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📄 __init__.py """ A sub-package for efficiently dealing with polynomials. Within the documentation for this sub-package, a "finite power series," i.e., a polynomial (also referred to simply as a "series") is represented by a 1-D numpy array of the polynomial's coefficients, ordered from lowest order term to highest. For example, array([1,2,3]) represents ``P_0 + 2P_1 + 3P_2``, where P_n is the n-th order basis polynomial applicable to the specific module in question, e.g., `polynomial` (which "wraps" the "standard" basis) or `chebyshev`. For optimal performance, all operations on polynomials, including evaluation at an argument, are implemented as operations on the coefficients. Additional (module-specific) information can be found in the docstring for the module of interest. This package provides convenience classes for each of six different kinds of polynomials: ======================== ================ Name Provides ======================== ================ `~polynomial.Polynomial` Power series `~chebyshev.Chebyshev` Chebyshev series `~legendre.Legendre` Legendre series `~laguerre.Laguerre` Laguerre series `~hermite.Hermite` Hermite series `~hermite_e.HermiteE` HermiteE series ======================== ================ These convenience classes provide a consistent interface for creating, manipulating, and fitting data with polynomials of different bases. The convenience classes are the preferred interface for the `~numpy.polynomial` package, and are available from the ``numpy.polynomial`` namespace. This eliminates the need to navigate to the corresponding submodules, e.g. ``np.polynomial.Polynomial`` or ``np.polynomial.Chebyshev`` instead of ``np.polynomial.polynomial.Polynomial`` or ``np.polynomial.chebyshev.Chebyshev``, respectively. The classes provide a more consistent and concise interface than the type-specific functions defined in the submodules for each type of polynomial. For example, to fit a Chebyshev polynomial with degree ``1`` to data given by arrays ``xdata`` and ``ydata``, the `~chebyshev.Chebyshev.fit` class method:: >>> from numpy.polynomial import Chebyshev >>> c = Chebyshev.fit(xdata, ydata, deg=1) is preferred over the `chebyshev.chebfit` function from the ``np.polynomial.chebyshev`` module:: >>> from numpy.polynomial.chebyshev import chebfit >>> c = chebfit(xdata, ydata, deg=1) See :doc:`routines.polynomials.classes` for more details. Convenience Classes =================== The following lists the various constants and methods common to all of the classes representing the various kinds of polynomials. In the following, the term ``Poly`` represents any one of the convenience classes (e.g. `~polynomial.Polynomial`, `~chebyshev.Chebyshev`, `~hermite.Hermite`, etc.) while the lowercase ``p`` represents an instance of a polynomial class. Constants --------- - ``Poly.domain`` -- Default domain - ``Poly.window`` -- Default window - ``Poly.basis_name`` -- String used to represent the basis - ``Poly.maxpower`` -- Maximum value ``n`` such that ``pn`` is allowed - ``Poly.nickname`` -- String used in printing Creation -------- Methods for creating polynomial instances. - ``Poly.basis(degree)`` -- Basis polynomial of given degree - ``Poly.identity()`` -- ``p`` where ``p(x) = x`` for all ``x`` - ``Poly.fit(x, y, deg)`` -- ``p`` of degree ``deg`` with coefficients determined by the least-squares fit to the data ``x``, ``y`` - ``Poly.fromroots(roots)`` -- ``p`` with specified roots - ``p.copy()`` -- Create a copy of ``p`` Conversion ---------- Methods for converting a polynomial instance of one kind to another. - ``p.cast(Poly)`` -- Convert ``p`` to instance of kind ``Poly`` - ``p.convert(Poly)`` -- Convert ``p`` to instance of kind ``Poly`` or map between ``domain`` and ``window`` Calculus -------- - ``p.deriv()`` -- Take the derivative of ``p`` - ``p.integ()`` -- Integrate ``p`` Validation ---------- - ``Poly.has_samecoef(p1, p2)`` -- Check if coefficients match - ``Poly.has_samedomain(p1, p2)`` -- Check if domains match - ``Poly.has_sametype(p1, p2)`` -- Check if types match - ``Poly.has_samewindow(p1, p2)`` -- Check if windows match Misc ---- - ``p.linspace()`` -- Return ``x, p(x)`` at equally-spaced points in ``domain`` - ``p.mapparms()`` -- Return the parameters for the linear mapping between ``domain`` and ``window``. - ``p.roots()`` -- Return the roots of `p`. - ``p.trim()`` -- Remove trailing coefficients. - ``p.cutdeg(degree)`` -- Truncate p to given degree - ``p.truncate(size)`` -- Truncate p to given size """ from .polynomial import Polynomial from .chebyshev import Chebyshev from .legendre import Legendre from .hermite import Hermite from .hermite_e import HermiteE from .laguerre import Laguerre __all__ = [ "set_default_printstyle", "polynomial", "Polynomial", "chebyshev", "Chebyshev", "legendre", "Legendre", "hermite", "Hermite", "hermite_e", "HermiteE", "laguerre", "Laguerre", ] def set_default_printstyle(style): """ Set the default format for the string representation of polynomials. Values for ``style`` must be valid inputs to ``__format__``, i.e. 'ascii' or 'unicode'. Parameters ---------- style : str Format string for default printing style. Must be either 'ascii' or 'unicode'. Notes ----- The default format depends on the platform: 'unicode' is used on Unix-based systems and 'ascii' on Windows. This determination is based on default font support for the unicode superscript and subscript ranges. Examples -------- >>> p = np.polynomial.Polynomial([1, 2, 3]) >>> c = np.polynomial.Chebyshev([1, 2, 3]) >>> np.polynomial.set_default_printstyle('unicode') >>> print(p) 1.0 + 2.0·x + 3.0·x² >>> print(c) 1.0 + 2.0·T₁(x) + 3.0·T₂(x) >>> np.polynomial.set_default_printstyle('ascii') >>> print(p) 1.0 + 2.0 x + 3.0 x2 >>> print(c) 1.0 + 2.0 T_1(x) + 3.0 T_2(x) >>> # Formatting supersedes all class/package-level defaults >>> print(f"{p:unicode}") 1.0 + 2.0·x + 3.0·x² """ if style not in ('unicode', 'ascii'): raise ValueError( f"Unsupported format string '{style}'. Valid options are 'ascii' " f"and 'unicode'" ) _use_unicode = True if style == 'ascii': _use_unicode = False from ._polybase import ABCPolyBase ABCPolyBase._use_unicode = _use_unicode from numpy._pytesttester import PytestTester test = PytestTester(__name__) del PytestTester	6.62 KB	2026-01-20 12:58
📄 __init__.pyi 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): ...	701 B	2026-01-20 12:58
📁 __pycache__	-	2026-03-21 15:45
📄 _polybase.py """ 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 ``1P_0(x) + 2P_1(x) + 3P_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 nm 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 + sclarg 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 + 7x + 4x2``: >>> 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 + sclx`` 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 + sclx``. 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 + scllbnd 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 + sclroots 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)	38.35 KB	2026-01-20 12:58
📄 _polybase.pyi import abc from typing import Any, ClassVar __all__: list[str] class ABCPolyBase(abc.ABC): __hash__: ClassVar[None] # type: ignore[assignment] __array_ufunc__: ClassVar[None] maxpower: ClassVar[int] coef: Any @property def symbol(self) -> str: ... @property @abc.abstractmethod def domain(self): ... @property @abc.abstractmethod def window(self): ... @property @abc.abstractmethod def basis_name(self): ... def has_samecoef(self, other): ... def has_samedomain(self, other): ... def has_samewindow(self, other): ... def has_sametype(self, other): ... def __init__(self, coef, domain=..., window=..., symbol: str = ...) -> None: ... def __format__(self, fmt_str): ... def __call__(self, arg): ... def __iter__(self): ... def __len__(self): ... def __neg__(self): ... def __pos__(self): ... def __add__(self, other): ... def __sub__(self, other): ... def __mul__(self, other): ... def __truediv__(self, other): ... def __floordiv__(self, other): ... def __mod__(self, other): ... def __divmod__(self, other): ... def __pow__(self, other): ... def __radd__(self, other): ... def __rsub__(self, other): ... def __rmul__(self, other): ... def __rdiv__(self, other): ... def __rtruediv__(self, other): ... def __rfloordiv__(self, other): ... def __rmod__(self, other): ... def __rdivmod__(self, other): ... def __eq__(self, other): ... def __ne__(self, other): ... def copy(self): ... def degree(self): ... def cutdeg(self, deg): ... def trim(self, tol=...): ... def truncate(self, size): ... def convert(self, domain=..., kind=..., window=...): ... def mapparms(self): ... def integ(self, m=..., k = ..., lbnd=...): ... def deriv(self, m=...): ... def roots(self): ... def linspace(self, n=..., domain=...): ... @classmethod def fit(cls, x, y, deg, domain=..., rcond=..., full=..., w=..., window=...): ... @classmethod def fromroots(cls, roots, domain = ..., window=...): ... @classmethod def identity(cls, domain=..., window=...): ... @classmethod def basis(cls, deg, domain=..., window=...): ... @classmethod def cast(cls, series, domain=..., window=...): ...	2.27 KB	2026-01-20 12:58
📄 chebyshev.py """ ==================================================== 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(2n-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: SS, AA -> S SA, AS -> 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 = rz2 z1[i:i+lc2] -= tmp z1[j:j+lc2] -= tmp i += 1 j -= 1 r = z1[i] quo[i] = r tmp = rz2 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 + sclx``. Returns ------- y : ndarray This module's representation of the Chebyshev series for ``off + sclx``. 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 + 2T_1 + 3T_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 + 2T_1 + 3T_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 + 2T_1 + 3T_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 + 2T_1 + 3T_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 + 2T_1 + 3T_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 ``1T_0 + 2T_1 + 3T_2`` while [[1,2],[1,2]] represents ``1T_0(x)T_0(y) + 1T_1(x)T_0(y) + 2T_0(x)T_1(y) + 2T_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 ``sclm``. 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] = (2j)c[j] c[j - 2] += (jc[j])/(j - 2) if n > 1: der[1] = 4c[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 + 2T_1 + 3T_2`` while [[1,2],[1,2]] represents ``1T_0(x)T_0(y) + 1T_1(x)T_0(y) + 2T_0(x)T_1(y) + 2T_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 = 2x c0 = c[-2] c1 = c[-1] for i in range(3, len(c) + 1): tmp = c0 c0 = c[-i] - c1 c1 = tmp + c1x2 return c0 + c1x 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] = x0 + 1 if ideg > 0: x2 = 2x 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:] /= 0.5order 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:`2deg - 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, 2ideg, 2) / (2.0ideg)) 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 = [pik/(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 ``1T_0(x) + 2T_1(x) + 3T_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'	61.32 KB	2026-01-20 12:58
📄 chebyshev.pyi 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	1.35 KB	2026-01-20 12:58
📄 hermite.py """ ============================================================== 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 + sclx``. Returns ------- y : ndarray This module's representation of the Hermite series for ``off + sclx``. 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 + 2P_1 + 3P_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 + 2P_1 + 3P_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 + iP_{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 + 2P_1 + 3P_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 + 2P_1 + 3P_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 + 2P_1 + 3P_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 ``1H_0 + 2H_1 + 3H_2`` while [[1,2],[1,2]] represents ``1H_0(x)H_0(y) + 1H_1(x)H_0(y) + 2H_0(x)H_1(y) + 2H_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 ``sclm``. 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] = (2j)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 + 2H_1 + 3H_2`` while [[1,2],[1,2]] represents ``1H_0(x)H_0(y) + 1H_1(x)H_0(y) + 2H_0(x)H_1(y) + 2H_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 = x2 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 + c1x2 return c0 + c1x2 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] = x0 + 1 if ideg > 0: x2 = x2 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([[-.5c[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(.5np.arange(1, n)) bot[...] = top mat[:, -1] -= sclc[:-1]/(2.0c[-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([-.5c[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 = -c1np.sqrt((nd - 1.)/nd) c1 = tmp + c1xnp.sqrt(2./nd) nd = nd - 1.0 return c0 + c1xnp.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:`2deg - 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(2ideg) 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(-x2) 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 ``1H_0(x) + 2H_1(X) + 3H_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'	51.28 KB	2026-01-20 12:58
📄 hermite.pyi from typing import Any from numpy import ndarray, dtype, int_, float_ from numpy.polynomial._polybase import ABCPolyBase from numpy.polynomial.polyutils import trimcoef __all__: list[str] hermtrim = trimcoef def poly2herm(pol): ... def herm2poly(c): ... hermdomain: ndarray[Any, dtype[int_]] hermzero: ndarray[Any, dtype[int_]] hermone: ndarray[Any, dtype[int_]] hermx: ndarray[Any, dtype[float_]] def hermline(off, scl): ... def hermfromroots(roots): ... def hermadd(c1, c2): ... def hermsub(c1, c2): ... def hermmulx(c): ... def hermmul(c1, c2): ... def hermdiv(c1, c2): ... def hermpow(c, pow, maxpower=...): ... def hermder(c, m=..., scl=..., axis=...): ... def hermint(c, m=..., k = ..., lbnd=..., scl=..., axis=...): ... def hermval(x, c, tensor=...): ... def hermval2d(x, y, c): ... def hermgrid2d(x, y, c): ... def hermval3d(x, y, z, c): ... def hermgrid3d(x, y, z, c): ... def hermvander(x, deg): ... def hermvander2d(x, y, deg): ... def hermvander3d(x, y, z, deg): ... def hermfit(x, y, deg, rcond=..., full=..., w=...): ... def hermcompanion(c): ... def hermroots(c): ... def hermgauss(deg): ... def hermweight(x): ... class Hermite(ABCPolyBase): domain: Any window: Any basis_name: Any	1.19 KB	2026-01-20 12:58
📄 hermite_e.py """ =================================================================== 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 + sclx``. Returns ------- y : ndarray This module's representation of the Hermite series for ``off + sclx``. 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 + 2P_1 + 3P_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 + 2P_1 + 3P_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 + 2P_1 + 3P_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 + 2P_1 + 3P_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 + 2P_1 + 3P_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 ``1He_0 + 2He_1 + 3He_2`` while [[1,2],[1,2]] represents ``1He_0(x)He_0(y) + 1He_1(x)He_0(y) + 2He_0(x)He_1(y) + 2He_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] = jc[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 + 2H_1 + 3H_2`` while [[1,2],[1,2]] represents ``1H_0(x)H_0(y) + 1H_1(x)H_0(y) + 2H_0(x)H_1(y) + 2H_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 + c1x return c0 + c1x 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] = x0 + 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] -= sclc[:-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(2np.pi))) c0 = 0. c1 = 1./np.sqrt(np.sqrt(2np.pi)) nd = float(n) for i in range(n - 1): tmp = c0 c0 = -c1np.sqrt((nd - 1.)/nd) c1 = tmp + c1xnp.sqrt(1./nd) nd = nd - 1.0 return c0 + c1x 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:`2deg - 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(2np.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(-.5x2) 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 ``1He_0(x) + 2He_1(X) + 3He_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'	51.41 KB	2026-01-20 12:58
📄 hermite_e.pyi 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	1.21 KB	2026-01-20 12:58
📄 laguerre.py """ ================================================== Laguerre Series (:mod:`numpy.polynomial.laguerre`) ================================================== This module provides a number of objects (mostly functions) useful for dealing with Laguerre series, including a `Laguerre` 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/ Laguerre Constants --------- .. autosummary:: :toctree: generated/ lagdomain lagzero lagone lagx Arithmetic ---------- .. autosummary:: :toctree: generated/ lagadd lagsub lagmulx lagmul lagdiv lagpow lagval lagval2d lagval3d laggrid2d laggrid3d Calculus -------- .. autosummary:: :toctree: generated/ lagder lagint Misc Functions -------------- .. autosummary:: :toctree: generated/ lagfromroots lagroots lagvander lagvander2d lagvander3d laggauss lagweight lagcompanion lagfit lagtrim lagline lag2poly poly2lag 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__ = [ 'lagzero', 'lagone', 'lagx', 'lagdomain', 'lagline', 'lagadd', 'lagsub', 'lagmulx', 'lagmul', 'lagdiv', 'lagpow', 'lagval', 'lagder', 'lagint', 'lag2poly', 'poly2lag', 'lagfromroots', 'lagvander', 'lagfit', 'lagtrim', 'lagroots', 'Laguerre', 'lagval2d', 'lagval3d', 'laggrid2d', 'laggrid3d', 'lagvander2d', 'lagvander3d', 'lagcompanion', 'laggauss', 'lagweight'] lagtrim = pu.trimcoef def poly2lag(pol): """ poly2lag(pol) Convert a polynomial to a Laguerre 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 Laguerre 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 Laguerre series. See Also -------- lag2poly 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.laguerre import poly2lag >>> poly2lag(np.arange(4)) array([ 23., -63., 58., -18.]) """ [pol] = pu.as_series([pol]) res = 0 for p in pol[::-1]: res = lagadd(lagmulx(res), p) return res def lag2poly(c): """ Convert a Laguerre series to a polynomial. Convert an array representing the coefficients of a Laguerre 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 Laguerre 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 -------- poly2lag 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.laguerre import lag2poly >>> lag2poly([ 23., -63., 58., -18.]) array([0., 1., 2., 3.]) """ from .polynomial import polyadd, polysub, polymulx [c] = pu.as_series([c]) n = len(c) if n == 1: 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, polysub((2i - 1)c1, polymulx(c1))/i) return polyadd(c0, polysub(c1, 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. # # Laguerre lagdomain = np.array([0, 1]) # Laguerre coefficients representing zero. lagzero = np.array([0]) # Laguerre coefficients representing one. lagone = np.array([1]) # Laguerre coefficients representing the identity x. lagx = np.array([1, -1]) def lagline(off, scl): """ Laguerre series whose graph is a straight line. Parameters ---------- off, scl : scalars The specified line is given by ``off + sclx``. Returns ------- y : ndarray This module's representation of the Laguerre series for ``off + sclx``. See Also -------- numpy.polynomial.polynomial.polyline numpy.polynomial.chebyshev.chebline numpy.polynomial.legendre.legline numpy.polynomial.hermite.hermline numpy.polynomial.hermite_e.hermeline Examples -------- >>> from numpy.polynomial.laguerre import lagline, lagval >>> lagval(0,lagline(3, 2)) 3.0 >>> lagval(1,lagline(3, 2)) 5.0 """ if scl != 0: return np.array([off + scl, -scl]) else: return np.array([off]) def lagfromroots(roots): """ Generate a Laguerre 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 Laguerre 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 Laguerre 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.chebyshev.chebfromroots numpy.polynomial.hermite.hermfromroots numpy.polynomial.hermite_e.hermefromroots Examples -------- >>> from numpy.polynomial.laguerre import lagfromroots, lagval >>> coef = lagfromroots((-1, 0, 1)) >>> lagval((-1, 0, 1), coef) array([0., 0., 0.]) >>> coef = lagfromroots((-1j, 1j)) >>> lagval((-1j, 1j), coef) array([0.+0.j, 0.+0.j]) """ return pu._fromroots(lagline, lagmul, roots) def lagadd(c1, c2): """ Add one Laguerre series to another. Returns the sum of two Laguerre 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 + 2P_1 + 3P_2``. Parameters ---------- c1, c2 : array_like 1-D arrays of Laguerre series coefficients ordered from low to high. Returns ------- out : ndarray Array representing the Laguerre series of their sum. See Also -------- lagsub, lagmulx, lagmul, lagdiv, lagpow Notes ----- Unlike multiplication, division, etc., the sum of two Laguerre series is a Laguerre 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.laguerre import lagadd >>> lagadd([1, 2, 3], [1, 2, 3, 4]) array([2., 4., 6., 4.]) """ return pu._add(c1, c2) def lagsub(c1, c2): """ Subtract one Laguerre series from another. Returns the difference of two Laguerre series `c1` - `c2`. The sequences of coefficients are from lowest order term to highest, i.e., [1,2,3] represents the series ``P_0 + 2P_1 + 3P_2``. Parameters ---------- c1, c2 : array_like 1-D arrays of Laguerre series coefficients ordered from low to high. Returns ------- out : ndarray Of Laguerre series coefficients representing their difference. See Also -------- lagadd, lagmulx, lagmul, lagdiv, lagpow Notes ----- Unlike multiplication, division, etc., the difference of two Laguerre series is a Laguerre 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.laguerre import lagsub >>> lagsub([1, 2, 3, 4], [1, 2, 3]) array([0., 0., 0., 4.]) """ return pu._sub(c1, c2) def lagmulx(c): """Multiply a Laguerre series by x. Multiply the Laguerre series `c` by x, where x is the independent variable. Parameters ---------- c : array_like 1-D array of Laguerre series coefficients ordered from low to high. Returns ------- out : ndarray Array representing the result of the multiplication. See Also -------- lagadd, lagsub, lagmul, lagdiv, lagpow Notes ----- The multiplication uses the recursion relationship for Laguerre polynomials in the form .. math:: xP_i(x) = (-(i + 1)P_{i + 1}(x) + (2i + 1)P_{i}(x) - iP_{i - 1}(x)) Examples -------- >>> from numpy.polynomial.laguerre import lagmulx >>> lagmulx([1, 2, 3]) array([-1., -1., 11., -9.]) """ # 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] prd[1] = -c[0] for i in range(1, len(c)): prd[i + 1] = -c[i](i + 1) prd[i] += c[i](2i + 1) prd[i - 1] -= c[i]i return prd def lagmul(c1, c2): """ Multiply one Laguerre series by another. Returns the product of two Laguerre 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 + 2P_1 + 3P_2``. Parameters ---------- c1, c2 : array_like 1-D arrays of Laguerre series coefficients ordered from low to high. Returns ------- out : ndarray Of Laguerre series coefficients representing their product. See Also -------- lagadd, lagsub, lagmulx, lagdiv, lagpow Notes ----- In general, the (polynomial) product of two C-series results in terms that are not in the Laguerre polynomial basis set. Thus, to express the product as a Laguerre 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.laguerre import lagmul >>> lagmul([1, 2, 3], [0, 1, 2]) array([ 8., -13., 38., -51., 36.]) """ # 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 = lagsub(c[-i]xs, (c1(nd - 1))/nd) c1 = lagadd(tmp, lagsub((2nd - 1)c1, lagmulx(c1))/nd) return lagadd(c0, lagsub(c1, lagmulx(c1))) def lagdiv(c1, c2): """ Divide one Laguerre series by another. Returns the quotient-with-remainder of two Laguerre 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 + 2P_1 + 3P_2``. Parameters ---------- c1, c2 : array_like 1-D arrays of Laguerre series coefficients ordered from low to high. Returns ------- [quo, rem] : ndarrays Of Laguerre series coefficients representing the quotient and remainder. See Also -------- lagadd, lagsub, lagmulx, lagmul, lagpow Notes ----- In general, the (polynomial) division of one Laguerre series by another results in quotient and remainder terms that are not in the Laguerre polynomial basis set. Thus, to express these results as a Laguerre series, it is necessary to "reproject" the results onto the Laguerre basis set, which may produce "unintuitive" (but correct) results; see Examples section below. Examples -------- >>> from numpy.polynomial.laguerre import lagdiv >>> lagdiv([ 8., -13., 38., -51., 36.], [0, 1, 2]) (array([1., 2., 3.]), array([0.])) >>> lagdiv([ 9., -12., 38., -51., 36.], [0, 1, 2]) (array([1., 2., 3.]), array([1., 1.])) """ return pu._div(lagmul, c1, c2) def lagpow(c, pow, maxpower=16): """Raise a Laguerre series to a power. Returns the Laguerre 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 + 2P_1 + 3P_2.`` Parameters ---------- c : array_like 1-D array of Laguerre 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 Laguerre series of power. See Also -------- lagadd, lagsub, lagmulx, lagmul, lagdiv Examples -------- >>> from numpy.polynomial.laguerre import lagpow >>> lagpow([1, 2, 3], 2) array([ 14., -16., 56., -72., 54.]) """ return pu._pow(lagmul, c, pow, maxpower) def lagder(c, m=1, scl=1, axis=0): """ Differentiate a Laguerre series. Returns the Laguerre 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 ``1L_0 + 2L_1 + 3L_2`` while [[1,2],[1,2]] represents ``1L_0(x)L_0(y) + 1L_1(x)L_0(y) + 2L_0(x)L_1(y) + 2L_1(x)L_1(y)`` if axis=0 is ``x`` and axis=1 is ``y``. Parameters ---------- c : array_like Array of Laguerre 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 Laguerre series of the derivative. See Also -------- lagint Notes ----- In general, the result of differentiating a Laguerre 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.laguerre import lagder >>> lagder([ 1., 1., 1., -3.]) array([1., 2., 3.]) >>> lagder([ 1., 0., 0., -4., 3.], 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, 1, -1): der[j - 1] = -c[j] c[j - 1] += c[j] der[0] = -c[1] c = der c = np.moveaxis(c, 0, iaxis) return c def lagint(c, m=1, k=[], lbnd=0, scl=1, axis=0): """ Integrate a Laguerre series. Returns the Laguerre 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 + 2L_1 + 3L_2`` while [[1,2],[1,2]] represents ``1L_0(x)L_0(y) + 1L_1(x)L_0(y) + 2L_0(x)L_1(y) + 2L_1(x)L_1(y)`` if axis=0 is ``x`` and axis=1 is ``y``. Parameters ---------- c : array_like Array of Laguerre 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 Laguerre series coefficients of the integral. Raises ------ ValueError If ``m < 0``, ``len(k) > m``, ``np.ndim(lbnd) != 0``, or ``np.ndim(scl) != 0``. See Also -------- lagder 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.laguerre import lagint >>> lagint([1,2,3]) array([ 1., 1., 1., -3.]) >>> lagint([1,2,3], m=2) array([ 1., 0., 0., -4., 3.]) >>> lagint([1,2,3], k=1) array([ 2., 1., 1., -3.]) >>> lagint([1,2,3], lbnd=-1) array([11.5, 1. , 1. , -3. ]) >>> lagint([1,2], m=2, k=[1,2], lbnd=-1) array([ 11.16666667, -5. , -3. , 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] tmp[1] = -c[0] for j in range(1, n): tmp[j] += c[j] tmp[j + 1] = -c[j] tmp[0] += k[i] - lagval(lbnd, tmp) c = tmp c = np.moveaxis(c, 0, iaxis) return c def lagval(x, c, tensor=True): """ Evaluate a Laguerre 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 -------- lagval2d, laggrid2d, lagval3d, laggrid3d Notes ----- The evaluation uses Clenshaw recursion, aka synthetic division. Examples -------- >>> from numpy.polynomial.laguerre import lagval >>> coef = [1,2,3] >>> lagval(1, coef) -0.5 >>> lagval([[1,2],[3,4]], coef) array([[-0.5, -4. ], [-4.5, -2. ]]) """ 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((2nd - 1) - x))/nd return c0 + c1(1 - x) def lagval2d(x, y, c): """ Evaluate a 2-D Laguerre 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 polynomial at points formed with pairs of corresponding values from `x` and `y`. See Also -------- lagval, laggrid2d, lagval3d, laggrid3d Notes ----- .. versionadded:: 1.7.0 """ return pu._valnd(lagval, c, x, y) def laggrid2d(x, y, c): """ Evaluate a 2-D Laguerre 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 -------- lagval, lagval2d, lagval3d, laggrid3d Notes ----- .. versionadded:: 1.7.0 """ return pu._gridnd(lagval, c, x, y) def lagval3d(x, y, z, c): """ Evaluate a 3-D Laguerre 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 -------- lagval, lagval2d, laggrid2d, laggrid3d Notes ----- .. versionadded:: 1.7.0 """ return pu._valnd(lagval, c, x, y, z) def laggrid3d(x, y, z, c): """ Evaluate a 3-D Laguerre 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 -------- lagval, lagval2d, laggrid2d, lagval3d Notes ----- .. versionadded:: 1.7.0 """ return pu._gridnd(lagval, c, x, y, z) def lagvander(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 Laguerre polynomial. If `c` is a 1-D array of coefficients of length `n + 1` and `V` is the array ``V = lagvander(x, n)``, then ``np.dot(V, c)`` and ``lagval(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 Laguerre 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 Laguerre polynomial. The dtype will be the same as the converted `x`. Examples -------- >>> from numpy.polynomial.laguerre import lagvander >>> x = np.array([0, 1, 2]) >>> lagvander(x, 3) array([[ 1. , 1. , 1. , 1. ], [ 1. , 0. , -0.5 , -0.66666667], [ 1. , -1. , -1. , -0.33333333]]) """ 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] = x0 + 1 if ideg > 0: v[1] = 1 - x for i in range(2, ideg + 1): v[i] = (v[i-1](2i - 1 - x) - v[i-2](i - 1))/i return np.moveaxis(v, 0, -1) def lagvander2d(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 Laguerre polynomials. If ``V = lagvander2d(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 ``lagval2d(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 Laguerre 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 -------- lagvander, lagvander3d, lagval2d, lagval3d Notes ----- .. versionadded:: 1.7.0 """ return pu._vander_nd_flat((lagvander, lagvander), (x, y), deg) def lagvander3d(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 Laguerre polynomials. If ``V = lagvander3d(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 ``lagval3d(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 Laguerre 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 -------- lagvander, lagvander3d, lagval2d, lagval3d Notes ----- .. versionadded:: 1.7.0 """ return pu._vander_nd_flat((lagvander, lagvander, lagvander), (x, y, z), deg) def lagfit(x, y, deg, rcond=None, full=False, w=None): """ Least squares fit of Laguerre series to data. Return the coefficients of a Laguerre 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. Returns ------- coef : ndarray, shape (M,) or (M, K) Laguerre 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.chebyshev.chebfit numpy.polynomial.hermite.hermfit numpy.polynomial.hermite_e.hermefit lagval : Evaluates a Laguerre series. lagvander : pseudo Vandermonde matrix of Laguerre series. lagweight : Laguerre 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 Laguerre 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 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 Laguerre series are probably most useful when the data can be approximated by ``sqrt(w(x)) * p(x)``, where ``w(x)`` is the Laguerre 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 `lagweight`. References ---------- .. [1] Wikipedia, "Curve fitting", https://en.wikipedia.org/wiki/Curve_fitting Examples -------- >>> from numpy.polynomial.laguerre import lagfit, lagval >>> x = np.linspace(0, 10) >>> err = np.random.randn(len(x))/10 >>> y = lagval(x, [1, 2, 3]) + err >>> lagfit(x, y, 2) array([ 0.96971004, 2.00193749, 3.00288744]) # may vary """ return pu._fit(lagvander, x, y, deg, rcond, full, w) def lagcompanion(c): """ Return the companion matrix of c. The usual companion matrix of the Laguerre polynomials is already symmetric when `c` is a basis Laguerre polynomial, so no scaling is applied. Parameters ---------- c : array_like 1-D array of Laguerre series 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([[1 + c[0]/c[1]]]) n = len(c) - 1 mat = np.zeros((n, n), dtype=c.dtype) top = mat.reshape(-1)[1::n+1] mid = mat.reshape(-1)[0::n+1] bot = mat.reshape(-1)[n::n+1] top[...] = -np.arange(1, n) mid[...] = 2.np.arange(n) + 1. bot[...] = top mat[:, -1] += (c[:-1]/c[-1])n return mat def lagroots(c): """ Compute the roots of a Laguerre 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.legendre.legroots numpy.polynomial.chebyshev.chebroots 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 Laguerre series basis polynomials aren't powers of `x` so the results of this function may seem unintuitive. Examples -------- >>> from numpy.polynomial.laguerre import lagroots, lagfromroots >>> coef = lagfromroots([0, 1, 2]) >>> coef array([ 2., -8., 12., -6.]) >>> lagroots(coef) array([-4.4408921e-16, 1.0000000e+00, 2.0000000e+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([1 + c[0]/c[1]]) # rotated companion matrix reduces error m = lagcompanion(c)[::-1,::-1] r = la.eigvals(m) r.sort() return r def laggauss(deg): """ Gauss-Laguerre quadrature. Computes the sample points and weights for Gauss-Laguerre quadrature. These sample points and weights will correctly integrate polynomials of degree :math:`2deg - 1` or less over the interval :math:`[0, \\inf]` with the weight function :math:`f(x) = \\exp(-x)`. 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 = lagcompanion(c) x = la.eigvalsh(m) # improve roots by one application of Newton dy = lagval(x, c) df = lagval(x, lagder(c)) x -= dy/df # compute the weights. We scale the factor to avoid possible numerical # overflow. fm = lagval(x, c[1:]) fm /= np.abs(fm).max() df /= np.abs(df).max() w = 1/(fm df) # scale w to get the right value, 1 in this case w /= w.sum() return x, w def lagweight(x): """Weight function of the Laguerre polynomials. The weight function is :math:`exp(-x)` and the interval of integration is :math:`[0, \\inf]`. The Laguerre 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) return w # # Laguerre series class # class Laguerre(ABCPolyBase): """A Laguerre series class. The Laguerre 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 Laguerre coefficients in order of increasing degree, i.e, ``(1, 2, 3)`` gives ``1L_0(x) + 2L_1(X) + 3L_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 [0, 1]. window : (2,) array_like, optional Window, see `domain` for its use. The default value is [0, 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(lagadd) _sub = staticmethod(lagsub) _mul = staticmethod(lagmul) _div = staticmethod(lagdiv) _pow = staticmethod(lagpow) _val = staticmethod(lagval) _int = staticmethod(lagint) _der = staticmethod(lagder) _fit = staticmethod(lagfit) _line = staticmethod(lagline) _roots = staticmethod(lagroots) _fromroots = staticmethod(lagfromroots) # Virtual properties domain = np.array(lagdomain) window = np.array(lagdomain) basis_name = 'L'	49.67 KB	2026-01-20 12:58
📄 laguerre.pyi 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] lagtrim = trimcoef def poly2lag(pol): ... def lag2poly(c): ... lagdomain: ndarray[Any, dtype[int_]] lagzero: ndarray[Any, dtype[int_]] lagone: ndarray[Any, dtype[int_]] lagx: ndarray[Any, dtype[int_]] def lagline(off, scl): ... def lagfromroots(roots): ... def lagadd(c1, c2): ... def lagsub(c1, c2): ... def lagmulx(c): ... def lagmul(c1, c2): ... def lagdiv(c1, c2): ... def lagpow(c, pow, maxpower=...): ... def lagder(c, m=..., scl=..., axis=...): ... def lagint(c, m=..., k = ..., lbnd=..., scl=..., axis=...): ... def lagval(x, c, tensor=...): ... def lagval2d(x, y, c): ... def laggrid2d(x, y, c): ... def lagval3d(x, y, z, c): ... def laggrid3d(x, y, z, c): ... def lagvander(x, deg): ... def lagvander2d(x, y, deg): ... def lagvander3d(x, y, z, deg): ... def lagfit(x, y, deg, rcond=..., full=..., w=...): ... def lagcompanion(c): ... def lagroots(c): ... def laggauss(deg): ... def lagweight(x): ... class Laguerre(ABCPolyBase): domain: Any window: Any basis_name: Any	1.15 KB	2026-01-20 12:58
📄 legendre.py """ ================================================== 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)(2i - 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 + sclx``. Returns ------- y : ndarray This module's representation of the Legendre series for ``off + sclx``. 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 + 2P_1 + 3P_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 + 2P_1 + 3P_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) + iP_{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 + 2P_1 + 3P_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)(2nd - 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 + 2P_1 + 3P_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 + 2P_1 + 3P_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 ``1L_0 + 2L_1 + 3L_2`` while [[1,2],[1,2]] represents ``1L_0(x)L_0(y) + 1L_1(x)L_0(y) + 2L_0(x)L_1(y) + 2L_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] = (2j - 1)c[j] c[j - 2] += c[j] if n > 1: der[1] = 3c[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 + 2L_1 + 3L_2`` while [[1,2],[1,2]] represents ``1L_0(x)L_0(y) + 1L_1(x)L_0(y) + 2L_0(x)L_1(y) + 2L_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]/(2j + 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 + (c1x(2nd - 1))/nd return c0 + c1x 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] = x0 + 1 if ideg > 0: v[1] = x for i in range(2, ideg + 1): v[i] = (v[i-1]x(2i - 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(2np.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/(2n - 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:`2deg - 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 = x0.0 + 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 ``1P_0(x) + 2P_1(x) + 3P_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'	50.34 KB	2026-01-20 12:58
📄 legendre.pyi 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	1.15 KB	2026-01-20 12:58
📄 polynomial.py """ ================================================= 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 + sclx``. 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 + 2x + 3x2``. 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(22) 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 + 2x + 3x*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 + 2x + 3x*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 + 2x + 3x2``. 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] -= c2c1[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 + 2x + 3x*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 + 2x + 3x2`` while [[1,2],[1,2]] represents ``1 + 1x + 2y + 2xy`` 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 ``sclm``. 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 + 3x2 + 4x3 >>> P.polyder(c) # (d/dx)(c) = 2 + 6x + 12x2 array([ 2., 6., 12.]) >>> P.polyder(c,3) # (d3/dx3)(c) = 24 array([24.]) >>> P.polyder(c,scl=-1) # (d/d(-x))(c) = -2 - 6x - 12x2 array([ -2., -6., -12.]) >>> P.polyder(c,2,-1) # (d2/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] = jc[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 + 2x + 3x*2`` while [[1,2],[1,2]] represents ``1 + 1x + 2y + 2xy`` 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] + x0 for i in range(2, len(c) + 1): c0 = c[-i] + c0x 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] = x0 + 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 = x3 - 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 = x3 - 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 + 2x + 3x2``. 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}}}"	47.96 KB	2026-01-20 12:58
📄 polynomial.pyi 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] polytrim = trimcoef polydomain: ndarray[Any, dtype[int_]] polyzero: ndarray[Any, dtype[int_]] polyone: ndarray[Any, dtype[int_]] polyx: ndarray[Any, dtype[int_]] def polyline(off, scl): ... def polyfromroots(roots): ... def polyadd(c1, c2): ... def polysub(c1, c2): ... def polymulx(c): ... def polymul(c1, c2): ... def polydiv(c1, c2): ... def polypow(c, pow, maxpower=...): ... def polyder(c, m=..., scl=..., axis=...): ... def polyint(c, m=..., k=..., lbnd=..., scl=..., axis=...): ... def polyval(x, c, tensor=...): ... def polyvalfromroots(x, r, tensor=...): ... def polyval2d(x, y, c): ... def polygrid2d(x, y, c): ... def polyval3d(x, y, z, c): ... def polygrid3d(x, y, z, c): ... def polyvander(x, deg): ... def polyvander2d(x, y, deg): ... def polyvander3d(x, y, z, deg): ... def polyfit(x, y, deg, rcond=..., full=..., w=...): ... def polyroots(c): ... class Polynomial(ABCPolyBase): domain: Any window: Any basis_name: Any	1.11 KB	2026-01-20 12:58
📄 polyutils.py """ Utility classes and functions for the polynomial modules. This module provides: error and warning objects; a polynomial base class; and some routines used in both the `polynomial` and `chebyshev` modules. Warning objects --------------- .. autosummary:: :toctree: generated/ RankWarning raised in least-squares fit for rank-deficient matrix. Functions --------- .. autosummary:: :toctree: generated/ as_series convert list of array_likes into 1-D arrays of common type. trimseq remove trailing zeros. trimcoef remove small trailing coefficients. getdomain return the domain appropriate for a given set of abscissae. mapdomain maps points between domains. mapparms parameters of the linear map between domains. """ import operator import functools import warnings import numpy as np from numpy.core.multiarray import dragon4_positional, dragon4_scientific from numpy.core.umath import absolute __all__ = [ 'RankWarning', 'as_series', 'trimseq', 'trimcoef', 'getdomain', 'mapdomain', 'mapparms', 'format_float'] # # Warnings and Exceptions # class RankWarning(UserWarning): """Issued by chebfit when the design matrix is rank deficient.""" pass # # Helper functions to convert inputs to 1-D arrays # def trimseq(seq): """Remove small Poly series coefficients. Parameters ---------- seq : sequence Sequence of Poly series coefficients. This routine fails for empty sequences. Returns ------- series : sequence Subsequence with trailing zeros removed. If the resulting sequence would be empty, return the first element. The returned sequence may or may not be a view. Notes ----- Do not lose the type info if the sequence contains unknown objects. """ if len(seq) == 0: return seq else: for i in range(len(seq) - 1, -1, -1): if seq[i] != 0: break return seq[:i+1] def as_series(alist, trim=True): """ Return argument as a list of 1-d arrays. The returned list contains array(s) of dtype double, complex double, or object. A 1-d argument of shape ``(N,)`` is parsed into ``N`` arrays of size one; a 2-d argument of shape ``(M,N)`` is parsed into ``M`` arrays of size ``N`` (i.e., is "parsed by row"); and a higher dimensional array raises a Value Error if it is not first reshaped into either a 1-d or 2-d array. Parameters ---------- alist : array_like A 1- or 2-d array_like trim : boolean, optional When True, trailing zeros are removed from the inputs. When False, the inputs are passed through intact. Returns ------- [a1, a2,...] : list of 1-D arrays A copy of the input data as a list of 1-d arrays. Raises ------ ValueError Raised when `as_series` cannot convert its input to 1-d arrays, or at least one of the resulting arrays is empty. Examples -------- >>> from numpy.polynomial import polyutils as pu >>> a = np.arange(4) >>> pu.as_series(a) [array([0.]), array([1.]), array([2.]), array([3.])] >>> b = np.arange(6).reshape((2,3)) >>> pu.as_series(b) [array([0., 1., 2.]), array([3., 4., 5.])] >>> pu.as_series((1, np.arange(3), np.arange(2, dtype=np.float16))) [array([1.]), array([0., 1., 2.]), array([0., 1.])] >>> pu.as_series([2, [1.1, 0.]]) [array([2.]), array([1.1])] >>> pu.as_series([2, [1.1, 0.]], trim=False) [array([2.]), array([1.1, 0. ])] """ arrays = [np.array(a, ndmin=1, copy=False) for a in alist] if min([a.size for a in arrays]) == 0: raise ValueError("Coefficient array is empty") if any(a.ndim != 1 for a in arrays): raise ValueError("Coefficient array is not 1-d") if trim: arrays = [trimseq(a) for a in arrays] if any(a.dtype == np.dtype(object) for a in arrays): ret = [] for a in arrays: if a.dtype != np.dtype(object): tmp = np.empty(len(a), dtype=np.dtype(object)) tmp[:] = a[:] ret.append(tmp) else: ret.append(a.copy()) else: try: dtype = np.common_type(arrays) except Exception as e: raise ValueError("Coefficient arrays have no common type") from e ret = [np.array(a, copy=True, dtype=dtype) for a in arrays] return ret def trimcoef(c, tol=0): """ Remove "small" "trailing" coefficients from a polynomial. "Small" means "small in absolute value" and is controlled by the parameter `tol`; "trailing" means highest order coefficient(s), e.g., in ``[0, 1, 1, 0, 0]`` (which represents ``0 + x + x2 + 0x*3 + 0x*4``) both the 3-rd and 4-th order coefficients would be "trimmed." Parameters ---------- c : array_like 1-d array of coefficients, ordered from lowest order to highest. tol : number, optional Trailing (i.e., highest order) elements with absolute value less than or equal to `tol` (default value is zero) are removed. Returns ------- trimmed : ndarray 1-d array with trailing zeros removed. If the resulting series would be empty, a series containing a single zero is returned. Raises ------ ValueError If `tol` < 0 See Also -------- trimseq Examples -------- >>> from numpy.polynomial import polyutils as pu >>> pu.trimcoef((0,0,3,0,5,0,0)) array([0., 0., 3., 0., 5.]) >>> pu.trimcoef((0,0,1e-3,0,1e-5,0,0),1e-3) # item == tol is trimmed array([0.]) >>> i = complex(0,1) # works for complex >>> pu.trimcoef((3e-4,1e-3(1-i),5e-4,2e-5(1+i)), 1e-3) array([0.0003+0.j , 0.001 -0.001j]) """ if tol < 0: raise ValueError("tol must be non-negative") [c] = as_series([c]) [ind] = np.nonzero(np.abs(c) > tol) if len(ind) == 0: return c[:1]0 else: return c[:ind[-1] + 1].copy() def getdomain(x): """ Return a domain suitable for given abscissae. Find a domain suitable for a polynomial or Chebyshev series defined at the values supplied. Parameters ---------- x : array_like 1-d array of abscissae whose domain will be determined. Returns ------- domain : ndarray 1-d array containing two values. If the inputs are complex, then the two returned points are the lower left and upper right corners of the smallest rectangle (aligned with the axes) in the complex plane containing the points `x`. If the inputs are real, then the two points are the ends of the smallest interval containing the points `x`. See Also -------- mapparms, mapdomain Examples -------- >>> from numpy.polynomial import polyutils as pu >>> points = np.arange(4)*2 - 5; points array([-5, -4, -1, 4]) >>> pu.getdomain(points) array([-5., 4.]) >>> c = np.exp(complex(0,1)np.pinp.arange(12)/6) # unit circle >>> pu.getdomain(c) array([-1.-1.j, 1.+1.j]) """ [x] = as_series([x], trim=False) if x.dtype.char in np.typecodes['Complex']: rmin, rmax = x.real.min(), x.real.max() imin, imax = x.imag.min(), x.imag.max() return np.array((complex(rmin, imin), complex(rmax, imax))) else: return np.array((x.min(), x.max())) def mapparms(old, new): """ Linear map parameters between domains. Return the parameters of the linear map ``offset + scalex`` that maps `old` to `new` such that ``old[i] -> new[i]``, ``i = 0, 1``. Parameters ---------- old, new : array_like Domains. Each domain must (successfully) convert to a 1-d array containing precisely two values. Returns ------- offset, scale : scalars The map ``L(x) = offset + scalex`` maps the first domain to the second. See Also -------- getdomain, mapdomain Notes ----- Also works for complex numbers, and thus can be used to calculate the parameters required to map any line in the complex plane to any other line therein. Examples -------- >>> from numpy.polynomial import polyutils as pu >>> pu.mapparms((-1,1),(-1,1)) (0.0, 1.0) >>> pu.mapparms((1,-1),(-1,1)) (-0.0, -1.0) >>> i = complex(0,1) >>> pu.mapparms((-i,-1),(1,i)) ((1+1j), (1-0j)) """ oldlen = old[1] - old[0] newlen = new[1] - new[0] off = (old[1]new[0] - old[0]new[1])/oldlen scl = newlen/oldlen return off, scl def mapdomain(x, old, new): """ Apply linear map to input points. The linear map ``offset + scalex`` that maps the domain `old` to the domain `new` is applied to the points `x`. Parameters ---------- x : array_like Points to be mapped. If `x` is a subtype of ndarray the subtype will be preserved. old, new : array_like The two domains that determine the map. Each must (successfully) convert to 1-d arrays containing precisely two values. Returns ------- x_out : ndarray Array of points of the same shape as `x`, after application of the linear map between the two domains. See Also -------- getdomain, mapparms Notes ----- Effectively, this implements: .. math:: x\\_out = new[0] + m(x - old[0]) where .. math:: m = \\frac{new[1]-new[0]}{old[1]-old[0]} Examples -------- >>> from numpy.polynomial import polyutils as pu >>> old_domain = (-1,1) >>> new_domain = (0,2np.pi) >>> x = np.linspace(-1,1,6); x array([-1. , -0.6, -0.2, 0.2, 0.6, 1. ]) >>> x_out = pu.mapdomain(x, old_domain, new_domain); x_out array([ 0. , 1.25663706, 2.51327412, 3.76991118, 5.02654825, # may vary 6.28318531]) >>> x - pu.mapdomain(x_out, new_domain, old_domain) array([0., 0., 0., 0., 0., 0.]) Also works for complex numbers (and thus can be used to map any line in the complex plane to any other line therein). >>> i = complex(0,1) >>> old = (-1 - i, 1 + i) >>> new = (-1 + i, 1 - i) >>> z = np.linspace(old[0], old[1], 6); z array([-1. -1.j , -0.6-0.6j, -0.2-0.2j, 0.2+0.2j, 0.6+0.6j, 1. +1.j ]) >>> new_z = pu.mapdomain(z, old, new); new_z array([-1.0+1.j , -0.6+0.6j, -0.2+0.2j, 0.2-0.2j, 0.6-0.6j, 1.0-1.j ]) # may vary """ x = np.asanyarray(x) off, scl = mapparms(old, new) return off + sclx def _nth_slice(i, ndim): sl = [np.newaxis] * ndim sl[i] = slice(None) return tuple(sl) def _vander_nd(vander_fs, points, degrees): r""" A generalization of the Vandermonde matrix for N dimensions The result is built by combining the results of 1d Vandermonde matrices, .. math:: W[i_0, \ldots, i_M, j_0, \ldots, j_N] = \prod_{k=0}^N{V_k(x_k)[i_0, \ldots, i_M, j_k]} where .. math:: N &= \texttt{len(points)} = \texttt{len(degrees)} = \texttt{len(vander\_fs)} \\ M &= \texttt{points[k].ndim} \\ V_k &= \texttt{vander\_fs[k]} \\ x_k &= \texttt{points[k]} \\ 0 \le j_k &\le \texttt{degrees[k]} Expanding the one-dimensional :math:`V_k` functions gives: .. math:: W[i_0, \ldots, i_M, j_0, \ldots, j_N] = \prod_{k=0}^N{B_{k, j_k}(x_k[i_0, \ldots, i_M])} where :math:`B_{k,m}` is the m'th basis of the polynomial construction used along dimension :math:`k`. For a regular polynomial, :math:`B_{k, m}(x) = P_m(x) = x^m`. Parameters ---------- vander_fs : Sequence[function(array_like, int) -> ndarray] The 1d vander function to use for each axis, such as ``polyvander`` points : Sequence[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. This must be the same length as `vander_fs`. degrees : Sequence[int] The maximum degree (inclusive) to use for each axis. This must be the same length as `vander_fs`. Returns ------- vander_nd : ndarray An array of shape ``points[0].shape + tuple(d + 1 for d in degrees)``. """ n_dims = len(vander_fs) if n_dims != len(points): raise ValueError( f"Expected {n_dims} dimensions of sample points, got {len(points)}") if n_dims != len(degrees): raise ValueError( f"Expected {n_dims} dimensions of degrees, got {len(degrees)}") if n_dims == 0: raise ValueError("Unable to guess a dtype or shape when no points are given") # convert to the same shape and type points = tuple(np.array(tuple(points), copy=False) + 0.0) # produce the vandermonde matrix for each dimension, placing the last # axis of each in an independent trailing axis of the output vander_arrays = ( vander_fs[i](points[i], degrees[i])[(...,) + _nth_slice(i, n_dims)] for i in range(n_dims) ) # we checked this wasn't empty already, so no `initial` needed return functools.reduce(operator.mul, vander_arrays) def _vander_nd_flat(vander_fs, points, degrees): """ Like `_vander_nd`, but flattens the last ``len(degrees)`` axes into a single axis Used to implement the public ``<type>vander<n>d`` functions. """ v = _vander_nd(vander_fs, points, degrees) return v.reshape(v.shape[:-len(degrees)] + (-1,)) def _fromroots(line_f, mul_f, roots): """ Helper function used to implement the ``<type>fromroots`` functions. Parameters ---------- line_f : function(float, float) -> ndarray The ``<type>line`` function, such as ``polyline`` mul_f : function(array_like, array_like) -> ndarray The ``<type>mul`` function, such as ``polymul`` roots See the ``<type>fromroots`` functions for more detail """ if len(roots) == 0: return np.ones(1) else: [roots] = as_series([roots], trim=False) roots.sort() p = [line_f(-r, 1) for r in roots] n = len(p) while n > 1: m, r = divmod(n, 2) tmp = [mul_f(p[i], p[i+m]) for i in range(m)] if r: tmp[0] = mul_f(tmp[0], p[-1]) p = tmp n = m return p[0] def _valnd(val_f, c, args): """ Helper function used to implement the ``<type>val<n>d`` functions. Parameters ---------- val_f : function(array_like, array_like, tensor: bool) -> array_like The ``<type>val`` function, such as ``polyval`` c, args See the ``<type>val<n>d`` functions for more detail """ args = [np.asanyarray(a) for a in args] shape0 = args[0].shape if not all((a.shape == shape0 for a in args[1:])): if len(args) == 3: raise ValueError('x, y, z are incompatible') elif len(args) == 2: raise ValueError('x, y are incompatible') else: raise ValueError('ordinates are incompatible') it = iter(args) x0 = next(it) # use tensor on only the first c = val_f(x0, c) for xi in it: c = val_f(xi, c, tensor=False) return c def _gridnd(val_f, c, args): """ Helper function used to implement the ``<type>grid<n>d`` functions. Parameters ---------- val_f : function(array_like, array_like, tensor: bool) -> array_like The ``<type>val`` function, such as ``polyval`` c, args See the ``<type>grid<n>d`` functions for more detail """ for xi in args: c = val_f(xi, c) return c def _div(mul_f, c1, c2): """ Helper function used to implement the ``<type>div`` functions. Implementation uses repeated subtraction of c2 multiplied by the nth basis. For some polynomial types, a more efficient approach may be possible. Parameters ---------- mul_f : function(array_like, array_like) -> array_like The ``<type>mul`` function, such as ``polymul`` c1, c2 See the ``<type>div`` functions for more detail """ # c1, c2 are trimmed copies [c1, c2] = as_series([c1, c2]) if c2[-1] == 0: raise ZeroDivisionError() 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: quo = np.empty(lc1 - lc2 + 1, dtype=c1.dtype) rem = c1 for i in range(lc1 - lc2, - 1, -1): p = mul_f([0]i + [1], c2) q = rem[-1]/p[-1] rem = rem[:-1] - qp[:-1] quo[i] = q return quo, trimseq(rem) def _add(c1, c2): """ Helper function used to implement the ``<type>add`` functions. """ # c1, c2 are trimmed copies [c1, c2] = as_series([c1, c2]) if len(c1) > len(c2): c1[:c2.size] += c2 ret = c1 else: c2[:c1.size] += c1 ret = c2 return trimseq(ret) def _sub(c1, c2): """ Helper function used to implement the ``<type>sub`` functions. """ # c1, c2 are trimmed copies [c1, c2] = as_series([c1, c2]) if len(c1) > len(c2): c1[:c2.size] -= c2 ret = c1 else: c2 = -c2 c2[:c1.size] += c1 ret = c2 return trimseq(ret) def _fit(vander_f, x, y, deg, rcond=None, full=False, w=None): """ Helper function used to implement the ``<type>fit`` functions. Parameters ---------- vander_f : function(array_like, int) -> ndarray The 1d vander function, such as ``polyvander`` c1, c2 See the ``<type>fit`` functions for more detail """ x = np.asarray(x) + 0.0 y = np.asarray(y) + 0.0 deg = np.asarray(deg) # check arguments. if deg.ndim > 1 or deg.dtype.kind not in 'iu' or deg.size == 0: raise TypeError("deg must be an int or non-empty 1-D array of int") if deg.min() < 0: raise ValueError("expected deg >= 0") if x.ndim != 1: raise TypeError("expected 1D vector for x") if x.size == 0: raise TypeError("expected non-empty vector for x") if y.ndim < 1 or y.ndim > 2: raise TypeError("expected 1D or 2D array for y") if len(x) != len(y): raise TypeError("expected x and y to have same length") if deg.ndim == 0: lmax = deg order = lmax + 1 van = vander_f(x, lmax) else: deg = np.sort(deg) lmax = deg[-1] order = len(deg) van = vander_f(x, lmax)[:, deg] # set up the least squares matrices in transposed form lhs = van.T rhs = y.T if w is not None: w = np.asarray(w) + 0.0 if w.ndim != 1: raise TypeError("expected 1D vector for w") if len(x) != len(w): raise TypeError("expected x and w to have same length") # apply weights. Don't use inplace operations as they # can cause problems with NA. lhs = lhs * w rhs = rhs * w # set rcond if rcond is None: rcond = len(x)np.finfo(x.dtype).eps # Determine the norms of the design matrix columns. if issubclass(lhs.dtype.type, np.complexfloating): scl = np.sqrt((np.square(lhs.real) + np.square(lhs.imag)).sum(1)) else: scl = np.sqrt(np.square(lhs).sum(1)) scl[scl == 0] = 1 # Solve the least squares problem. c, resids, rank, s = np.linalg.lstsq(lhs.T/scl, rhs.T, rcond) c = (c.T/scl).T # Expand c to include non-fitted coefficients which are set to zero if deg.ndim > 0: if c.ndim == 2: cc = np.zeros((lmax+1, c.shape[1]), dtype=c.dtype) else: cc = np.zeros(lmax+1, dtype=c.dtype) cc[deg] = c c = cc # warn on rank reduction if rank != order and not full: msg = "The fit may be poorly conditioned" warnings.warn(msg, RankWarning, stacklevel=2) if full: return c, [resids, rank, s, rcond] else: return c def _pow(mul_f, c, pow, maxpower): """ Helper function used to implement the ``<type>pow`` functions. Parameters ---------- mul_f : function(array_like, array_like) -> ndarray The ``<type>mul`` function, such as ``polymul`` c : array_like 1-D array of array of series coefficients pow, maxpower See the ``<type>pow`` functions for more detail """ # c is a trimmed copy [c] = 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. prd = c for i in range(2, power + 1): prd = mul_f(prd, c) return prd def _deprecate_as_int(x, desc): """ Like `operator.index`, but emits a deprecation warning when passed a float Parameters ---------- x : int-like, or float with integral value Value to interpret as an integer desc : str description to include in any error message Raises ------ TypeError : if x is a non-integral float or non-numeric DeprecationWarning : if x is an integral float """ try: return operator.index(x) except TypeError as e: # Numpy 1.17.0, 2019-03-11 try: ix = int(x) except TypeError: pass else: if ix == x: warnings.warn( f"In future, this will raise TypeError, as {desc} will " "need to be an integer not just an integral float.", DeprecationWarning, stacklevel=3 ) return ix raise TypeError(f"{desc} must be an integer") from e def format_float(x, parens=False): if not np.issubdtype(type(x), np.floating): return str(x) opts = np.get_printoptions() if np.isnan(x): return opts['nanstr'] elif np.isinf(x): return opts['infstr'] exp_format = False if x != 0: a = absolute(x) if a >= 1.e8 or a < 10*min(0, -(opts['precision']-1)//2): exp_format = True trim, unique = '0', True if opts['floatmode'] == 'fixed': trim, unique = 'k', False if exp_format: s = dragon4_scientific(x, precision=opts['precision'], unique=unique, trim=trim, sign=opts['sign'] == '+') if parens: s = '(' + s + ')' else: s = dragon4_positional(x, precision=opts['precision'], fractional=True, unique=unique, trim=trim, sign=opts['sign'] == '+') return s	22.69 KB	2026-01-20 12:58
📄 polyutils.pyi __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=...): ...	264 B	2026-01-20 12:58
📄 setup.py 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)	373 B	2026-01-20 12:58
📁 tests	-	2026-03-21 15:45

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