1854 lines
56 KiB
Python
1854 lines
56 KiB
Python
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"""
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Objects for dealing with Hermite series.
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This module provides a number of objects (mostly functions) useful for
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dealing with Hermite series, including a `Hermite` class that
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encapsulates the usual arithmetic operations. (General information
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on how this module represents and works with such polynomials is in the
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docstring for its "parent" sub-package, `numpy.polynomial`).
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Constants
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---------
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- `hermdomain` -- Hermite series default domain, [-1,1].
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- `hermzero` -- Hermite series that evaluates identically to 0.
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- `hermone` -- Hermite series that evaluates identically to 1.
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- `hermx` -- Hermite series for the identity map, ``f(x) = x``.
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Arithmetic
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----------
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- `hermmulx` -- multiply a Hermite series in ``P_i(x)`` by ``x``.
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- `hermadd` -- add two Hermite series.
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- `hermsub` -- subtract one Hermite series from another.
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- `hermmul` -- multiply two Hermite series.
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- `hermdiv` -- divide one Hermite series by another.
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- `hermval` -- evaluate a Hermite series at given points.
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- `hermval2d` -- evaluate a 2D Hermite series at given points.
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- `hermval3d` -- evaluate a 3D Hermite series at given points.
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- `hermgrid2d` -- evaluate a 2D Hermite series on a Cartesian product.
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- `hermgrid3d` -- evaluate a 3D Hermite series on a Cartesian product.
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Calculus
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--------
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- `hermder` -- differentiate a Hermite series.
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- `hermint` -- integrate a Hermite series.
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Misc Functions
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--------------
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- `hermfromroots` -- create a Hermite series with specified roots.
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- `hermroots` -- find the roots of a Hermite series.
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- `hermvander` -- Vandermonde-like matrix for Hermite polynomials.
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- `hermvander2d` -- Vandermonde-like matrix for 2D power series.
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- `hermvander3d` -- Vandermonde-like matrix for 3D power series.
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- `hermgauss` -- Gauss-Hermite quadrature, points and weights.
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- `hermweight` -- Hermite weight function.
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- `hermcompanion` -- symmetrized companion matrix in Hermite form.
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- `hermfit` -- least-squares fit returning a Hermite series.
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- `hermtrim` -- trim leading coefficients from a Hermite series.
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- `hermline` -- Hermite series of given straight line.
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- `herm2poly` -- convert a Hermite series to a polynomial.
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- `poly2herm` -- convert a polynomial to a Hermite series.
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Classes
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-------
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- `Hermite` -- A Hermite series class.
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See also
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--------
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`numpy.polynomial`
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"""
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from __future__ import division, absolute_import, print_function
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import warnings
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import numpy as np
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import numpy.linalg as la
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from numpy.core.multiarray import normalize_axis_index
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from . import polyutils as pu
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from ._polybase import ABCPolyBase
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__all__ = [
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'hermzero', 'hermone', 'hermx', 'hermdomain', 'hermline', 'hermadd',
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'hermsub', 'hermmulx', 'hermmul', 'hermdiv', 'hermpow', 'hermval',
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'hermder', 'hermint', 'herm2poly', 'poly2herm', 'hermfromroots',
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'hermvander', 'hermfit', 'hermtrim', 'hermroots', 'Hermite',
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'hermval2d', 'hermval3d', 'hermgrid2d', 'hermgrid3d', 'hermvander2d',
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'hermvander3d', 'hermcompanion', 'hermgauss', 'hermweight']
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hermtrim = pu.trimcoef
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def poly2herm(pol):
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"""
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poly2herm(pol)
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Convert a polynomial to a Hermite series.
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Convert an array representing the coefficients of a polynomial (relative
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to the "standard" basis) ordered from lowest degree to highest, to an
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array of the coefficients of the equivalent Hermite series, ordered
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from lowest to highest degree.
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Parameters
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----------
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pol : array_like
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1-D array containing the polynomial coefficients
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Returns
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-------
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c : ndarray
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1-D array containing the coefficients of the equivalent Hermite
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series.
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See Also
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--------
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herm2poly
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Notes
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-----
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The easy way to do conversions between polynomial basis sets
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is to use the convert method of a class instance.
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Examples
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--------
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>>> from numpy.polynomial.hermite import poly2herm
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>>> poly2herm(np.arange(4))
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array([ 1. , 2.75 , 0.5 , 0.375])
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"""
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[pol] = pu.as_series([pol])
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deg = len(pol) - 1
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res = 0
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for i in range(deg, -1, -1):
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res = hermadd(hermmulx(res), pol[i])
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return res
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def herm2poly(c):
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"""
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Convert a Hermite series to a polynomial.
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Convert an array representing the coefficients of a Hermite series,
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ordered from lowest degree to highest, to an array of the coefficients
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of the equivalent polynomial (relative to the "standard" basis) ordered
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from lowest to highest degree.
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Parameters
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----------
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c : array_like
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1-D array containing the Hermite series coefficients, ordered
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from lowest order term to highest.
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Returns
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-------
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pol : ndarray
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1-D array containing the coefficients of the equivalent polynomial
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(relative to the "standard" basis) ordered from lowest order term
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to highest.
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See Also
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--------
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poly2herm
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Notes
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-----
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The easy way to do conversions between polynomial basis sets
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is to use the convert method of a class instance.
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Examples
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--------
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>>> from numpy.polynomial.hermite import herm2poly
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>>> herm2poly([ 1. , 2.75 , 0.5 , 0.375])
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array([ 0., 1., 2., 3.])
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"""
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from .polynomial import polyadd, polysub, polymulx
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[c] = pu.as_series([c])
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n = len(c)
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if n == 1:
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return c
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if n == 2:
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c[1] *= 2
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return c
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else:
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c0 = c[-2]
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c1 = c[-1]
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# i is the current degree of c1
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for i in range(n - 1, 1, -1):
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tmp = c0
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c0 = polysub(c[i - 2], c1*(2*(i - 1)))
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c1 = polyadd(tmp, polymulx(c1)*2)
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return polyadd(c0, polymulx(c1)*2)
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#
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# These are constant arrays are of integer type so as to be compatible
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# with the widest range of other types, such as Decimal.
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#
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# Hermite
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hermdomain = np.array([-1, 1])
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# Hermite coefficients representing zero.
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hermzero = np.array([0])
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# Hermite coefficients representing one.
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hermone = np.array([1])
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# Hermite coefficients representing the identity x.
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hermx = np.array([0, 1/2])
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def hermline(off, scl):
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"""
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Hermite series whose graph is a straight line.
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Parameters
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----------
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off, scl : scalars
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The specified line is given by ``off + scl*x``.
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Returns
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-------
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y : ndarray
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This module's representation of the Hermite series for
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``off + scl*x``.
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See Also
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--------
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polyline, chebline
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Examples
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--------
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>>> from numpy.polynomial.hermite import hermline, hermval
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>>> hermval(0,hermline(3, 2))
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3.0
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>>> hermval(1,hermline(3, 2))
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5.0
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"""
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if scl != 0:
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return np.array([off, scl/2])
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else:
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return np.array([off])
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def hermfromroots(roots):
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"""
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Generate a Hermite series with given roots.
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The function returns the coefficients of the polynomial
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.. math:: p(x) = (x - r_0) * (x - r_1) * ... * (x - r_n),
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in Hermite form, where the `r_n` are the roots specified in `roots`.
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If a zero has multiplicity n, then it must appear in `roots` n times.
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For instance, if 2 is a root of multiplicity three and 3 is a root of
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multiplicity 2, then `roots` looks something like [2, 2, 2, 3, 3]. The
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roots can appear in any order.
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If the returned coefficients are `c`, then
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.. math:: p(x) = c_0 + c_1 * H_1(x) + ... + c_n * H_n(x)
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The coefficient of the last term is not generally 1 for monic
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polynomials in Hermite form.
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Parameters
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----------
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roots : array_like
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Sequence containing the roots.
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Returns
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-------
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out : ndarray
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1-D array of coefficients. If all roots are real then `out` is a
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real array, if some of the roots are complex, then `out` is complex
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even if all the coefficients in the result are real (see Examples
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below).
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See Also
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--------
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polyfromroots, legfromroots, lagfromroots, chebfromroots,
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hermefromroots.
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Examples
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--------
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>>> from numpy.polynomial.hermite import hermfromroots, hermval
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>>> coef = hermfromroots((-1, 0, 1))
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>>> hermval((-1, 0, 1), coef)
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array([ 0., 0., 0.])
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>>> coef = hermfromroots((-1j, 1j))
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>>> hermval((-1j, 1j), coef)
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array([ 0.+0.j, 0.+0.j])
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"""
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if len(roots) == 0:
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return np.ones(1)
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else:
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[roots] = pu.as_series([roots], trim=False)
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roots.sort()
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p = [hermline(-r, 1) for r in roots]
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n = len(p)
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while n > 1:
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m, r = divmod(n, 2)
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tmp = [hermmul(p[i], p[i+m]) for i in range(m)]
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if r:
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tmp[0] = hermmul(tmp[0], p[-1])
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p = tmp
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n = m
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return p[0]
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def hermadd(c1, c2):
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"""
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Add one Hermite series to another.
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Returns the sum of two Hermite series `c1` + `c2`. The arguments
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are sequences of coefficients ordered from lowest order term to
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highest, i.e., [1,2,3] represents the series ``P_0 + 2*P_1 + 3*P_2``.
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Parameters
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----------
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c1, c2 : array_like
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1-D arrays of Hermite series coefficients ordered from low to
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high.
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Returns
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-------
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out : ndarray
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Array representing the Hermite series of their sum.
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See Also
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--------
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hermsub, hermmul, hermdiv, hermpow
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Notes
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-----
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Unlike multiplication, division, etc., the sum of two Hermite series
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is a Hermite series (without having to "reproject" the result onto
|
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the basis set) so addition, just like that of "standard" polynomials,
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is simply "component-wise."
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Examples
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--------
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>>> from numpy.polynomial.hermite import hermadd
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>>> hermadd([1, 2, 3], [1, 2, 3, 4])
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array([ 2., 4., 6., 4.])
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"""
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# c1, c2 are trimmed copies
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[c1, c2] = pu.as_series([c1, c2])
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if len(c1) > len(c2):
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c1[:c2.size] += c2
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ret = c1
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else:
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c2[:c1.size] += c1
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ret = c2
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return pu.trimseq(ret)
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def hermsub(c1, c2):
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"""
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Subtract one Hermite series from another.
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|
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Returns the difference of two Hermite series `c1` - `c2`. The
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sequences of coefficients are from lowest order term to highest, i.e.,
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[1,2,3] represents the series ``P_0 + 2*P_1 + 3*P_2``.
|
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|
|
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Parameters
|
||
|
----------
|
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|
c1, c2 : array_like
|
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1-D arrays of Hermite series coefficients ordered from low to
|
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high.
|
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|
|
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|
Returns
|
||
|
-------
|
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out : ndarray
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Of Hermite series coefficients representing their difference.
|
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|
|
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|
See Also
|
||
|
--------
|
||
|
hermadd, 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"
|
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|
polynomials, is simply "component-wise."
|
||
|
|
||
|
Examples
|
||
|
--------
|
||
|
>>> from numpy.polynomial.hermite import hermsub
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>>> hermsub([1, 2, 3, 4], [1, 2, 3])
|
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array([ 0., 0., 0., 4.])
|
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|
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|
"""
|
||
|
# c1, c2 are trimmed copies
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[c1, c2] = pu.as_series([c1, c2])
|
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|
if len(c1) > len(c2):
|
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|
c1[:c2.size] -= c2
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ret = c1
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|
else:
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|
c2 = -c2
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|
c2[:c1.size] += c1
|
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|
ret = c2
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return pu.trimseq(ret)
|
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|
|
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|
|
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|
def hermmulx(c):
|
||
|
"""Multiply a Hermite series by x.
|
||
|
|
||
|
Multiply the Hermite series `c` by x, where x is the independent
|
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|
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)/2 + i*P_{i - 1}(x))
|
||
|
|
||
|
Examples
|
||
|
--------
|
||
|
>>> from numpy.polynomial.hermite import hermmulx
|
||
|
>>> hermmulx([1, 2, 3])
|
||
|
array([ 2. , 6.5, 1. , 1.5])
|
||
|
|
||
|
"""
|
||
|
# c is a trimmed copy
|
||
|
[c] = pu.as_series([c])
|
||
|
# The zero series needs special treatment
|
||
|
if len(c) == 1 and c[0] == 0:
|
||
|
return c
|
||
|
|
||
|
prd = np.empty(len(c) + 1, dtype=c.dtype)
|
||
|
prd[0] = c[0]*0
|
||
|
prd[1] = c[0]/2
|
||
|
for i in range(1, len(c)):
|
||
|
prd[i + 1] = c[i]/2
|
||
|
prd[i - 1] += c[i]*i
|
||
|
return prd
|
||
|
|
||
|
|
||
|
def hermmul(c1, c2):
|
||
|
"""
|
||
|
Multiply one Hermite series by another.
|
||
|
|
||
|
Returns the product of two Hermite series `c1` * `c2`. The arguments
|
||
|
are sequences of coefficients, from lowest order "term" to highest,
|
||
|
e.g., [1,2,3] represents the series ``P_0 + 2*P_1 + 3*P_2``.
|
||
|
|
||
|
Parameters
|
||
|
----------
|
||
|
c1, c2 : array_like
|
||
|
1-D arrays of Hermite series coefficients ordered from low to
|
||
|
high.
|
||
|
|
||
|
Returns
|
||
|
-------
|
||
|
out : ndarray
|
||
|
Of Hermite series coefficients representing their product.
|
||
|
|
||
|
See Also
|
||
|
--------
|
||
|
hermadd, hermsub, hermdiv, hermpow
|
||
|
|
||
|
Notes
|
||
|
-----
|
||
|
In general, the (polynomial) product of two C-series results in terms
|
||
|
that are not in the Hermite polynomial basis set. Thus, to express
|
||
|
the product as a Hermite series, it is necessary to "reproject" the
|
||
|
product onto said basis set, which may produce "unintuitive" (but
|
||
|
correct) results; see Examples section below.
|
||
|
|
||
|
Examples
|
||
|
--------
|
||
|
>>> from numpy.polynomial.hermite import hermmul
|
||
|
>>> hermmul([1, 2, 3], [0, 1, 2])
|
||
|
array([ 52., 29., 52., 7., 6.])
|
||
|
|
||
|
"""
|
||
|
# s1, s2 are trimmed copies
|
||
|
[c1, c2] = pu.as_series([c1, c2])
|
||
|
|
||
|
if len(c1) > len(c2):
|
||
|
c = c2
|
||
|
xs = c1
|
||
|
else:
|
||
|
c = c1
|
||
|
xs = c2
|
||
|
|
||
|
if len(c) == 1:
|
||
|
c0 = c[0]*xs
|
||
|
c1 = 0
|
||
|
elif len(c) == 2:
|
||
|
c0 = c[0]*xs
|
||
|
c1 = c[1]*xs
|
||
|
else:
|
||
|
nd = len(c)
|
||
|
c0 = c[-2]*xs
|
||
|
c1 = c[-1]*xs
|
||
|
for i in range(3, len(c) + 1):
|
||
|
tmp = c0
|
||
|
nd = nd - 1
|
||
|
c0 = hermsub(c[-i]*xs, c1*(2*(nd - 1)))
|
||
|
c1 = hermadd(tmp, hermmulx(c1)*2)
|
||
|
return hermadd(c0, hermmulx(c1)*2)
|
||
|
|
||
|
|
||
|
def hermdiv(c1, c2):
|
||
|
"""
|
||
|
Divide one Hermite series by another.
|
||
|
|
||
|
Returns the quotient-with-remainder of two Hermite series
|
||
|
`c1` / `c2`. The arguments are sequences of coefficients from lowest
|
||
|
order "term" to highest, e.g., [1,2,3] represents the series
|
||
|
``P_0 + 2*P_1 + 3*P_2``.
|
||
|
|
||
|
Parameters
|
||
|
----------
|
||
|
c1, c2 : array_like
|
||
|
1-D arrays of Hermite series coefficients ordered from low to
|
||
|
high.
|
||
|
|
||
|
Returns
|
||
|
-------
|
||
|
[quo, rem] : ndarrays
|
||
|
Of Hermite series coefficients representing the quotient and
|
||
|
remainder.
|
||
|
|
||
|
See Also
|
||
|
--------
|
||
|
hermadd, hermsub, 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.]))
|
||
|
|
||
|
"""
|
||
|
# c1, c2 are trimmed copies
|
||
|
[c1, c2] = pu.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 = hermmul([0]*i + [1], c2)
|
||
|
q = rem[-1]/p[-1]
|
||
|
rem = rem[:-1] - q*p[:-1]
|
||
|
quo[i] = q
|
||
|
return quo, pu.trimseq(rem)
|
||
|
|
||
|
|
||
|
def hermpow(c, pow, maxpower=16):
|
||
|
"""Raise a Hermite series to a power.
|
||
|
|
||
|
Returns the Hermite series `c` raised to the power `pow`. The
|
||
|
argument `c` is a sequence of coefficients ordered from low to high.
|
||
|
i.e., [1,2,3] is the series ``P_0 + 2*P_1 + 3*P_2.``
|
||
|
|
||
|
Parameters
|
||
|
----------
|
||
|
c : array_like
|
||
|
1-D array of Hermite series coefficients ordered from low to
|
||
|
high.
|
||
|
pow : integer
|
||
|
Power to which the series will be raised
|
||
|
maxpower : integer, optional
|
||
|
Maximum power allowed. This is mainly to limit growth of the series
|
||
|
to unmanageable size. Default is 16
|
||
|
|
||
|
Returns
|
||
|
-------
|
||
|
coef : ndarray
|
||
|
Hermite series of power.
|
||
|
|
||
|
See Also
|
||
|
--------
|
||
|
hermadd, hermsub, hermmul, hermdiv
|
||
|
|
||
|
Examples
|
||
|
--------
|
||
|
>>> from numpy.polynomial.hermite import hermpow
|
||
|
>>> hermpow([1, 2, 3], 2)
|
||
|
array([ 81., 52., 82., 12., 9.])
|
||
|
|
||
|
"""
|
||
|
# 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.
|
||
|
prd = c
|
||
|
for i in range(2, power + 1):
|
||
|
prd = hermmul(prd, c)
|
||
|
return prd
|
||
|
|
||
|
|
||
|
def hermder(c, m=1, scl=1, axis=0):
|
||
|
"""
|
||
|
Differentiate a Hermite series.
|
||
|
|
||
|
Returns the Hermite series coefficients `c` differentiated `m` times
|
||
|
along `axis`. At each iteration the result is multiplied by `scl` (the
|
||
|
scaling factor is for use in a linear change of variable). The argument
|
||
|
`c` is an array of coefficients from low to high degree along each
|
||
|
axis, e.g., [1,2,3] represents the series ``1*H_0 + 2*H_1 + 3*H_2``
|
||
|
while [[1,2],[1,2]] represents ``1*H_0(x)*H_0(y) + 1*H_1(x)*H_0(y) +
|
||
|
2*H_0(x)*H_1(y) + 2*H_1(x)*H_1(y)`` if axis=0 is ``x`` and axis=1 is
|
||
|
``y``.
|
||
|
|
||
|
Parameters
|
||
|
----------
|
||
|
c : array_like
|
||
|
Array of Hermite series coefficients. If `c` is multidimensional the
|
||
|
different axis correspond to different variables with the degree in
|
||
|
each axis given by the corresponding index.
|
||
|
m : int, optional
|
||
|
Number of derivatives taken, must be non-negative. (Default: 1)
|
||
|
scl : scalar, optional
|
||
|
Each differentiation is multiplied by `scl`. The end result is
|
||
|
multiplication by ``scl**m``. This is for use in a linear change of
|
||
|
variable. (Default: 1)
|
||
|
axis : int, optional
|
||
|
Axis over which the derivative is taken. (Default: 0).
|
||
|
|
||
|
.. versionadded:: 1.7.0
|
||
|
|
||
|
Returns
|
||
|
-------
|
||
|
der : ndarray
|
||
|
Hermite series of the derivative.
|
||
|
|
||
|
See Also
|
||
|
--------
|
||
|
hermint
|
||
|
|
||
|
Notes
|
||
|
-----
|
||
|
In general, the result of differentiating a Hermite series does not
|
||
|
resemble the same operation on a power series. Thus the result of this
|
||
|
function may be "unintuitive," albeit correct; see Examples section
|
||
|
below.
|
||
|
|
||
|
Examples
|
||
|
--------
|
||
|
>>> from numpy.polynomial.hermite import hermder
|
||
|
>>> hermder([ 1. , 0.5, 0.5, 0.5])
|
||
|
array([ 1., 2., 3.])
|
||
|
>>> hermder([-0.5, 1./2., 1./8., 1./12., 1./16.], m=2)
|
||
|
array([ 1., 2., 3.])
|
||
|
|
||
|
"""
|
||
|
c = np.array(c, ndmin=1, copy=1)
|
||
|
if c.dtype.char in '?bBhHiIlLqQpP':
|
||
|
c = c.astype(np.double)
|
||
|
cnt, iaxis = [int(t) for t in [m, axis]]
|
||
|
|
||
|
if cnt != m:
|
||
|
raise ValueError("The order of derivation must be integer")
|
||
|
if cnt < 0:
|
||
|
raise ValueError("The order of derivation must be non-negative")
|
||
|
if iaxis != axis:
|
||
|
raise ValueError("The axis must be integer")
|
||
|
iaxis = normalize_axis_index(iaxis, c.ndim)
|
||
|
|
||
|
if cnt == 0:
|
||
|
return c
|
||
|
|
||
|
c = np.moveaxis(c, iaxis, 0)
|
||
|
n = len(c)
|
||
|
if cnt >= n:
|
||
|
c = c[:1]*0
|
||
|
else:
|
||
|
for i in range(cnt):
|
||
|
n = n - 1
|
||
|
c *= scl
|
||
|
der = np.empty((n,) + c.shape[1:], dtype=c.dtype)
|
||
|
for j in range(n, 0, -1):
|
||
|
der[j - 1] = (2*j)*c[j]
|
||
|
c = der
|
||
|
c = np.moveaxis(c, 0, iaxis)
|
||
|
return c
|
||
|
|
||
|
|
||
|
def hermint(c, m=1, k=[], lbnd=0, scl=1, axis=0):
|
||
|
"""
|
||
|
Integrate a Hermite series.
|
||
|
|
||
|
Returns the Hermite series coefficients `c` integrated `m` times from
|
||
|
`lbnd` along `axis`. At each iteration the resulting series is
|
||
|
**multiplied** by `scl` and an integration constant, `k`, is added.
|
||
|
The scaling factor is for use in a linear change of variable. ("Buyer
|
||
|
beware": note that, depending on what one is doing, one may want `scl`
|
||
|
to be the reciprocal of what one might expect; for more information,
|
||
|
see the Notes section below.) The argument `c` is an array of
|
||
|
coefficients from low to high degree along each axis, e.g., [1,2,3]
|
||
|
represents the series ``H_0 + 2*H_1 + 3*H_2`` while [[1,2],[1,2]]
|
||
|
represents ``1*H_0(x)*H_0(y) + 1*H_1(x)*H_0(y) + 2*H_0(x)*H_1(y) +
|
||
|
2*H_1(x)*H_1(y)`` if axis=0 is ``x`` and axis=1 is ``y``.
|
||
|
|
||
|
Parameters
|
||
|
----------
|
||
|
c : array_like
|
||
|
Array of Hermite series coefficients. If c is multidimensional the
|
||
|
different axis correspond to different variables with the degree in
|
||
|
each axis given by the corresponding index.
|
||
|
m : int, optional
|
||
|
Order of integration, must be positive. (Default: 1)
|
||
|
k : {[], list, scalar}, optional
|
||
|
Integration constant(s). The value of the first integral at
|
||
|
``lbnd`` is the first value in the list, the value of the second
|
||
|
integral at ``lbnd`` is the second value, etc. If ``k == []`` (the
|
||
|
default), all constants are set to zero. If ``m == 1``, a single
|
||
|
scalar can be given instead of a list.
|
||
|
lbnd : scalar, optional
|
||
|
The lower bound of the integral. (Default: 0)
|
||
|
scl : scalar, optional
|
||
|
Following each integration the result is *multiplied* by `scl`
|
||
|
before the integration constant is added. (Default: 1)
|
||
|
axis : int, optional
|
||
|
Axis over which the integral is taken. (Default: 0).
|
||
|
|
||
|
.. versionadded:: 1.7.0
|
||
|
|
||
|
Returns
|
||
|
-------
|
||
|
S : ndarray
|
||
|
Hermite series coefficients of the integral.
|
||
|
|
||
|
Raises
|
||
|
------
|
||
|
ValueError
|
||
|
If ``m < 0``, ``len(k) > m``, ``np.ndim(lbnd) != 0``, or
|
||
|
``np.ndim(scl) != 0``.
|
||
|
|
||
|
See Also
|
||
|
--------
|
||
|
hermder
|
||
|
|
||
|
Notes
|
||
|
-----
|
||
|
Note that the result of each integration is *multiplied* by `scl`.
|
||
|
Why is this important to note? Say one is making a linear change of
|
||
|
variable :math:`u = ax + b` in an integral relative to `x`. Then
|
||
|
:math:`dx = du/a`, so one will need to set `scl` equal to
|
||
|
:math:`1/a` - perhaps not what one would have first thought.
|
||
|
|
||
|
Also note that, in general, the result of integrating a C-series needs
|
||
|
to be "reprojected" onto the C-series basis set. Thus, typically,
|
||
|
the result of this function is "unintuitive," albeit correct; see
|
||
|
Examples section below.
|
||
|
|
||
|
Examples
|
||
|
--------
|
||
|
>>> from numpy.polynomial.hermite import hermint
|
||
|
>>> hermint([1,2,3]) # integrate once, value 0 at 0.
|
||
|
array([ 1. , 0.5, 0.5, 0.5])
|
||
|
>>> hermint([1,2,3], m=2) # integrate twice, value & deriv 0 at 0
|
||
|
array([-0.5 , 0.5 , 0.125 , 0.08333333, 0.0625 ])
|
||
|
>>> 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 ])
|
||
|
|
||
|
"""
|
||
|
c = np.array(c, ndmin=1, copy=1)
|
||
|
if c.dtype.char in '?bBhHiIlLqQpP':
|
||
|
c = c.astype(np.double)
|
||
|
if not np.iterable(k):
|
||
|
k = [k]
|
||
|
cnt, iaxis = [int(t) for t in [m, axis]]
|
||
|
|
||
|
if cnt != m:
|
||
|
raise ValueError("The order of integration must be integer")
|
||
|
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.")
|
||
|
if iaxis != axis:
|
||
|
raise ValueError("The axis must be integer")
|
||
|
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
|
||
|
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=0)
|
||
|
if c.dtype.char in '?bBhHiIlLqQpP':
|
||
|
c = c.astype(np.double)
|
||
|
if isinstance(x, (tuple, list)):
|
||
|
x = np.asarray(x)
|
||
|
if isinstance(x, np.ndarray) and tensor:
|
||
|
c = c.reshape(c.shape + (1,)*x.ndim)
|
||
|
|
||
|
x2 = x*2
|
||
|
if len(c) == 1:
|
||
|
c0 = c[0]
|
||
|
c1 = 0
|
||
|
elif len(c) == 2:
|
||
|
c0 = c[0]
|
||
|
c1 = c[1]
|
||
|
else:
|
||
|
nd = len(c)
|
||
|
c0 = c[-2]
|
||
|
c1 = c[-1]
|
||
|
for i in range(3, len(c) + 1):
|
||
|
tmp = c0
|
||
|
nd = nd - 1
|
||
|
c0 = c[-i] - c1*(2*(nd - 1))
|
||
|
c1 = tmp + c1*x2
|
||
|
return c0 + c1*x2
|
||
|
|
||
|
|
||
|
def hermval2d(x, y, c):
|
||
|
"""
|
||
|
Evaluate a 2-D Hermite series at points (x, y).
|
||
|
|
||
|
This function returns the values:
|
||
|
|
||
|
.. math:: p(x,y) = \\sum_{i,j} c_{i,j} * H_i(x) * H_j(y)
|
||
|
|
||
|
The parameters `x` and `y` are converted to arrays only if they are
|
||
|
tuples or a lists, otherwise they are treated as a scalars and they
|
||
|
must have the same shape after conversion. In either case, either `x`
|
||
|
and `y` or their elements must support multiplication and addition both
|
||
|
with themselves and with the elements of `c`.
|
||
|
|
||
|
If `c` is a 1-D array a one is implicitly appended to its shape to make
|
||
|
it 2-D. The shape of the result will be c.shape[2:] + x.shape.
|
||
|
|
||
|
Parameters
|
||
|
----------
|
||
|
x, y : array_like, compatible objects
|
||
|
The two dimensional series is evaluated at the points `(x, y)`,
|
||
|
where `x` and `y` must have the same shape. If `x` or `y` is a list
|
||
|
or tuple, it is first converted to an ndarray, otherwise it is left
|
||
|
unchanged and if it isn't an ndarray it is treated as a scalar.
|
||
|
c : array_like
|
||
|
Array of coefficients ordered so that the coefficient of the term
|
||
|
of multi-degree i,j is contained in ``c[i,j]``. If `c` has
|
||
|
dimension greater than two the remaining indices enumerate multiple
|
||
|
sets of coefficients.
|
||
|
|
||
|
Returns
|
||
|
-------
|
||
|
values : ndarray, compatible object
|
||
|
The values of the two dimensional polynomial at points formed with
|
||
|
pairs of corresponding values from `x` and `y`.
|
||
|
|
||
|
See Also
|
||
|
--------
|
||
|
hermval, hermgrid2d, hermval3d, hermgrid3d
|
||
|
|
||
|
Notes
|
||
|
-----
|
||
|
|
||
|
.. versionadded:: 1.7.0
|
||
|
|
||
|
"""
|
||
|
try:
|
||
|
x, y = np.array((x, y), copy=0)
|
||
|
except Exception:
|
||
|
raise ValueError('x, y are incompatible')
|
||
|
|
||
|
c = hermval(x, c)
|
||
|
c = hermval(y, c, tensor=False)
|
||
|
return c
|
||
|
|
||
|
|
||
|
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
|
||
|
|
||
|
"""
|
||
|
c = hermval(x, c)
|
||
|
c = hermval(y, c)
|
||
|
return c
|
||
|
|
||
|
|
||
|
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
|
||
|
|
||
|
"""
|
||
|
try:
|
||
|
x, y, z = np.array((x, y, z), copy=0)
|
||
|
except Exception:
|
||
|
raise ValueError('x, y, z are incompatible')
|
||
|
|
||
|
c = hermval(x, c)
|
||
|
c = hermval(y, c, tensor=False)
|
||
|
c = hermval(z, c, tensor=False)
|
||
|
return c
|
||
|
|
||
|
|
||
|
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
|
||
|
|
||
|
"""
|
||
|
c = hermval(x, c)
|
||
|
c = hermval(y, c)
|
||
|
c = hermval(z, c)
|
||
|
return c
|
||
|
|
||
|
|
||
|
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 = int(deg)
|
||
|
if ideg != deg:
|
||
|
raise ValueError("deg must be integer")
|
||
|
if ideg < 0:
|
||
|
raise ValueError("deg must be non-negative")
|
||
|
|
||
|
x = np.array(x, copy=0, ndmin=1) + 0.0
|
||
|
dims = (ideg + 1,) + x.shape
|
||
|
dtyp = x.dtype
|
||
|
v = np.empty(dims, dtype=dtyp)
|
||
|
v[0] = x*0 + 1
|
||
|
if ideg > 0:
|
||
|
x2 = x*2
|
||
|
v[1] = x2
|
||
|
for i in range(2, ideg + 1):
|
||
|
v[i] = (v[i-1]*x2 - v[i-2]*(2*(i - 1)))
|
||
|
return np.moveaxis(v, 0, -1)
|
||
|
|
||
|
|
||
|
def hermvander2d(x, y, deg):
|
||
|
"""Pseudo-Vandermonde matrix of given degrees.
|
||
|
|
||
|
Returns the pseudo-Vandermonde matrix of degrees `deg` and sample
|
||
|
points `(x, y)`. The pseudo-Vandermonde matrix is defined by
|
||
|
|
||
|
.. math:: V[..., (deg[1] + 1)*i + j] = H_i(x) * H_j(y),
|
||
|
|
||
|
where `0 <= i <= deg[0]` and `0 <= j <= deg[1]`. The leading indices of
|
||
|
`V` index the points `(x, y)` and the last index encodes the degrees of
|
||
|
the Hermite polynomials.
|
||
|
|
||
|
If ``V = hermvander2d(x, y, [xdeg, ydeg])``, then the columns of `V`
|
||
|
correspond to the elements of a 2-D coefficient array `c` of shape
|
||
|
(xdeg + 1, ydeg + 1) in the order
|
||
|
|
||
|
.. math:: c_{00}, c_{01}, c_{02} ... , c_{10}, c_{11}, c_{12} ...
|
||
|
|
||
|
and ``np.dot(V, c.flat)`` and ``hermval2d(x, y, c)`` will be the same
|
||
|
up to roundoff. This equivalence is useful both for least squares
|
||
|
fitting and for the evaluation of a large number of 2-D Hermite
|
||
|
series of the same degrees and sample points.
|
||
|
|
||
|
Parameters
|
||
|
----------
|
||
|
x, y : array_like
|
||
|
Arrays of point coordinates, all of the same shape. The dtypes
|
||
|
will be converted to either float64 or complex128 depending on
|
||
|
whether any of the elements are complex. Scalars are converted to 1-D
|
||
|
arrays.
|
||
|
deg : list of ints
|
||
|
List of maximum degrees of the form [x_deg, y_deg].
|
||
|
|
||
|
Returns
|
||
|
-------
|
||
|
vander2d : ndarray
|
||
|
The shape of the returned matrix is ``x.shape + (order,)``, where
|
||
|
:math:`order = (deg[0]+1)*(deg([1]+1)`. The dtype will be the same
|
||
|
as the converted `x` and `y`.
|
||
|
|
||
|
See Also
|
||
|
--------
|
||
|
hermvander, hermvander3d. hermval2d, hermval3d
|
||
|
|
||
|
Notes
|
||
|
-----
|
||
|
|
||
|
.. versionadded:: 1.7.0
|
||
|
|
||
|
"""
|
||
|
ideg = [int(d) for d in deg]
|
||
|
is_valid = [id == d and id >= 0 for id, d in zip(ideg, deg)]
|
||
|
if is_valid != [1, 1]:
|
||
|
raise ValueError("degrees must be non-negative integers")
|
||
|
degx, degy = ideg
|
||
|
x, y = np.array((x, y), copy=0) + 0.0
|
||
|
|
||
|
vx = hermvander(x, degx)
|
||
|
vy = hermvander(y, degy)
|
||
|
v = vx[..., None]*vy[..., None,:]
|
||
|
return v.reshape(v.shape[:-2] + (-1,))
|
||
|
|
||
|
|
||
|
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
|
||
|
|
||
|
"""
|
||
|
ideg = [int(d) for d in deg]
|
||
|
is_valid = [id == d and id >= 0 for id, d in zip(ideg, deg)]
|
||
|
if is_valid != [1, 1, 1]:
|
||
|
raise ValueError("degrees must be non-negative integers")
|
||
|
degx, degy, degz = ideg
|
||
|
x, y, z = np.array((x, y, z), copy=0) + 0.0
|
||
|
|
||
|
vx = hermvander(x, degx)
|
||
|
vy = hermvander(y, degy)
|
||
|
vz = hermvander(z, degz)
|
||
|
v = vx[..., None, None]*vy[..., None,:, None]*vz[..., None, None,:]
|
||
|
return v.reshape(v.shape[:-3] + (-1,))
|
||
|
|
||
|
|
||
|
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 contribution of each point
|
||
|
``(x[i],y[i])`` to the fit is weighted by `w[i]`. Ideally the
|
||
|
weights are chosen so that the errors of the products ``w[i]*y[i]``
|
||
|
all have the same variance. 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
|
||
|
|
||
|
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`.
|
||
|
|
||
|
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', RankWarning)
|
||
|
|
||
|
See Also
|
||
|
--------
|
||
|
chebfit, legfit, lagfit, polyfit, hermefit
|
||
|
hermval : Evaluates a Hermite series.
|
||
|
hermvander : Vandermonde matrix of Hermite series.
|
||
|
hermweight : Hermite weight function
|
||
|
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",
|
||
|
http://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([ 0.97902637, 1.99849131, 3.00006 ])
|
||
|
|
||
|
"""
|
||
|
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 = hermvander(x, lmax)
|
||
|
else:
|
||
|
deg = np.sort(deg)
|
||
|
lmax = deg[-1]
|
||
|
order = len(deg)
|
||
|
van = hermvander(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 = la.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, pu.RankWarning, stacklevel=2)
|
||
|
|
||
|
if full:
|
||
|
return c, [resids, rank, s, rcond]
|
||
|
else:
|
||
|
return c
|
||
|
|
||
|
|
||
|
def hermcompanion(c):
|
||
|
"""Return the scaled companion matrix of c.
|
||
|
|
||
|
The basis polynomials are scaled so that the companion matrix is
|
||
|
symmetric when `c` is an Hermite basis polynomial. This provides
|
||
|
better eigenvalue estimates than the unscaled case and for basis
|
||
|
polynomials the eigenvalues are guaranteed to be real if
|
||
|
`numpy.linalg.eigvalsh` is used to obtain them.
|
||
|
|
||
|
Parameters
|
||
|
----------
|
||
|
c : array_like
|
||
|
1-D array of Hermite series coefficients ordered from low to high
|
||
|
degree.
|
||
|
|
||
|
Returns
|
||
|
-------
|
||
|
mat : ndarray
|
||
|
Scaled companion matrix of dimensions (deg, deg).
|
||
|
|
||
|
Notes
|
||
|
-----
|
||
|
|
||
|
.. versionadded:: 1.7.0
|
||
|
|
||
|
"""
|
||
|
# c is a trimmed copy
|
||
|
[c] = pu.as_series([c])
|
||
|
if len(c) < 2:
|
||
|
raise ValueError('Series must have maximum degree of at least 1.')
|
||
|
if len(c) == 2:
|
||
|
return np.array([[-.5*c[0]/c[1]]])
|
||
|
|
||
|
n = len(c) - 1
|
||
|
mat = np.zeros((n, n), dtype=c.dtype)
|
||
|
scl = np.hstack((1., 1./np.sqrt(2.*np.arange(n - 1, 0, -1))))
|
||
|
scl = np.multiply.accumulate(scl)[::-1]
|
||
|
top = mat.reshape(-1)[1::n+1]
|
||
|
bot = mat.reshape(-1)[n::n+1]
|
||
|
top[...] = np.sqrt(.5*np.arange(1, n))
|
||
|
bot[...] = top
|
||
|
mat[:, -1] -= scl*c[:-1]/(2.0*c[-1])
|
||
|
return mat
|
||
|
|
||
|
|
||
|
def hermroots(c):
|
||
|
"""
|
||
|
Compute the roots of a Hermite series.
|
||
|
|
||
|
Return the roots (a.k.a. "zeros") of the polynomial
|
||
|
|
||
|
.. math:: p(x) = \\sum_i c[i] * H_i(x).
|
||
|
|
||
|
Parameters
|
||
|
----------
|
||
|
c : 1-D array_like
|
||
|
1-D array of coefficients.
|
||
|
|
||
|
Returns
|
||
|
-------
|
||
|
out : ndarray
|
||
|
Array of the roots of the series. If all the roots are real,
|
||
|
then `out` is also real, otherwise it is complex.
|
||
|
|
||
|
See Also
|
||
|
--------
|
||
|
polyroots, legroots, lagroots, chebroots, hermeroots
|
||
|
|
||
|
Notes
|
||
|
-----
|
||
|
The root estimates are obtained as the eigenvalues of the companion
|
||
|
matrix, Roots far from the origin of the complex plane may have large
|
||
|
errors due to the numerical instability of the series for such
|
||
|
values. Roots with multiplicity greater than 1 will also show larger
|
||
|
errors as the value of the series near such points is relatively
|
||
|
insensitive to errors in the roots. Isolated roots near the origin can
|
||
|
be improved by a few iterations of Newton's method.
|
||
|
|
||
|
The Hermite series basis polynomials aren't powers of `x` so the
|
||
|
results of this function may seem unintuitive.
|
||
|
|
||
|
Examples
|
||
|
--------
|
||
|
>>> from numpy.polynomial.hermite import hermroots, hermfromroots
|
||
|
>>> coef = hermfromroots([-1, 0, 1])
|
||
|
>>> coef
|
||
|
array([ 0. , 0.25 , 0. , 0.125])
|
||
|
>>> hermroots(coef)
|
||
|
array([ -1.00000000e+00, -1.38777878e-17, 1.00000000e+00])
|
||
|
|
||
|
"""
|
||
|
# c is a trimmed copy
|
||
|
[c] = pu.as_series([c])
|
||
|
if len(c) <= 1:
|
||
|
return np.array([], dtype=c.dtype)
|
||
|
if len(c) == 2:
|
||
|
return np.array([-.5*c[0]/c[1]])
|
||
|
|
||
|
m = hermcompanion(c)
|
||
|
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.ones(x.shape)/np.sqrt(np.sqrt(np.pi))
|
||
|
|
||
|
c0 = 0.
|
||
|
c1 = 1./np.sqrt(np.sqrt(np.pi))
|
||
|
nd = float(n)
|
||
|
for i in range(n - 1):
|
||
|
tmp = c0
|
||
|
c0 = -c1*np.sqrt((nd - 1.)/nd)
|
||
|
c1 = tmp + c1*x*np.sqrt(2./nd)
|
||
|
nd = nd - 1.0
|
||
|
return c0 + c1*x*np.sqrt(2)
|
||
|
|
||
|
|
||
|
def hermgauss(deg):
|
||
|
"""
|
||
|
Gauss-Hermite quadrature.
|
||
|
|
||
|
Computes the sample points and weights for Gauss-Hermite quadrature.
|
||
|
These sample points and weights will correctly integrate polynomials of
|
||
|
degree :math:`2*deg - 1` or less over the interval :math:`[-\\inf, \\inf]`
|
||
|
with the weight function :math:`f(x) = \\exp(-x^2)`.
|
||
|
|
||
|
Parameters
|
||
|
----------
|
||
|
deg : int
|
||
|
Number of sample points and weights. It must be >= 1.
|
||
|
|
||
|
Returns
|
||
|
-------
|
||
|
x : ndarray
|
||
|
1-D ndarray containing the sample points.
|
||
|
y : ndarray
|
||
|
1-D ndarray containing the weights.
|
||
|
|
||
|
Notes
|
||
|
-----
|
||
|
|
||
|
.. versionadded:: 1.7.0
|
||
|
|
||
|
The results have only been tested up to degree 100, higher degrees may
|
||
|
be problematic. The weights are determined by using the fact that
|
||
|
|
||
|
.. math:: w_k = c / (H'_n(x_k) * H_{n-1}(x_k))
|
||
|
|
||
|
where :math:`c` is a constant independent of :math:`k` and :math:`x_k`
|
||
|
is the k'th root of :math:`H_n`, and then scaling the results to get
|
||
|
the right value when integrating 1.
|
||
|
|
||
|
"""
|
||
|
ideg = int(deg)
|
||
|
if ideg != deg or ideg < 1:
|
||
|
raise ValueError("deg must be a non-negative integer")
|
||
|
|
||
|
# first approximation of roots. We use the fact that the companion
|
||
|
# matrix is symmetric in this case in order to obtain better zeros.
|
||
|
c = np.array([0]*deg + [1], dtype=np.float64)
|
||
|
m = hermcompanion(c)
|
||
|
x = la.eigvalsh(m)
|
||
|
|
||
|
# improve roots by one application of Newton
|
||
|
dy = _normed_hermite_n(x, ideg)
|
||
|
df = _normed_hermite_n(x, ideg - 1) * np.sqrt(2*ideg)
|
||
|
x -= dy/df
|
||
|
|
||
|
# compute the weights. We scale the factor to avoid possible numerical
|
||
|
# overflow.
|
||
|
fm = _normed_hermite_n(x, ideg - 1)
|
||
|
fm /= np.abs(fm).max()
|
||
|
w = 1/(fm * fm)
|
||
|
|
||
|
# for Hermite we can also symmetrize
|
||
|
w = (w + w[::-1])/2
|
||
|
x = (x - x[::-1])/2
|
||
|
|
||
|
# scale w to get the right value
|
||
|
w *= np.sqrt(np.pi) / w.sum()
|
||
|
|
||
|
return x, w
|
||
|
|
||
|
|
||
|
def hermweight(x):
|
||
|
"""
|
||
|
Weight function of the Hermite polynomials.
|
||
|
|
||
|
The weight function is :math:`\\exp(-x^2)` and the interval of
|
||
|
integration is :math:`[-\\inf, \\inf]`. the Hermite polynomials are
|
||
|
orthogonal, but not normalized, with respect to this weight function.
|
||
|
|
||
|
Parameters
|
||
|
----------
|
||
|
x : array_like
|
||
|
Values at which the weight function will be computed.
|
||
|
|
||
|
Returns
|
||
|
-------
|
||
|
w : ndarray
|
||
|
The weight function at `x`.
|
||
|
|
||
|
Notes
|
||
|
-----
|
||
|
|
||
|
.. versionadded:: 1.7.0
|
||
|
|
||
|
"""
|
||
|
w = np.exp(-x**2)
|
||
|
return w
|
||
|
|
||
|
|
||
|
#
|
||
|
# Hermite series class
|
||
|
#
|
||
|
|
||
|
class Hermite(ABCPolyBase):
|
||
|
"""An Hermite series class.
|
||
|
|
||
|
The Hermite class provides the standard Python numerical methods
|
||
|
'+', '-', '*', '//', '%', 'divmod', '**', and '()' as well as the
|
||
|
attributes and methods listed in the `ABCPolyBase` documentation.
|
||
|
|
||
|
Parameters
|
||
|
----------
|
||
|
coef : array_like
|
||
|
Hermite coefficients in order of increasing degree, i.e,
|
||
|
``(1, 2, 3)`` gives ``1*H_0(x) + 2*H_1(X) + 3*H_2(x)``.
|
||
|
domain : (2,) array_like, optional
|
||
|
Domain to use. The interval ``[domain[0], domain[1]]`` is mapped
|
||
|
to the interval ``[window[0], window[1]]`` by shifting and scaling.
|
||
|
The default value is [-1, 1].
|
||
|
window : (2,) array_like, optional
|
||
|
Window, see `domain` for its use. The default value is [-1, 1].
|
||
|
|
||
|
.. versionadded:: 1.6.0
|
||
|
|
||
|
"""
|
||
|
# 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
|
||
|
nickname = 'herm'
|
||
|
domain = np.array(hermdomain)
|
||
|
window = np.array(hermdomain)
|