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diff DEPENDENCIES/mingw32/Python27/Lib/site-packages/numpy/polynomial/hermite.py @ 87:2a2c65a20a8b
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| author | Chris Cannam |
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| date | Wed, 25 Feb 2015 14:05:22 +0000 |
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--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/DEPENDENCIES/mingw32/Python27/Lib/site-packages/numpy/polynomial/hermite.py Wed Feb 25 14:05:22 2015 +0000 @@ -0,0 +1,1789 @@ +""" +Objects for dealing with Hermite series. + +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`). + +Constants +--------- +- `hermdomain` -- Hermite series default domain, [-1,1]. +- `hermzero` -- Hermite series that evaluates identically to 0. +- `hermone` -- Hermite series that evaluates identically to 1. +- `hermx` -- Hermite series for the identity map, ``f(x) = x``. + +Arithmetic +---------- +- `hermmulx` -- multiply a Hermite series in ``P_i(x)`` by ``x``. +- `hermadd` -- add two Hermite series. +- `hermsub` -- subtract one Hermite series from another. +- `hermmul` -- multiply two Hermite series. +- `hermdiv` -- divide one Hermite series by another. +- `hermval` -- evaluate a Hermite series at given points. +- `hermval2d` -- evaluate a 2D Hermite series at given points. +- `hermval3d` -- evaluate a 3D Hermite series at given points. +- `hermgrid2d` -- evaluate a 2D Hermite series on a Cartesian product. +- `hermgrid3d` -- evaluate a 3D Hermite series on a Cartesian product. + +Calculus +-------- +- `hermder` -- differentiate a Hermite series. +- `hermint` -- integrate a Hermite series. + +Misc Functions +-------------- +- `hermfromroots` -- create a Hermite series with specified roots. +- `hermroots` -- find the roots of a Hermite series. +- `hermvander` -- Vandermonde-like matrix for Hermite polynomials. +- `hermvander2d` -- Vandermonde-like matrix for 2D power series. +- `hermvander3d` -- Vandermonde-like matrix for 3D power series. +- `hermgauss` -- Gauss-Hermite quadrature, points and weights. +- `hermweight` -- Hermite weight function. +- `hermcompanion` -- symmetrized companion matrix in Hermite form. +- `hermfit` -- least-squares fit returning a Hermite series. +- `hermtrim` -- trim leading coefficients from a Hermite series. +- `hermline` -- Hermite series of given straight line. +- `herm2poly` -- convert a Hermite series to a polynomial. +- `poly2herm` -- convert a polynomial to a Hermite series. + +Classes +------- +- `Hermite` -- A Hermite series class. + +See also +-------- +`numpy.polynomial` + +""" +from __future__ import division, absolute_import, print_function + +import warnings +import numpy as np +import numpy.linalg as la + +from . import polyutils as pu +from ._polybase import ABCPolyBase + +__all__ = [ + 'hermzero', 'hermone', 'hermx', 'hermdomain', 'hermline', 'hermadd', + 'hermsub', 'hermmulx', 'hermmul', 'hermdiv', 'hermpow', 'hermval', + 'hermder', 'hermint', 'herm2poly', 'poly2herm', 'hermfromroots', + 'hermvander', 'hermfit', 'hermtrim', 'hermroots', 'Hermite', + 'hermval2d', 'hermval3d', 'hermgrid2d', 'hermgrid3d', 'hermvander2d', + 'hermvander3d', 'hermcompanion', 'hermgauss', 'hermweight'] + +hermtrim = pu.trimcoef + + +def poly2herm(pol): + """ + poly2herm(pol) + + Convert a polynomial to a Hermite series. + + Convert an array representing the coefficients of a polynomial (relative + to the "standard" basis) ordered from lowest degree to highest, to an + array of the coefficients of the equivalent Hermite series, ordered + from lowest to highest degree. + + Parameters + ---------- + pol : array_like + 1-D array containing the polynomial coefficients + + Returns + ------- + c : ndarray + 1-D array containing the coefficients of the equivalent Hermite + series. + + See Also + -------- + herm2poly + + Notes + ----- + The easy way to do conversions between polynomial basis sets + is to use the convert method of a class instance. + + Examples + -------- + >>> from numpy.polynomial.hermite import poly2herm + >>> poly2herm(np.arange(4)) + array([ 1. , 2.75 , 0.5 , 0.375]) + + """ + [pol] = pu.as_series([pol]) + deg = len(pol) - 1 + res = 0 + for i in range(deg, -1, -1): + res = hermadd(hermmulx(res), pol[i]) + return res + + +def herm2poly(c): + """ + Convert a Hermite series to a polynomial. + + Convert an array representing the coefficients of a Hermite series, + ordered from lowest degree to highest, to an array of the coefficients + of the equivalent polynomial (relative to the "standard" basis) ordered + from lowest to highest degree. + + Parameters + ---------- + c : array_like + 1-D array containing the Hermite series coefficients, ordered + from lowest order term to highest. + + Returns + ------- + pol : ndarray + 1-D array containing the coefficients of the equivalent polynomial + (relative to the "standard" basis) ordered from lowest order term + to highest. + + See Also + -------- + poly2herm + + Notes + ----- + The easy way to do conversions between polynomial basis sets + is to use the convert method of a class instance. + + Examples + -------- + >>> from numpy.polynomial.hermite import herm2poly + >>> herm2poly([ 1. , 2.75 , 0.5 , 0.375]) + array([ 0., 1., 2., 3.]) + + """ + from .polynomial import polyadd, polysub, polymulx + + [c] = pu.as_series([c]) + n = len(c) + if n == 1: + return c + if n == 2: + c[1] *= 2 + return c + else: + c0 = c[-2] + c1 = c[-1] + # i is the current degree of c1 + for i in range(n - 1, 1, -1): + tmp = c0 + c0 = polysub(c[i - 2], c1*(2*(i - 1))) + c1 = polyadd(tmp, polymulx(c1)*2) + return polyadd(c0, polymulx(c1)*2) + +# +# These are constant arrays are of integer type so as to be compatible +# with the widest range of other types, such as Decimal. +# + +# Hermite +hermdomain = np.array([-1, 1]) + +# Hermite coefficients representing zero. +hermzero = np.array([0]) + +# Hermite coefficients representing one. +hermone = np.array([1]) + +# Hermite coefficients representing the identity x. +hermx = np.array([0, 1/2]) + + +def hermline(off, scl): + """ + Hermite series whose graph is a straight line. + + + + Parameters + ---------- + off, scl : scalars + The specified line is given by ``off + scl*x``. + + Returns + ------- + y : ndarray + This module's representation of the Hermite series for + ``off + scl*x``. + + See Also + -------- + polyline, chebline + + 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 + -------- + polyfromroots, legfromroots, lagfromroots, chebfromroots, + 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]) + + """ + if len(roots) == 0: + return np.ones(1) + else: + [roots] = pu.as_series([roots], trim=False) + roots.sort() + p = [hermline(-r, 1) for r in roots] + n = len(p) + while n > 1: + m, r = divmod(n, 2) + tmp = [hermmul(p[i], p[i+m]) for i in range(m)] + if r: + tmp[0] = hermmul(tmp[0], p[-1]) + p = tmp + n = m + return p[0] + + +def hermadd(c1, c2): + """ + Add one Hermite series to another. + + Returns the sum of two Hermite series `c1` + `c2`. The arguments + are sequences of coefficients ordered from lowest order term to + highest, i.e., [1,2,3] represents the series ``P_0 + 2*P_1 + 3*P_2``. + + Parameters + ---------- + c1, c2 : array_like + 1-D arrays of Hermite series coefficients ordered from low to + high. + + Returns + ------- + out : ndarray + Array representing the Hermite series of their sum. + + See Also + -------- + hermsub, 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.]) + + """ + # c1, c2 are trimmed copies + [c1, c2] = pu.as_series([c1, c2]) + if len(c1) > len(c2): + c1[:c2.size] += c2 + ret = c1 + else: + c2[:c1.size] += c1 + ret = c2 + return pu.trimseq(ret) + + +def hermsub(c1, c2): + """ + Subtract one Hermite series from another. + + Returns the difference of two Hermite series `c1` - `c2`. The + sequences of coefficients are from lowest order term to highest, i.e., + [1,2,3] represents the series ``P_0 + 2*P_1 + 3*P_2``. + + Parameters + ---------- + c1, c2 : array_like + 1-D arrays of Hermite series coefficients ordered from low to + high. + + Returns + ------- + out : ndarray + Of Hermite series coefficients representing their difference. + + See Also + -------- + hermadd, 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.]) + + """ + # c1, c2 are trimmed copies + [c1, c2] = pu.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 pu.trimseq(ret) + + +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. + + 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") + if not -c.ndim <= iaxis < c.ndim: + raise ValueError("The axis is out of range") + if iaxis < 0: + iaxis += c.ndim + + if cnt == 0: + return c + + c = np.rollaxis(c, iaxis) + 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.rollaxis(c, 0, iaxis + 1) + 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.isscalar(lbnd) == False``, or + ``np.isscalar(scl) == False``. + + 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 iaxis != axis: + raise ValueError("The axis must be integer") + if not -c.ndim <= iaxis < c.ndim: + raise ValueError("The axis is out of range") + if iaxis < 0: + iaxis += c.ndim + + if cnt == 0: + return c + + c = np.rollaxis(c, iaxis) + 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.rollaxis(c, 0, iaxis + 1) + 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: + 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: + 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.rollaxis(v, 0, v.ndim) + + +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]*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 + Degree of the fitting polynomial + 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 ]) + + """ + order = int(deg) + 1 + x = np.asarray(x) + 0.0 + y = np.asarray(y) + 0.0 + + # check arguments. + if deg < 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") + + # set up the least squares matrices in transposed form + lhs = hermvander(x, deg).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 + + # warn on rank reduction + if rank != order and not full: + msg = "The fit may be poorly conditioned" + warnings.warn(msg, pu.RankWarning) + + 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., np.sqrt(2.*np.arange(1, n)))) + scl = np.multiply.accumulate(scl) + 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] -= (c[:-1]/c[-1])*(scl/scl[-1])*.5 + 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 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]) + m = hermcompanion(c) + x = la.eigvals(m) + x.sort() + + # improve roots by one application of Newton + dy = hermval(x, c) + df = hermval(x, hermder(c)) + x -= dy/df + + # compute the weights. We scale the factor to avoid possible numerical + # overflow. + fm = hermval(x, c[1:]) + fm /= np.abs(fm).max() + df /= np.abs(df).max() + w = 1/(fm * df) + + # 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 + Laguerre 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)