # type: ignore + raise Exception('You need "yield from" syntax for iterate in a Promise.') diff --git a/infer_4_33_0/lib/python3.10/site-packages/wandb/vendor/promise-2.3.0/wandb_promise/schedulers/__pycache__/gevent.cpython-310.pyc b/infer_4_33_0/lib/python3.10/site-packages/wandb/vendor/promise-2.3.0/wandb_promise/schedulers/__pycache__/gevent.cpython-310.pyc new file mode 100644 index 0000000000000000000000000000000000000000..614c3fe22d5dc3ad17f261cb02069c32d0c4f3ab Binary files /dev/null and b/infer_4_33_0/lib/python3.10/site-packages/wandb/vendor/promise-2.3.0/wandb_promise/schedulers/__pycache__/gevent.cpython-310.pyc differ diff --git a/infer_4_33_0/lib/python3.10/site-packages/wandb/vendor/promise-2.3.0/wandb_promise/schedulers/__pycache__/immediate.cpython-310.pyc b/infer_4_33_0/lib/python3.10/site-packages/wandb/vendor/promise-2.3.0/wandb_promise/schedulers/__pycache__/immediate.cpython-310.pyc new file mode 100644 index 0000000000000000000000000000000000000000..3bab31701541a4a86b03dceeff5171f4fb57a2d2 Binary files /dev/null and b/infer_4_33_0/lib/python3.10/site-packages/wandb/vendor/promise-2.3.0/wandb_promise/schedulers/__pycache__/immediate.cpython-310.pyc differ diff --git a/infer_4_33_0/lib/python3.10/site-packages/wandb/vendor/promise-2.3.0/wandb_promise/schedulers/gevent.py b/infer_4_33_0/lib/python3.10/site-packages/wandb/vendor/promise-2.3.0/wandb_promise/schedulers/gevent.py new file mode 100644 index 0000000000000000000000000000000000000000..6b84c4c8b579f9c6c663161ed3e7a5c6d6d83dac --- /dev/null +++ b/infer_4_33_0/lib/python3.10/site-packages/wandb/vendor/promise-2.3.0/wandb_promise/schedulers/gevent.py @@ -0,0 +1,21 @@ +from __future__ import absolute_import + +from gevent.event import Event # type: ignore +import gevent # type: ignore + + +class GeventScheduler(object): + def call(self, fn): + # print fn + gevent.spawn(fn) + + def wait(self, promise, timeout=None): + e = Event() + + def on_resolve_or_reject(_): + e.set() + + promise._then(on_resolve_or_reject, on_resolve_or_reject) + waited = e.wait(timeout) + if not waited: + raise Exception("Timeout") diff --git a/infer_4_33_0/lib/python3.10/site-packages/wandb/vendor/promise-2.3.0/wandb_promise/schedulers/immediate.py b/infer_4_33_0/lib/python3.10/site-packages/wandb/vendor/promise-2.3.0/wandb_promise/schedulers/immediate.py new file mode 100644 index 0000000000000000000000000000000000000000..5e1dd4c11a1fb198e63e90eaa7f784544effebb8 --- /dev/null +++ b/infer_4_33_0/lib/python3.10/site-packages/wandb/vendor/promise-2.3.0/wandb_promise/schedulers/immediate.py @@ -0,0 +1,27 @@ +from threading import Event + +if False: + from ..promise import Promise + from typing import Callable, Any, Optional # flake8: noqa + + +class ImmediateScheduler(object): + def call(self, fn): + # type: (Callable) -> None + try: + fn() + except: + pass + + def wait(self, promise, timeout=None): + # type: (Promise, Optional[float]) -> None + e = Event() + + def on_resolve_or_reject(_): + # type: (Any) -> None + e.set() + + promise._then(on_resolve_or_reject, on_resolve_or_reject) + waited = e.wait(timeout) + if not waited: + raise Exception("Timeout") diff --git a/infer_4_33_0/lib/python3.10/site-packages/wandb/vendor/promise-2.3.0/wandb_promise/utils.py b/infer_4_33_0/lib/python3.10/site-packages/wandb/vendor/promise-2.3.0/wandb_promise/utils.py new file mode 100644 index 0000000000000000000000000000000000000000..ca695776f5b1913993dd958097dc23c6f8f4e9c2 --- /dev/null +++ b/infer_4_33_0/lib/python3.10/site-packages/wandb/vendor/promise-2.3.0/wandb_promise/utils.py @@ -0,0 +1,56 @@ +import functools +import inspect +import types +import warnings +import sys + + +def warn(msg): + # type: (str) -> None + warnings.simplefilter("always", DeprecationWarning) # turn off filter + warnings.warn(msg, category=DeprecationWarning, stacklevel=2) + warnings.simplefilter("default", DeprecationWarning) # reset filter + + +class deprecated(object): + def __init__(self, reason, name=None): + if inspect.isclass(reason) or inspect.isfunction(reason): + raise TypeError("Reason for deprecation must be supplied") + self.reason = reason + self.name = name + + def __call__(self, cls_or_func): + if inspect.isfunction(cls_or_func): + fmt = "Call to deprecated function or method {name} ({reason})." + + elif inspect.isclass(cls_or_func): + fmt = "Call to deprecated class {name} ({reason})." + + else: + raise TypeError(type(cls_or_func)) + + msg = fmt.format(name=self.name or cls_or_func.__name__, reason=self.reason) + + @functools.wraps(cls_or_func) + def new_func(*args, **kwargs): + warn(msg) + return cls_or_func(*args, **kwargs) + + return new_func + + +PY2 = sys.version_info[0] == 2 +PY3 = sys.version_info[0] == 3 + +if PY3: + string_types = (str,) # type: tuple + integer_types = (int,) # type: tuple + class_types = (type,) # type: tuple + text_type = str + binary_type = bytes +else: + string_types = (basestring,) # type: tuple + integer_types = (int, long) # type: tuple + class_types = (type, types.ClassType) # type: tuple + text_type = unicode + binary_type = str diff --git a/infer_4_33_0/lib/python3.10/site-packages/wandb/vendor/watchdog_0_9_0/wandb_watchdog/__pycache__/__init__.cpython-310.pyc b/infer_4_33_0/lib/python3.10/site-packages/wandb/vendor/watchdog_0_9_0/wandb_watchdog/__pycache__/__init__.cpython-310.pyc new file mode 100644 index 0000000000000000000000000000000000000000..e2a56f579aee16b3c63f573fb9aa4117a031beab Binary files /dev/null and b/infer_4_33_0/lib/python3.10/site-packages/wandb/vendor/watchdog_0_9_0/wandb_watchdog/__pycache__/__init__.cpython-310.pyc differ diff --git a/infer_4_33_0/lib/python3.10/site-packages/wandb/vendor/watchdog_0_9_0/wandb_watchdog/__pycache__/events.cpython-310.pyc b/infer_4_33_0/lib/python3.10/site-packages/wandb/vendor/watchdog_0_9_0/wandb_watchdog/__pycache__/events.cpython-310.pyc new file mode 100644 index 0000000000000000000000000000000000000000..6d698b4204eff95a08479979286231f363a3499a Binary files /dev/null and b/infer_4_33_0/lib/python3.10/site-packages/wandb/vendor/watchdog_0_9_0/wandb_watchdog/__pycache__/events.cpython-310.pyc differ diff --git a/infer_4_33_0/lib/python3.10/site-packages/wandb/vendor/watchdog_0_9_0/wandb_watchdog/__pycache__/patterns.cpython-310.pyc b/infer_4_33_0/lib/python3.10/site-packages/wandb/vendor/watchdog_0_9_0/wandb_watchdog/__pycache__/patterns.cpython-310.pyc new file mode 100644 index 0000000000000000000000000000000000000000..d78be1f9cdbe88c3e3bb20f0f645f5536c851635 Binary files /dev/null and b/infer_4_33_0/lib/python3.10/site-packages/wandb/vendor/watchdog_0_9_0/wandb_watchdog/__pycache__/patterns.cpython-310.pyc differ diff --git a/infer_4_33_0/lib/python3.10/site-packages/wandb/vendor/watchdog_0_9_0/wandb_watchdog/__pycache__/version.cpython-310.pyc b/infer_4_33_0/lib/python3.10/site-packages/wandb/vendor/watchdog_0_9_0/wandb_watchdog/__pycache__/version.cpython-310.pyc new file mode 100644 index 0000000000000000000000000000000000000000..691b7387a096c092c48875573d67f01427be89a0 Binary files /dev/null and b/infer_4_33_0/lib/python3.10/site-packages/wandb/vendor/watchdog_0_9_0/wandb_watchdog/__pycache__/version.cpython-310.pyc differ diff --git a/infer_4_33_0/lib/python3.10/site-packages/wandb/vendor/watchdog_0_9_0/wandb_watchdog/__pycache__/watchmedo.cpython-310.pyc b/infer_4_33_0/lib/python3.10/site-packages/wandb/vendor/watchdog_0_9_0/wandb_watchdog/__pycache__/watchmedo.cpython-310.pyc new file mode 100644 index 0000000000000000000000000000000000000000..7bf7005b9f9c5564b8b2e56bc32ae0fa6f7cec7c Binary files /dev/null and b/infer_4_33_0/lib/python3.10/site-packages/wandb/vendor/watchdog_0_9_0/wandb_watchdog/__pycache__/watchmedo.cpython-310.pyc differ diff --git a/infer_4_33_0/lib/python3.10/site-packages/wandb/vendor/watchdog_0_9_0/wandb_watchdog/observers/__pycache__/fsevents2.cpython-310.pyc b/infer_4_33_0/lib/python3.10/site-packages/wandb/vendor/watchdog_0_9_0/wandb_watchdog/observers/__pycache__/fsevents2.cpython-310.pyc new file mode 100644 index 0000000000000000000000000000000000000000..f7b9d4e5bf210c9be6dfa058222e74ad605782e5 Binary files /dev/null and b/infer_4_33_0/lib/python3.10/site-packages/wandb/vendor/watchdog_0_9_0/wandb_watchdog/observers/__pycache__/fsevents2.cpython-310.pyc differ diff --git a/infer_4_33_0/lib/python3.10/site-packages/wandb/vendor/watchdog_0_9_0/wandb_watchdog/observers/__pycache__/inotify_c.cpython-310.pyc b/infer_4_33_0/lib/python3.10/site-packages/wandb/vendor/watchdog_0_9_0/wandb_watchdog/observers/__pycache__/inotify_c.cpython-310.pyc new file mode 100644 index 0000000000000000000000000000000000000000..c4d4be3fd5b3e93440c9e6347507f6fe83d5205b Binary files /dev/null and b/infer_4_33_0/lib/python3.10/site-packages/wandb/vendor/watchdog_0_9_0/wandb_watchdog/observers/__pycache__/inotify_c.cpython-310.pyc differ diff --git a/infer_4_33_0/lib/python3.10/site-packages/wandb/vendor/watchdog_0_9_0/wandb_watchdog/observers/__pycache__/read_directory_changes.cpython-310.pyc b/infer_4_33_0/lib/python3.10/site-packages/wandb/vendor/watchdog_0_9_0/wandb_watchdog/observers/__pycache__/read_directory_changes.cpython-310.pyc new file mode 100644 index 0000000000000000000000000000000000000000..d396c8413f72452b48d88964fc8de19b4735380d Binary files /dev/null and b/infer_4_33_0/lib/python3.10/site-packages/wandb/vendor/watchdog_0_9_0/wandb_watchdog/observers/__pycache__/read_directory_changes.cpython-310.pyc differ diff --git a/infer_4_33_0/lib/python3.10/site-packages/wandb/vendor/watchdog_0_9_0/wandb_watchdog/observers/fsevents.py b/infer_4_33_0/lib/python3.10/site-packages/wandb/vendor/watchdog_0_9_0/wandb_watchdog/observers/fsevents.py new file mode 100644 index 0000000000000000000000000000000000000000..3b7f007ae1e70315853cbfb1473f40a7424a52bc --- /dev/null +++ b/infer_4_33_0/lib/python3.10/site-packages/wandb/vendor/watchdog_0_9_0/wandb_watchdog/observers/fsevents.py @@ -0,0 +1,172 @@ +#!/usr/bin/env python +# -*- coding: utf-8 -*- +# +# Copyright 2011 Yesudeep Mangalapilly +# Copyright 2012 Google, Inc. +# +# Licensed under the Apache License, Version 2.0 (the "License"); +# you may not use this file except in compliance with the License. +# You may obtain a copy of the License at +# +# http://www.apache.org/licenses/LICENSE-2.0 +# +# Unless required by applicable law or agreed to in writing, software +# distributed under the License is distributed on an "AS IS" BASIS, +# WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied. +# See the License for the specific language governing permissions and +# limitations under the License. + +""" +:module: watchdog.observers.fsevents +:synopsis: FSEvents based emitter implementation. +:author: yesudeep@google.com (Yesudeep Mangalapilly) +:platforms: Mac OS X +""" + +from __future__ import with_statement + +import sys +import threading +import unicodedata +import _watchdog_fsevents as _fsevents + +from wandb_watchdog.events import ( + FileDeletedEvent, + FileModifiedEvent, + FileCreatedEvent, + FileMovedEvent, + DirDeletedEvent, + DirModifiedEvent, + DirCreatedEvent, + DirMovedEvent +) + +from wandb_watchdog.utils.dirsnapshot import DirectorySnapshot +from wandb_watchdog.observers.api import ( + BaseObserver, + EventEmitter, + DEFAULT_EMITTER_TIMEOUT, + DEFAULT_OBSERVER_TIMEOUT +) + + +class FSEventsEmitter(EventEmitter): + + """ + Mac OS X FSEvents Emitter class. + + :param event_queue: + The event queue to fill with events. + :param watch: + A watch object representing the directory to monitor. + :type watch: + :class:`watchdog.observers.api.ObservedWatch` + :param timeout: + Read events blocking timeout (in seconds). + :type timeout: + ``float`` + """ + + def __init__(self, event_queue, watch, timeout=DEFAULT_EMITTER_TIMEOUT): + EventEmitter.__init__(self, event_queue, watch, timeout) + self._lock = threading.Lock() + self.snapshot = DirectorySnapshot(watch.path, watch.is_recursive) + + def on_thread_stop(self): + _fsevents.remove_watch(self.watch) + _fsevents.stop(self) + + def queue_events(self, timeout): + with self._lock: + if not self.watch.is_recursive\ + and self.watch.path not in self.pathnames: + return + new_snapshot = DirectorySnapshot(self.watch.path, + self.watch.is_recursive) + events = new_snapshot - self.snapshot + self.snapshot = new_snapshot + + # Files. + for src_path in events.files_deleted: + self.queue_event(FileDeletedEvent(src_path)) + for src_path in events.files_modified: + self.queue_event(FileModifiedEvent(src_path)) + for src_path in events.files_created: + self.queue_event(FileCreatedEvent(src_path)) + for src_path, dest_path in events.files_moved: + self.queue_event(FileMovedEvent(src_path, dest_path)) + + # Directories. + for src_path in events.dirs_deleted: + self.queue_event(DirDeletedEvent(src_path)) + for src_path in events.dirs_modified: + self.queue_event(DirModifiedEvent(src_path)) + for src_path in events.dirs_created: + self.queue_event(DirCreatedEvent(src_path)) + for src_path, dest_path in events.dirs_moved: + self.queue_event(DirMovedEvent(src_path, dest_path)) + + def run(self): + try: + def callback(pathnames, flags, emitter=self): + emitter.queue_events(emitter.timeout) + + # for pathname, flag in zip(pathnames, flags): + # if emitter.watch.is_recursive: # and pathname != emitter.watch.path: + # new_sub_snapshot = DirectorySnapshot(pathname, True) + # old_sub_snapshot = self.snapshot.copy(pathname) + # diff = new_sub_snapshot - old_sub_snapshot + # self.snapshot += new_subsnapshot + # else: + # new_snapshot = DirectorySnapshot(emitter.watch.path, False) + # diff = new_snapshot - emitter.snapshot + # emitter.snapshot = new_snapshot + + # INFO: FSEvents reports directory notifications recursively + # by default, so we do not need to add subdirectory paths. + #pathnames = set([self.watch.path]) + # if self.watch.is_recursive: + # for root, directory_names, _ in os.walk(self.watch.path): + # for directory_name in directory_names: + # full_path = absolute_path( + # os.path.join(root, directory_name)) + # pathnames.add(full_path) + self.pathnames = [self.watch.path] + _fsevents.add_watch(self, + self.watch, + callback, + self.pathnames) + _fsevents.read_events(self) + except: + pass + + +class FSEventsObserver(BaseObserver): + + def __init__(self, timeout=DEFAULT_OBSERVER_TIMEOUT): + BaseObserver.__init__(self, emitter_class=FSEventsEmitter, + timeout=timeout) + + def schedule(self, event_handler, path, recursive=False): + # Python 2/3 compat + try: + str_class = unicode + except NameError: + str_class = str + + # Fix for issue #26: Trace/BPT error when given a unicode path + # string. https://github.com/gorakhargosh/watchdog/issues#issue/26 + if isinstance(path, str_class): + #path = unicode(path, 'utf-8') + path = unicodedata.normalize('NFC', path) + # We only encode the path in Python 2 for backwards compatibility. + # On Python 3 we want the path to stay as unicode if possible for + # the sake of path matching not having to be rewritten to use the + # bytes API instead of strings. The _watchdog_fsevent.so code for + # Python 3 can handle both str and bytes paths, which is why we + # do not HAVE to encode it with Python 3. The Python 2 code in + # _watchdog_fsevents.so was not changed for the sake of backwards + # compatibility. + if sys.version_info < (3,): + path = path.encode('utf-8') + return BaseObserver.schedule(self, event_handler, path, recursive) diff --git a/infer_4_33_0/lib/python3.10/site-packages/wandb/vendor/watchdog_0_9_0/wandb_watchdog/observers/polling.py b/infer_4_33_0/lib/python3.10/site-packages/wandb/vendor/watchdog_0_9_0/wandb_watchdog/observers/polling.py new file mode 100644 index 0000000000000000000000000000000000000000..f0019fdbf3f8bd85df90fb71dbdf6656189daf3b --- /dev/null +++ b/infer_4_33_0/lib/python3.10/site-packages/wandb/vendor/watchdog_0_9_0/wandb_watchdog/observers/polling.py @@ -0,0 +1,145 @@ +#!/usr/bin/env python +# -*- coding: utf-8 -*- +# +# Copyright 2011 Yesudeep Mangalapilly +# Copyright 2012 Google, Inc. +# +# Licensed under the Apache License, Version 2.0 (the "License"); +# you may not use this file except in compliance with the License. +# You may obtain a copy of the License at +# +# http://www.apache.org/licenses/LICENSE-2.0 +# +# Unless required by applicable law or agreed to in writing, software +# distributed under the License is distributed on an "AS IS" BASIS, +# WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied. +# See the License for the specific language governing permissions and +# limitations under the License. + + +""" +:module: watchdog.observers.polling +:synopsis: Polling emitter implementation. +:author: yesudeep@google.com (Yesudeep Mangalapilly) + +Classes +------- +.. autoclass:: PollingObserver + :members: + :show-inheritance: + +.. autoclass:: PollingObserverVFS + :members: + :show-inheritance: + :special-members: +""" + +from __future__ import with_statement +import os +import threading +from functools import partial +from wandb_watchdog.utils import stat as default_stat +from wandb_watchdog.utils.dirsnapshot import DirectorySnapshot, DirectorySnapshotDiff +from wandb_watchdog.observers.api import ( + EventEmitter, + BaseObserver, + DEFAULT_OBSERVER_TIMEOUT, + DEFAULT_EMITTER_TIMEOUT +) + +from wandb_watchdog.events import ( + DirMovedEvent, + DirDeletedEvent, + DirCreatedEvent, + DirModifiedEvent, + FileMovedEvent, + FileDeletedEvent, + FileCreatedEvent, + FileModifiedEvent +) + + +class PollingEmitter(EventEmitter): + """ + Platform-independent emitter that polls a directory to detect file + system changes. + """ + + def __init__(self, event_queue, watch, timeout=DEFAULT_EMITTER_TIMEOUT, + stat=default_stat, listdir=os.listdir): + EventEmitter.__init__(self, event_queue, watch, timeout) + self._snapshot = None + self._lock = threading.Lock() + self._take_snapshot = lambda: DirectorySnapshot( + self.watch.path, self.watch.is_recursive, stat=stat, listdir=listdir) + + def on_thread_start(self): + self._snapshot = self._take_snapshot() + + def queue_events(self, timeout): + # We don't want to hit the disk continuously. + # timeout behaves like an interval for polling emitters. + if self.stopped_event.wait(timeout): + return + + with self._lock: + if not self.should_keep_running(): + return + + # Get event diff between fresh snapshot and previous snapshot. + # Update snapshot. + try: + new_snapshot = self._take_snapshot() + except OSError: + self.queue_event(DirDeletedEvent(self.watch.path)) + self.stop() + return + + events = DirectorySnapshotDiff(self._snapshot, new_snapshot) + self._snapshot = new_snapshot + + # Files. + for src_path in events.files_deleted: + self.queue_event(FileDeletedEvent(src_path)) + for src_path in events.files_modified: + self.queue_event(FileModifiedEvent(src_path)) + for src_path in events.files_created: + self.queue_event(FileCreatedEvent(src_path)) + for src_path, dest_path in events.files_moved: + self.queue_event(FileMovedEvent(src_path, dest_path)) + + # Directories. + for src_path in events.dirs_deleted: + self.queue_event(DirDeletedEvent(src_path)) + for src_path in events.dirs_modified: + self.queue_event(DirModifiedEvent(src_path)) + for src_path in events.dirs_created: + self.queue_event(DirCreatedEvent(src_path)) + for src_path, dest_path in events.dirs_moved: + self.queue_event(DirMovedEvent(src_path, dest_path)) + + +class PollingObserver(BaseObserver): + """ + Platform-independent observer that polls a directory to detect file + system changes. + """ + + def __init__(self, timeout=DEFAULT_OBSERVER_TIMEOUT): + BaseObserver.__init__(self, emitter_class=PollingEmitter, timeout=timeout) + + +class PollingObserverVFS(BaseObserver): + """ + File system independent observer that polls a directory to detect changes. + """ + + def __init__(self, stat, listdir, polling_interval=1): + """ + :param stat: stat function. See ``os.stat`` for details. + :param listdir: listdir function. See ``os.listdir`` for details. + :type polling_interval: float + :param polling_interval: interval in seconds between polling the file system. + """ + emitter_cls = partial(PollingEmitter, stat=stat, listdir=listdir) + BaseObserver.__init__(self, emitter_class=emitter_cls, timeout=polling_interval) diff --git a/infer_4_33_0/lib/python3.10/site-packages/wandb/vendor/watchdog_0_9_0/wandb_watchdog/tricks/__pycache__/__init__.cpython-310.pyc b/infer_4_33_0/lib/python3.10/site-packages/wandb/vendor/watchdog_0_9_0/wandb_watchdog/tricks/__pycache__/__init__.cpython-310.pyc new file mode 100644 index 0000000000000000000000000000000000000000..60b16e61c8ca448e6be30bf21dacd8e76953e460 Binary files /dev/null and b/infer_4_33_0/lib/python3.10/site-packages/wandb/vendor/watchdog_0_9_0/wandb_watchdog/tricks/__pycache__/__init__.cpython-310.pyc differ diff --git a/infer_4_33_0/lib/python3.10/site-packages/wandb/vendor/watchdog_0_9_0/wandb_watchdog/utils/__init__.py b/infer_4_33_0/lib/python3.10/site-packages/wandb/vendor/watchdog_0_9_0/wandb_watchdog/utils/__init__.py new file mode 100644 index 0000000000000000000000000000000000000000..0e1b25803132190a2338f1f95ae6910295d2e473 --- /dev/null +++ b/infer_4_33_0/lib/python3.10/site-packages/wandb/vendor/watchdog_0_9_0/wandb_watchdog/utils/__init__.py @@ -0,0 +1,151 @@ +#!/usr/bin/env python +# -*- coding: utf-8 -*- +# +# Copyright 2011 Yesudeep Mangalapilly +# Copyright 2012 Google, Inc. +# +# Licensed under the Apache License, Version 2.0 (the "License"); +# you may not use this file except in compliance with the License. +# You may obtain a copy of the License at +# +# http://www.apache.org/licenses/LICENSE-2.0 +# +# Unless required by applicable law or agreed to in writing, software +# distributed under the License is distributed on an "AS IS" BASIS, +# WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied. +# See the License for the specific language governing permissions and +# limitations under the License. + + +""" +:module: watchdog.utils +:synopsis: Utility classes and functions. +:author: yesudeep@google.com (Yesudeep Mangalapilly) + +Classes +------- +.. autoclass:: BaseThread + :members: + :show-inheritance: + :inherited-members: + +""" +import os +import sys +import threading +from wandb_watchdog.utils import platform +from wandb_watchdog.utils.compat import Event + + +if sys.version_info[0] == 2 and platform.is_windows(): + # st_ino is not implemented in os.stat on this platform + import win32stat + stat = win32stat.stat +else: + stat = os.stat + + +def has_attribute(ob, attribute): + """ + :func:`hasattr` swallows exceptions. :func:`has_attribute` tests a Python object for the + presence of an attribute. + + :param ob: + object to inspect + :param attribute: + ``str`` for the name of the attribute. + """ + return getattr(ob, attribute, None) is not None + + +class UnsupportedLibc(Exception): + pass + + +class BaseThread(threading.Thread): + """ Convenience class for creating stoppable threads. """ + + def __init__(self): + threading.Thread.__init__(self) + self.daemon = True + self._stopped_event = Event() + + @property + def stopped_event(self): + return self._stopped_event + + def should_keep_running(self): + """Determines whether the thread should continue running.""" + return not self._stopped_event.is_set() + + def on_thread_stop(self): + """Override this method instead of :meth:`stop()`. + :meth:`stop()` calls this method. + + This method is called immediately after the thread is signaled to stop. + """ + pass + + def stop(self): + """Signals the thread to stop.""" + self._stopped_event.set() + self.on_thread_stop() + + def on_thread_start(self): + """Override this method instead of :meth:`start()`. :meth:`start()` + calls this method. + + This method is called right before this thread is started and this + object’s run() method is invoked. + """ + pass + + def start(self): + self.on_thread_start() + threading.Thread.start(self) + + +def load_module(module_name): + """Imports a module given its name and returns a handle to it.""" + try: + __import__(module_name) + except ImportError: + raise ImportError('No module named %s' % module_name) + return sys.modules[module_name] + + +def load_class(dotted_path): + """Loads and returns a class definition provided a dotted path + specification the last part of the dotted path is the class name + and there is at least one module name preceding the class name. + + Notes: + You will need to ensure that the module you are trying to load + exists in the Python path. + + Examples: + - module.name.ClassName # Provided module.name is in the Python path. + - module.ClassName # Provided module is in the Python path. + + What won't work: + - ClassName + - modle.name.ClassName # Typo in module name. + - module.name.ClasNam # Typo in classname. + """ + dotted_path_split = dotted_path.split('.') + if len(dotted_path_split) > 1: + klass_name = dotted_path_split[-1] + module_name = '.'.join(dotted_path_split[:-1]) + + module = load_module(module_name) + if has_attribute(module, klass_name): + klass = getattr(module, klass_name) + return klass + # Finally create and return an instance of the class + # return klass(*args, **kwargs) + else: + raise AttributeError('Module %s does not have class attribute %s' % ( + module_name, klass_name)) + else: + raise ValueError( + 'Dotted module path %s must contain a module name and a classname' % dotted_path) diff --git a/infer_4_33_0/lib/python3.10/site-packages/wandb/vendor/watchdog_0_9_0/wandb_watchdog/utils/__pycache__/__init__.cpython-310.pyc b/infer_4_33_0/lib/python3.10/site-packages/wandb/vendor/watchdog_0_9_0/wandb_watchdog/utils/__pycache__/__init__.cpython-310.pyc new file mode 100644 index 0000000000000000000000000000000000000000..45ca039540692a03cc2c7a65fef3c6f74e875d73 Binary files /dev/null and b/infer_4_33_0/lib/python3.10/site-packages/wandb/vendor/watchdog_0_9_0/wandb_watchdog/utils/__pycache__/__init__.cpython-310.pyc differ diff --git a/infer_4_33_0/lib/python3.10/site-packages/wandb/vendor/watchdog_0_9_0/wandb_watchdog/utils/__pycache__/bricks.cpython-310.pyc b/infer_4_33_0/lib/python3.10/site-packages/wandb/vendor/watchdog_0_9_0/wandb_watchdog/utils/__pycache__/bricks.cpython-310.pyc new file mode 100644 index 0000000000000000000000000000000000000000..5551d0367dc1536d4402f0f8ad19bf7b7643745a Binary files /dev/null and b/infer_4_33_0/lib/python3.10/site-packages/wandb/vendor/watchdog_0_9_0/wandb_watchdog/utils/__pycache__/bricks.cpython-310.pyc differ diff --git a/infer_4_33_0/lib/python3.10/site-packages/wandb/vendor/watchdog_0_9_0/wandb_watchdog/utils/__pycache__/compat.cpython-310.pyc b/infer_4_33_0/lib/python3.10/site-packages/wandb/vendor/watchdog_0_9_0/wandb_watchdog/utils/__pycache__/compat.cpython-310.pyc new file mode 100644 index 0000000000000000000000000000000000000000..0fec8fb5e79ed637e8197a964234330bbeee1e4e Binary files /dev/null and b/infer_4_33_0/lib/python3.10/site-packages/wandb/vendor/watchdog_0_9_0/wandb_watchdog/utils/__pycache__/compat.cpython-310.pyc differ diff --git a/infer_4_33_0/lib/python3.10/site-packages/wandb/vendor/watchdog_0_9_0/wandb_watchdog/utils/__pycache__/decorators.cpython-310.pyc b/infer_4_33_0/lib/python3.10/site-packages/wandb/vendor/watchdog_0_9_0/wandb_watchdog/utils/__pycache__/decorators.cpython-310.pyc new file mode 100644 index 0000000000000000000000000000000000000000..8e799b699ae271bff684baf679c251f0d6cd1edd Binary files /dev/null and b/infer_4_33_0/lib/python3.10/site-packages/wandb/vendor/watchdog_0_9_0/wandb_watchdog/utils/__pycache__/decorators.cpython-310.pyc differ diff --git a/infer_4_33_0/lib/python3.10/site-packages/wandb/vendor/watchdog_0_9_0/wandb_watchdog/utils/__pycache__/dirsnapshot.cpython-310.pyc b/infer_4_33_0/lib/python3.10/site-packages/wandb/vendor/watchdog_0_9_0/wandb_watchdog/utils/__pycache__/dirsnapshot.cpython-310.pyc new file mode 100644 index 0000000000000000000000000000000000000000..5485bbd0462009fc3e6600fa5430f1d5e2d79f3b Binary files /dev/null and b/infer_4_33_0/lib/python3.10/site-packages/wandb/vendor/watchdog_0_9_0/wandb_watchdog/utils/__pycache__/dirsnapshot.cpython-310.pyc differ diff --git a/infer_4_33_0/lib/python3.10/site-packages/wandb/vendor/watchdog_0_9_0/wandb_watchdog/utils/__pycache__/event_backport.cpython-310.pyc b/infer_4_33_0/lib/python3.10/site-packages/wandb/vendor/watchdog_0_9_0/wandb_watchdog/utils/__pycache__/event_backport.cpython-310.pyc new file mode 100644 index 0000000000000000000000000000000000000000..57cce2cc194fd7f76372c2535d599aa067e0665b Binary files /dev/null and b/infer_4_33_0/lib/python3.10/site-packages/wandb/vendor/watchdog_0_9_0/wandb_watchdog/utils/__pycache__/event_backport.cpython-310.pyc differ diff --git a/infer_4_33_0/lib/python3.10/site-packages/wandb/vendor/watchdog_0_9_0/wandb_watchdog/utils/__pycache__/importlib2.cpython-310.pyc b/infer_4_33_0/lib/python3.10/site-packages/wandb/vendor/watchdog_0_9_0/wandb_watchdog/utils/__pycache__/importlib2.cpython-310.pyc new file mode 100644 index 0000000000000000000000000000000000000000..daba20b0653d3b8e2401440ac6515ddc5254798a Binary files /dev/null and b/infer_4_33_0/lib/python3.10/site-packages/wandb/vendor/watchdog_0_9_0/wandb_watchdog/utils/__pycache__/importlib2.cpython-310.pyc differ diff --git a/infer_4_33_0/lib/python3.10/site-packages/wandb/vendor/watchdog_0_9_0/wandb_watchdog/utils/__pycache__/platform.cpython-310.pyc b/infer_4_33_0/lib/python3.10/site-packages/wandb/vendor/watchdog_0_9_0/wandb_watchdog/utils/__pycache__/platform.cpython-310.pyc new file mode 100644 index 0000000000000000000000000000000000000000..fbb1e83a2219a7b30be31b8592e7cf5a1df9cc95 Binary files /dev/null and b/infer_4_33_0/lib/python3.10/site-packages/wandb/vendor/watchdog_0_9_0/wandb_watchdog/utils/__pycache__/platform.cpython-310.pyc differ diff --git a/infer_4_33_0/lib/python3.10/site-packages/wandb/vendor/watchdog_0_9_0/wandb_watchdog/utils/__pycache__/unicode_paths.cpython-310.pyc b/infer_4_33_0/lib/python3.10/site-packages/wandb/vendor/watchdog_0_9_0/wandb_watchdog/utils/__pycache__/unicode_paths.cpython-310.pyc new file mode 100644 index 0000000000000000000000000000000000000000..d2041e052b9b4bac3dceb1c5a9c602a881dd2a5c Binary files /dev/null and b/infer_4_33_0/lib/python3.10/site-packages/wandb/vendor/watchdog_0_9_0/wandb_watchdog/utils/__pycache__/unicode_paths.cpython-310.pyc differ diff --git a/infer_4_33_0/lib/python3.10/site-packages/wandb/vendor/watchdog_0_9_0/wandb_watchdog/utils/compat.py b/infer_4_33_0/lib/python3.10/site-packages/wandb/vendor/watchdog_0_9_0/wandb_watchdog/utils/compat.py new file mode 100644 index 0000000000000000000000000000000000000000..0f6e7947b924a0bd07a99b7e6523c8a161beac42 --- /dev/null +++ b/infer_4_33_0/lib/python3.10/site-packages/wandb/vendor/watchdog_0_9_0/wandb_watchdog/utils/compat.py @@ -0,0 +1,29 @@ +# -*- coding: utf-8 -*- +# +# Copyright 2014 Thomas Amland +# +# Licensed under the Apache License, Version 2.0 (the "License"); +# you may not use this file except in compliance with the License. +# You may obtain a copy of the License at +# +# http://www.apache.org/licenses/LICENSE-2.0 +# +# Unless required by applicable law or agreed to in writing, software +# distributed under the License is distributed on an "AS IS" BASIS, +# WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied. +# See the License for the specific language governing permissions and +# limitations under the License. +import sys + +__all__ = ['queue', 'Event'] + +try: + import queue +except ImportError: + import Queue as queue + + +if sys.version_info < (2, 7): + from watchdog.utils.event_backport import Event +else: + from threading import Event \ No newline at end of file diff --git a/infer_4_33_0/lib/python3.10/site-packages/wandb/vendor/watchdog_0_9_0/wandb_watchdog/utils/delayed_queue.py b/infer_4_33_0/lib/python3.10/site-packages/wandb/vendor/watchdog_0_9_0/wandb_watchdog/utils/delayed_queue.py new file mode 100644 index 0000000000000000000000000000000000000000..6d98a50469b4bbe89f20639005fe9ba1e3f8d3cc --- /dev/null +++ b/infer_4_33_0/lib/python3.10/site-packages/wandb/vendor/watchdog_0_9_0/wandb_watchdog/utils/delayed_queue.py @@ -0,0 +1,88 @@ +# -*- coding: utf-8 -*- +# +# Copyright 2014 Thomas Amland +# +# Licensed under the Apache License, Version 2.0 (the "License"); +# you may not use this file except in compliance with the License. +# You may obtain a copy of the License at +# +# http://www.apache.org/licenses/LICENSE-2.0 +# +# Unless required by applicable law or agreed to in writing, software +# distributed under the License is distributed on an "AS IS" BASIS, +# WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied. +# See the License for the specific language governing permissions and +# limitations under the License. + +import time +import threading +from collections import deque + + +class DelayedQueue(object): + + def __init__(self, delay): + self.delay = delay + self._lock = threading.Lock() + self._not_empty = threading.Condition(self._lock) + self._queue = deque() + self._closed = False + + def put(self, element): + """Add element to queue.""" + self._lock.acquire() + self._queue.append((element, time.time())) + self._not_empty.notify() + self._lock.release() + + def close(self): + """Close queue, indicating no more items will be added.""" + self._closed = True + # Interrupt the blocking _not_empty.wait() call in get + self._not_empty.acquire() + self._not_empty.notify() + self._not_empty.release() + + def get(self): + """Remove and return an element from the queue, or this queue has been + closed raise the Closed exception. + """ + while True: + # wait for element to be added to queue + self._not_empty.acquire() + while len(self._queue) == 0 and not self._closed: + self._not_empty.wait() + + if self._closed: + self._not_empty.release() + return None + head, insert_time = self._queue[0] + self._not_empty.release() + + # wait for delay + time_left = insert_time + self.delay - time.time() + while time_left > 0: + time.sleep(time_left) + time_left = insert_time + self.delay - time.time() + + # return element if it's still in the queue + self._lock.acquire() + try: + if len(self._queue) > 0 and self._queue[0][0] is head: + self._queue.popleft() + return head + finally: + self._lock.release() + + def remove(self, predicate): + """Remove and return the first items for which predicate is True, + ignoring delay.""" + try: + self._lock.acquire() + for i, (elem, t) in enumerate(self._queue): + if predicate(elem): + del self._queue[i] + return elem + finally: + self._lock.release() + return None diff --git a/infer_4_33_0/lib/python3.10/site-packages/wandb/vendor/watchdog_0_9_0/wandb_watchdog/utils/dirsnapshot.py b/infer_4_33_0/lib/python3.10/site-packages/wandb/vendor/watchdog_0_9_0/wandb_watchdog/utils/dirsnapshot.py new file mode 100644 index 0000000000000000000000000000000000000000..c321d0ffe45c3b943e4649c1dbeec9931bf0db5d --- /dev/null +++ b/infer_4_33_0/lib/python3.10/site-packages/wandb/vendor/watchdog_0_9_0/wandb_watchdog/utils/dirsnapshot.py @@ -0,0 +1,293 @@ +#!/usr/bin/env python +# -*- coding: utf-8 -*- +# +# Copyright 2011 Yesudeep Mangalapilly +# Copyright 2012 Google, Inc. +# Copyright 2014 Thomas Amland +# +# Licensed under the Apache License, Version 2.0 (the "License"); +# you may not use this file except in compliance with the License. +# You may obtain a copy of the License at +# +# http://www.apache.org/licenses/LICENSE-2.0 +# +# Unless required by applicable law or agreed to in writing, software +# distributed under the License is distributed on an "AS IS" BASIS, +# WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied. +# See the License for the specific language governing permissions and +# limitations under the License. + +""" +:module: watchdog.utils.dirsnapshot +:synopsis: Directory snapshots and comparison. +:author: yesudeep@google.com (Yesudeep Mangalapilly) + +.. ADMONITION:: Where are the moved events? They "disappeared" + + This implementation does not take partition boundaries + into consideration. It will only work when the directory + tree is entirely on the same file system. More specifically, + any part of the code that depends on inode numbers can + break if partition boundaries are crossed. In these cases, + the snapshot diff will represent file/directory movement as + created and deleted events. + +Classes +------- +.. autoclass:: DirectorySnapshot + :members: + :show-inheritance: + +.. autoclass:: DirectorySnapshotDiff + :members: + :show-inheritance: + +""" + +import errno +import os +from stat import S_ISDIR +from wandb_watchdog.utils import stat as default_stat + + +class DirectorySnapshotDiff(object): + """ + Compares two directory snapshots and creates an object that represents + the difference between the two snapshots. + + :param ref: + The reference directory snapshot. + :type ref: + :class:`DirectorySnapshot` + :param snapshot: + The directory snapshot which will be compared + with the reference snapshot. + :type snapshot: + :class:`DirectorySnapshot` + """ + + def __init__(self, ref, snapshot): + created = snapshot.paths - ref.paths + deleted = ref.paths - snapshot.paths + + # check that all unchanged paths have the same inode + for path in ref.paths & snapshot.paths: + if ref.inode(path) != snapshot.inode(path): + created.add(path) + deleted.add(path) + + # find moved paths + moved = set() + for path in set(deleted): + inode = ref.inode(path) + new_path = snapshot.path(inode) + if new_path: + # file is not deleted but moved + deleted.remove(path) + moved.add((path, new_path)) + + for path in set(created): + inode = snapshot.inode(path) + old_path = ref.path(inode) + if old_path: + created.remove(path) + moved.add((old_path, path)) + + # find modified paths + # first check paths that have not moved + modified = set() + for path in ref.paths & snapshot.paths: + if ref.inode(path) == snapshot.inode(path): + if ref.mtime(path) != snapshot.mtime(path): + modified.add(path) + + for (old_path, new_path) in moved: + if ref.mtime(old_path) != snapshot.mtime(new_path): + modified.add(old_path) + + self._dirs_created = [path for path in created if snapshot.isdir(path)] + self._dirs_deleted = [path for path in deleted if ref.isdir(path)] + self._dirs_modified = [path for path in modified if ref.isdir(path)] + self._dirs_moved = [(frm, to) for (frm, to) in moved if ref.isdir(frm)] + + self._files_created = list(created - set(self._dirs_created)) + self._files_deleted = list(deleted - set(self._dirs_deleted)) + self._files_modified = list(modified - set(self._dirs_modified)) + self._files_moved = list(moved - set(self._dirs_moved)) + + @property + def files_created(self): + """List of files that were created.""" + return self._files_created + + @property + def files_deleted(self): + """List of files that were deleted.""" + return self._files_deleted + + @property + def files_modified(self): + """List of files that were modified.""" + return self._files_modified + + @property + def files_moved(self): + """ + List of files that were moved. + + Each event is a two-tuple the first item of which is the path + that has been renamed to the second item in the tuple. + """ + return self._files_moved + + @property + def dirs_modified(self): + """ + List of directories that were modified. + """ + return self._dirs_modified + + @property + def dirs_moved(self): + """ + List of directories that were moved. + + Each event is a two-tuple the first item of which is the path + that has been renamed to the second item in the tuple. + """ + return self._dirs_moved + + @property + def dirs_deleted(self): + """ + List of directories that were deleted. + """ + return self._dirs_deleted + + @property + def dirs_created(self): + """ + List of directories that were created. + """ + return self._dirs_created + +class DirectorySnapshot(object): + """ + A snapshot of stat information of files in a directory. + + :param path: + The directory path for which a snapshot should be taken. + :type path: + ``str`` + :param recursive: + ``True`` if the entire directory tree should be included in the + snapshot; ``False`` otherwise. + :type recursive: + ``bool`` + :param walker_callback: + .. deprecated:: 0.7.2 + :param stat: + Use custom stat function that returns a stat structure for path. + Currently only st_dev, st_ino, st_mode and st_mtime are needed. + + A function with the signature ``walker_callback(path, stat_info)`` + which will be called for every entry in the directory tree. + :param listdir: + Use custom listdir function. See ``os.listdir`` for details. + """ + + def __init__(self, path, recursive=True, + walker_callback=(lambda p, s: None), + stat=default_stat, + listdir=os.listdir): + self._stat_info = {} + self._inode_to_path = {} + + st = stat(path) + self._stat_info[path] = st + self._inode_to_path[(st.st_ino, st.st_dev)] = path + + def walk(root): + try: + paths = [os.path.join(root, name) for name in listdir(root)] + except OSError as e: + # Directory may have been deleted between finding it in the directory + # list of its parent and trying to delete its contents. If this + # happens we treat it as empty. + if e.errno == errno.ENOENT: + return + else: + raise + entries = [] + for p in paths: + try: + entries.append((p, stat(p))) + except OSError: + continue + for _ in entries: + yield _ + if recursive: + for path, st in entries: + if S_ISDIR(st.st_mode): + for _ in walk(path): + yield _ + + for p, st in walk(path): + i = (st.st_ino, st.st_dev) + self._inode_to_path[i] = p + self._stat_info[p] = st + walker_callback(p, st) + + @property + def paths(self): + """ + Set of file/directory paths in the snapshot. + """ + return set(self._stat_info.keys()) + + def path(self, id): + """ + Returns path for id. None if id is unknown to this snapshot. + """ + return self._inode_to_path.get(id) + + def inode(self, path): + """ Returns an id for path. """ + st = self._stat_info[path] + return (st.st_ino, st.st_dev) + + def isdir(self, path): + return S_ISDIR(self._stat_info[path].st_mode) + + def mtime(self, path): + return self._stat_info[path].st_mtime + + def stat_info(self, path): + """ + Returns a stat information object for the specified path from + the snapshot. + + Attached information is subject to change. Do not use unless + you specify `stat` in constructor. Use :func:`inode`, :func:`mtime`, + :func:`isdir` instead. + + :param path: + The path for which stat information should be obtained + from a snapshot. + """ + return self._stat_info[path] + + def __sub__(self, previous_dirsnap): + """Allow subtracting a DirectorySnapshot object instance from + another. + + :returns: + A :class:`DirectorySnapshotDiff` object. + """ + return DirectorySnapshotDiff(previous_dirsnap, self) + + def __str__(self): + return self.__repr__() + + def __repr__(self): + return str(self._stat_info) diff --git a/infer_4_33_0/lib/python3.10/site-packages/wandb/vendor/watchdog_0_9_0/wandb_watchdog/utils/echo.py b/infer_4_33_0/lib/python3.10/site-packages/wandb/vendor/watchdog_0_9_0/wandb_watchdog/utils/echo.py new file mode 100644 index 0000000000000000000000000000000000000000..12803e030d5c75308657c5f6717f76059907774c --- /dev/null +++ b/infer_4_33_0/lib/python3.10/site-packages/wandb/vendor/watchdog_0_9_0/wandb_watchdog/utils/echo.py @@ -0,0 +1,157 @@ +#!/usr/bin/env python +# -*- coding: utf-8 -*- +# echo.py: Tracing function calls using Python decorators. +# +# Written by Thomas Guest +# Please see http://wordaligned.org/articles/echo +# +# Place into the public domain. + +""" Echo calls made to functions and methods in a module. + +"Echoing" a function call means printing out the name of the function +and the values of its arguments before making the call (which is more +commonly referred to as "tracing", but Python already has a trace module). + +Example: to echo calls made to functions in "my_module" do: + + import echo + import my_module + echo.echo_module(my_module) + +Example: to echo calls made to functions in "my_module.my_class" do: + + echo.echo_class(my_module.my_class) + +Alternatively, echo.echo can be used to decorate functions. Calls to the +decorated function will be echoed. + +Example: + + @echo.echo + def my_function(args): + pass +""" +import inspect +import sys + + +def name(item): + " Return an item's name. " + return item.__name__ + + +def is_classmethod(instancemethod, klass): + " Determine if an instancemethod is a classmethod. " + return inspect.ismethod(instancemethod) and instancemethod.__self__ is klass + +def is_static_method(method, klass): + """Returns True if method is an instance method of klass.""" + for c in klass.mro(): + if name(method) in c.__dict__: + return isinstance(c.__dict__[name(method)], staticmethod) + else: + return False + +def is_class_private_name(name): + " Determine if a name is a class private name. " + # Exclude system defined names such as __init__, __add__ etc + return name.startswith("__") and not name.endswith("__") + + +def method_name(method): + """ Return a method's name. + + This function returns the name the method is accessed by from + outside the class (i.e. it prefixes "private" methods appropriately). + """ + mname = name(method) + if is_class_private_name(mname): + mname = "_%s%s" % (name(method.__self__.__class__), mname) + return mname + + +def format_arg_value(arg_val): + """ Return a string representing a (name, value) pair. + + >>> format_arg_value(('x', (1, 2, 3))) + 'x=(1, 2, 3)' + """ + arg, val = arg_val + return "%s=%r" % (arg, val) + + +def echo(fn, write=sys.stdout.write): + """ Echo calls to a function. + + Returns a decorated version of the input function which "echoes" calls + made to it by writing out the function's name and the arguments it was + called with. + """ + import functools + # Unpack function's arg count, arg names, arg defaults + code = fn.__code__ + argcount = code.co_argcount + argnames = code.co_varnames[:argcount] + fn_defaults = fn.__defaults__ or list() + argdefs = dict(list(zip(argnames[-len(fn_defaults):], fn_defaults))) + + @functools.wraps(fn) + def wrapped(*v, **k): + # Collect function arguments by chaining together positional, + # defaulted, extra positional and keyword arguments. + positional = list(map(format_arg_value, list(zip(argnames, v)))) + defaulted = [format_arg_value((a, argdefs[a])) + for a in argnames[len(v):] if a not in k] + nameless = list(map(repr, v[argcount:])) + keyword = list(map(format_arg_value, list(k.items()))) + args = positional + defaulted + nameless + keyword + write("%s(%s)\n" % (name(fn), ", ".join(args))) + return fn(*v, **k) + + return wrapped + + +def echo_instancemethod(klass, method, write=sys.stdout.write): + """ Change an instancemethod so that calls to it are echoed. + + Replacing a classmethod is a little more tricky. + See: http://www.python.org/doc/current/ref/types.html + """ + mname = method_name(method) + never_echo = "__str__", "__repr__", # Avoid recursion printing method calls + if mname in never_echo: + pass + elif is_classmethod(method, klass): + setattr(klass, mname, classmethod(echo(method.__func__, write))) + else: + setattr(klass, mname, echo(method, write)) + +def echo_class(klass, write=sys.stdout.write): + """ Echo calls to class methods and static functions + """ + for _, method in inspect.getmembers(klass, inspect.ismethod): + #In python 3 only class methods are returned here, but in python2 instance methods are too. + echo_instancemethod(klass, method, write) + for _, fn in inspect.getmembers(klass, inspect.isfunction): + if is_static_method(fn, klass): + setattr(klass, name(fn), staticmethod(echo(fn, write))) + else: + #It's not a class or a static method, so it must be an instance method. + #This should only be called in python 3, because in python 3 instance methods are considered functions. + echo_instancemethod(klass, fn, write) + +def echo_module(mod, write=sys.stdout.write): + """ Echo calls to functions and methods in a module. + """ + for fname, fn in inspect.getmembers(mod, inspect.isfunction): + setattr(mod, fname, echo(fn, write)) + for _, klass in inspect.getmembers(mod, inspect.isclass): + echo_class(klass, write) + +if __name__ == "__main__": + import doctest + + optionflags = doctest.ELLIPSIS + doctest.testfile('echoexample.txt', optionflags=optionflags) + doctest.testmod(optionflags=optionflags) diff --git a/infer_4_33_0/lib/python3.10/site-packages/wandb/vendor/watchdog_0_9_0/wandb_watchdog/utils/event_backport.py b/infer_4_33_0/lib/python3.10/site-packages/wandb/vendor/watchdog_0_9_0/wandb_watchdog/utils/event_backport.py new file mode 100644 index 0000000000000000000000000000000000000000..5c136e46d54839347c36e7f3ff81ff64f51b0d5b --- /dev/null +++ b/infer_4_33_0/lib/python3.10/site-packages/wandb/vendor/watchdog_0_9_0/wandb_watchdog/utils/event_backport.py @@ -0,0 +1,41 @@ +#!/usr/bin/env python +# -*- coding: utf-8 -*- +# Backport of Event from py2.7 (method wait in py2.6 returns None) + +from threading import Condition, Lock + + +class Event(object): + + def __init__(self,): + self.__cond = Condition(Lock()) + self.__flag = False + + def isSet(self): + return self.__flag + + is_set = isSet + + def set(self): + self.__cond.acquire() + try: + self.__flag = True + self.__cond.notify_all() + finally: + self.__cond.release() + + def clear(self): + self.__cond.acquire() + try: + self.__flag = False + finally: + self.__cond.release() + + def wait(self, timeout=None): + self.__cond.acquire() + try: + if not self.__flag: + self.__cond.wait(timeout) + return self.__flag + finally: + self.__cond.release() diff --git a/infer_4_33_0/lib/python3.10/site-packages/wandb/vendor/watchdog_0_9_0/wandb_watchdog/utils/platform.py b/infer_4_33_0/lib/python3.10/site-packages/wandb/vendor/watchdog_0_9_0/wandb_watchdog/utils/platform.py new file mode 100644 index 0000000000000000000000000000000000000000..239c6a25829dde1b70cfab092678820ca91a326d --- /dev/null +++ b/infer_4_33_0/lib/python3.10/site-packages/wandb/vendor/watchdog_0_9_0/wandb_watchdog/utils/platform.py @@ -0,0 +1,57 @@ +#!/usr/bin/env python +# -*- coding: utf-8 -*- +# +# Copyright 2011 Yesudeep Mangalapilly +# Copyright 2012 Google, Inc. +# +# Licensed under the Apache License, Version 2.0 (the "License"); +# you may not use this file except in compliance with the License. +# You may obtain a copy of the License at +# +# http://www.apache.org/licenses/LICENSE-2.0 +# +# Unless required by applicable law or agreed to in writing, software +# distributed under the License is distributed on an "AS IS" BASIS, +# WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied. +# See the License for the specific language governing permissions and +# limitations under the License. + + +import sys + +PLATFORM_WINDOWS = 'windows' +PLATFORM_LINUX = 'linux' +PLATFORM_BSD = 'bsd' +PLATFORM_DARWIN = 'darwin' +PLATFORM_UNKNOWN = 'unknown' + + +def get_platform_name(): + if sys.platform.startswith("win"): + return PLATFORM_WINDOWS + elif sys.platform.startswith('darwin'): + return PLATFORM_DARWIN + elif sys.platform.startswith('linux'): + return PLATFORM_LINUX + elif sys.platform.startswith(('dragonfly', 'freebsd', 'netbsd', 'openbsd', )): + return PLATFORM_BSD + else: + return PLATFORM_UNKNOWN + +__platform__ = get_platform_name() + + +def is_linux(): + return __platform__ == PLATFORM_LINUX + + +def is_bsd(): + return __platform__ == PLATFORM_BSD + + +def is_darwin(): + return __platform__ == PLATFORM_DARWIN + + +def is_windows(): + return __platform__ == PLATFORM_WINDOWS diff --git a/infer_4_33_0/lib/python3.10/site-packages/wandb/vendor/watchdog_0_9_0/wandb_watchdog/utils/unicode_paths.py b/infer_4_33_0/lib/python3.10/site-packages/wandb/vendor/watchdog_0_9_0/wandb_watchdog/utils/unicode_paths.py new file mode 100644 index 0000000000000000000000000000000000000000..9e4b4b425fd9c7420d2a01ff5ddf0c7eadc60b06 --- /dev/null +++ b/infer_4_33_0/lib/python3.10/site-packages/wandb/vendor/watchdog_0_9_0/wandb_watchdog/utils/unicode_paths.py @@ -0,0 +1,64 @@ +#!/usr/bin/env python +# -*- coding: utf-8 -*- +# +# Copyright (c) 2013 Will Bond +# +# Permission is hereby granted, free of charge, to any person obtaining a copy +# of this software and associated documentation files (the "Software"), to deal +# in the Software without restriction, including without limitation the rights +# to use, copy, modify, merge, publish, distribute, sublicense, and/or sell +# copies of the Software, and to permit persons to whom the Software is +# furnished to do so, subject to the following conditions: +# +# The above copyright notice and this permission notice shall be included in +# all copies or substantial portions of the Software. +# +# THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR +# IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY, +# FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL THE +# AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER +# LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING FROM, +# OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN THE +# SOFTWARE. + + +import sys + +from wandb_watchdog.utils import platform + +try: + # Python 2 + str_cls = unicode + bytes_cls = str +except NameError: + # Python 3 + str_cls = str + bytes_cls = bytes + + +# This is used by Linux when the locale seems to be improperly set. UTF-8 tends +# to be the encoding used by all distros, so this is a good fallback. +fs_fallback_encoding = 'utf-8' +fs_encoding = sys.getfilesystemencoding() or fs_fallback_encoding + + +def encode(path): + if isinstance(path, str_cls): + try: + path = path.encode(fs_encoding, 'strict') + except UnicodeEncodeError: + if not platform.is_linux(): + raise + path = path.encode(fs_fallback_encoding, 'strict') + return path + + +def decode(path): + if isinstance(path, bytes_cls): + try: + path = path.decode(fs_encoding, 'strict') + except UnicodeDecodeError: + if not platform.is_linux(): + raise + path = path.decode(fs_fallback_encoding, 'strict') + return path