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 """ Objects for dealing with Hermite_e series. This module provides a number of objects (mostly functions) useful for dealing with Hermite_e series, including a HermiteE class that encapsulates the usual arithmetic operations. (General information on how this module represents and works with such polynomials is in the docstring for its "parent" sub-package, numpy.polynomial). Constants --------- - hermedomain -- Hermite_e series default domain, [-1,1]. - hermezero -- Hermite_e series that evaluates identically to 0. - hermeone -- Hermite_e series that evaluates identically to 1. - hermex -- Hermite_e series for the identity map, f(x) = x. Arithmetic ---------- - hermemulx -- multiply a Hermite_e series in P_i(x) by x. - hermeadd -- add two Hermite_e series. - hermesub -- subtract one Hermite_e series from another. - hermemul -- multiply two Hermite_e series. - hermediv -- divide one Hermite_e series by another. - hermeval -- evaluate a Hermite_e series at given points. - hermeval2d -- evaluate a 2D Hermite_e series at given points. - hermeval3d -- evaluate a 3D Hermite_e series at given points. - hermegrid2d -- evaluate a 2D Hermite_e series on a Cartesian product. - hermegrid3d -- evaluate a 3D Hermite_e series on a Cartesian product. Calculus -------- - hermeder -- differentiate a Hermite_e series. - hermeint -- integrate a Hermite_e series. Misc Functions -------------- - hermefromroots -- create a Hermite_e series with specified roots. - hermeroots -- find the roots of a Hermite_e series. - hermevander -- Vandermonde-like matrix for Hermite_e polynomials. - hermevander2d -- Vandermonde-like matrix for 2D power series. - hermevander3d -- Vandermonde-like matrix for 3D power series. - hermegauss -- Gauss-Hermite_e quadrature, points and weights. - hermeweight -- Hermite_e weight function. - hermecompanion -- symmetrized companion matrix in Hermite_e form. - hermefit -- least-squares fit returning a Hermite_e series. - hermetrim -- trim leading coefficients from a Hermite_e series. - hermeline -- Hermite_e series of given straight line. - herme2poly -- convert a Hermite_e series to a polynomial. - poly2herme -- convert a polynomial to a Hermite_e series. Classes ------- - HermiteE -- A Hermite_e series class. See also -------- numpy.polynomial """ from __future__ import division, absolute_import, print_function import numpy as np import numpy.linalg as la from . import polyutils as pu import warnings from .polytemplate import polytemplate __all__ = ['hermezero', 'hermeone', 'hermex', 'hermedomain', 'hermeline', 'hermeadd', 'hermesub', 'hermemulx', 'hermemul', 'hermediv', 'hermpow', 'hermeval', 'hermeder', 'hermeint', 'herme2poly', 'poly2herme', 'hermefromroots', 'hermevander', 'hermefit', 'hermetrim', 'hermeroots', 'HermiteE', 'hermeval2d', 'hermeval3d', 'hermegrid2d', 'hermegrid3d', 'hermevander2d', 'hermevander3d', 'hermecompanion', 'hermegauss', 'hermeweight'] hermetrim = pu.trimcoef def poly2herme(pol) : """ poly2herme(pol) Convert a polynomial to a Hermite series. Convert an array representing the coefficients of a polynomial (relative to the "standard" basis) ordered from lowest degree to highest, to an array of the coefficients of the equivalent Hermite series, ordered from lowest to highest degree. Parameters ---------- pol : array_like 1-D array containing the polynomial coefficients Returns ------- c : ndarray 1-D array containing the coefficients of the equivalent Hermite series. See Also -------- herme2poly Notes ----- The easy way to do conversions between polynomial basis sets is to use the convert method of a class instance. Examples -------- >>> from numpy.polynomial.hermite_e import poly2herme >>> poly2herme(np.arange(4)) array([ 2., 10., 2., 3.]) """ [pol] = pu.as_series([pol]) deg = len(pol) - 1 res = 0 for i in range(deg, -1, -1) : res = hermeadd(hermemulx(res), pol[i]) return res def herme2poly(c) : """ Convert a Hermite series to a polynomial. Convert an array representing the coefficients of a Hermite series, ordered from lowest degree to highest, to an array of the coefficients of the equivalent polynomial (relative to the "standard" basis) ordered from lowest to highest degree. Parameters ---------- c : array_like 1-D array containing the Hermite series coefficients, ordered from lowest order term to highest. Returns ------- pol : ndarray 1-D array containing the coefficients of the equivalent polynomial (relative to the "standard" basis) ordered from lowest order term to highest. See Also -------- poly2herme Notes ----- The easy way to do conversions between polynomial basis sets is to use the convert method of a class instance. Examples -------- >>> from numpy.polynomial.hermite_e import herme2poly >>> herme2poly([ 2., 10., 2., 3.]) array([ 0., 1., 2., 3.]) """ from .polynomial import polyadd, polysub, polymulx [c] = pu.as_series([c]) n = len(c) if n == 1: return c if n == 2: return c else: c0 = c[-2] c1 = c[-1] # i is the current degree of c1 for i in range(n - 1, 1, -1) : tmp = c0 c0 = polysub(c[i - 2], c1*(i - 1)) c1 = polyadd(tmp, polymulx(c1)) return polyadd(c0, polymulx(c1)) # # These are constant arrays are of integer type so as to be compatible # with the widest range of other types, such as Decimal. # # Hermite hermedomain = np.array([-1, 1]) # Hermite coefficients representing zero. hermezero = np.array([0]) # Hermite coefficients representing one. hermeone = np.array([1]) # Hermite coefficients representing the identity x. hermex = np.array([0, 1]) def hermeline(off, scl) : """ Hermite series whose graph is a straight line. Parameters ---------- off, scl : scalars The specified line is given by off + scl*x. Returns ------- y : ndarray This module's representation of the Hermite series for off + scl*x. See Also -------- polyline, chebline Examples -------- >>> from numpy.polynomial.hermite_e import hermeline >>> from numpy.polynomial.hermite_e import hermeline, hermeval >>> hermeval(0,hermeline(3, 2)) 3.0 >>> hermeval(1,hermeline(3, 2)) 5.0 """ if scl != 0 : return np.array([off, scl]) else : return np.array([off]) def hermefromroots(roots) : """ Generate a HermiteE series with given roots. The function returns the coefficients of the polynomial .. math:: p(x) = (x - r_0) * (x - r_1) * ... * (x - r_n), in HermiteE form, where the r_n are the roots specified in roots. If a zero has multiplicity n, then it must appear in roots n times. For instance, if 2 is a root of multiplicity three and 3 is a root of multiplicity 2, then roots looks something like [2, 2, 2, 3, 3]. The roots can appear in any order. If the returned coefficients are c, then .. math:: p(x) = c_0 + c_1 * He_1(x) + ... + c_n * He_n(x) The coefficient of the last term is not generally 1 for monic polynomials in HermiteE form. Parameters ---------- roots : array_like Sequence containing the roots. Returns ------- out : ndarray 1-D array of coefficients. If all roots are real then out is a real array, if some of the roots are complex, then out is complex even if all the coefficients in the result are real (see Examples below). See Also -------- polyfromroots, legfromroots, lagfromroots, hermfromroots, chebfromroots. Examples -------- >>> from numpy.polynomial.hermite_e import hermefromroots, hermeval >>> coef = hermefromroots((-1, 0, 1)) >>> hermeval((-1, 0, 1), coef) array([ 0., 0., 0.]) >>> coef = hermefromroots((-1j, 1j)) >>> hermeval((-1j, 1j), coef) array([ 0.+0.j, 0.+0.j]) """ if len(roots) == 0 : return np.ones(1) else : [roots] = pu.as_series([roots], trim=False) roots.sort() p = [hermeline(-r, 1) for r in roots] n = len(p) while n > 1: m, r = divmod(n, 2) tmp = [hermemul(p[i], p[i+m]) for i in range(m)] if r: tmp[0] = hermemul(tmp[0], p[-1]) p = tmp n = m return p[0] def hermeadd(c1, c2): """ Add one Hermite series to another. Returns the sum of two Hermite series c1 + c2. The arguments are sequences of coefficients ordered from lowest order term to highest, i.e., [1,2,3] represents the series P_0 + 2*P_1 + 3*P_2. Parameters ---------- c1, c2 : array_like 1-D arrays of Hermite series coefficients ordered from low to high. Returns ------- out : ndarray Array representing the Hermite series of their sum. See Also -------- hermesub, hermemul, hermediv, hermepow Notes ----- Unlike multiplication, division, etc., the sum of two Hermite series is a Hermite series (without having to "reproject" the result onto the basis set) so addition, just like that of "standard" polynomials, is simply "component-wise." Examples -------- >>> from numpy.polynomial.hermite_e import hermeadd >>> hermeadd([1, 2, 3], [1, 2, 3, 4]) array([ 2., 4., 6., 4.]) """ # 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 hermesub(c1, c2): """ Subtract one Hermite series from another. Returns the difference of two Hermite series c1 - c2. The sequences of coefficients are from lowest order term to highest, i.e., [1,2,3] represents the series P_0 + 2*P_1 + 3*P_2. Parameters ---------- c1, c2 : array_like 1-D arrays of Hermite series coefficients ordered from low to high. Returns ------- out : ndarray Of Hermite series coefficients representing their difference. See Also -------- hermeadd, hermemul, hermediv, hermepow Notes ----- Unlike multiplication, division, etc., the difference of two Hermite series is a Hermite series (without having to "reproject" the result onto the basis set) so subtraction, just like that of "standard" polynomials, is simply "component-wise." Examples -------- >>> from numpy.polynomial.hermite_e import hermesub >>> hermesub([1, 2, 3, 4], [1, 2, 3]) array([ 0., 0., 0., 4.]) """ # 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 hermemulx(c): """Multiply a Hermite series by x. Multiply the Hermite series c by x, where x is the independent variable. Parameters ---------- c : array_like 1-D array of Hermite series coefficients ordered from low to high. Returns ------- out : ndarray Array representing the result of the multiplication. Notes ----- The multiplication uses the recursion relationship for Hermite polynomials in the form .. math:: xP_i(x) = (P_{i + 1}(x) + iP_{i - 1}(x))) Examples -------- >>> from numpy.polynomial.hermite_e import hermemulx >>> hermemulx([1, 2, 3]) array([ 2., 7., 2., 3.]) """ # c is a trimmed copy [c] = pu.as_series([c]) # The zero series needs special treatment if len(c) == 1 and c[0] == 0: return c prd = np.empty(len(c) + 1, dtype=c.dtype) prd[0] = c[0]*0 prd[1] = c[0] for i in range(1, len(c)): prd[i + 1] = c[i] prd[i - 1] += c[i]*i return prd def hermemul(c1, c2): """ Multiply one Hermite series by another. Returns the product of two Hermite series c1 * c2. The arguments are sequences of coefficients, from lowest order "term" to highest, e.g., [1,2,3] represents the series P_0 + 2*P_1 + 3*P_2. Parameters ---------- c1, c2 : array_like 1-D arrays of Hermite series coefficients ordered from low to high. Returns ------- out : ndarray Of Hermite series coefficients representing their product. See Also -------- hermeadd, hermesub, hermediv, hermepow Notes ----- In general, the (polynomial) product of two C-series results in terms that are not in the Hermite polynomial basis set. Thus, to express the product as a Hermite series, it is necessary to "reproject" the product onto said basis set, which may produce "unintuitive" (but correct) results; see Examples section below. Examples -------- >>> from numpy.polynomial.hermite_e import hermemul >>> hermemul([1, 2, 3], [0, 1, 2]) array([ 14., 15., 28., 7., 6.]) """ # s1, s2 are trimmed copies [c1, c2] = pu.as_series([c1, c2]) if len(c1) > len(c2): c = c2 xs = c1 else: c = c1 xs = c2 if len(c) == 1: c0 = c[0]*xs c1 = 0 elif len(c) == 2: c0 = c[0]*xs c1 = c[1]*xs else : nd = len(c) c0 = c[-2]*xs c1 = c[-1]*xs for i in range(3, len(c) + 1) : tmp = c0 nd = nd - 1 c0 = hermesub(c[-i]*xs, c1*(nd - 1)) c1 = hermeadd(tmp, hermemulx(c1)) return hermeadd(c0, hermemulx(c1)) def hermediv(c1, c2): """ Divide one Hermite series by another. Returns the quotient-with-remainder of two Hermite series c1 / c2. The arguments are sequences of coefficients from lowest order "term" to highest, e.g., [1,2,3] represents the series P_0 + 2*P_1 + 3*P_2. Parameters ---------- c1, c2 : array_like 1-D arrays of Hermite series coefficients ordered from low to high. Returns ------- [quo, rem] : ndarrays Of Hermite series coefficients representing the quotient and remainder. See Also -------- hermeadd, hermesub, hermemul, hermepow Notes ----- In general, the (polynomial) division of one Hermite series by another results in quotient and remainder terms that are not in the Hermite polynomial basis set. Thus, to express these results as a Hermite series, it is necessary to "reproject" the results onto the Hermite basis set, which may produce "unintuitive" (but correct) results; see Examples section below. Examples -------- >>> from numpy.polynomial.hermite_e import hermediv >>> hermediv([ 14., 15., 28., 7., 6.], [0, 1, 2]) (array([ 1., 2., 3.]), array([ 0.])) >>> hermediv([ 15., 17., 28., 7., 6.], [0, 1, 2]) (array([ 1., 2., 3.]), array([ 1., 2.])) """ # 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 = hermemul([0]*i + [1], c2) q = rem[-1]/p[-1] rem = rem[:-1] - q*p[:-1] quo[i] = q return quo, pu.trimseq(rem) def hermepow(c, pow, maxpower=16) : """Raise a Hermite series to a power. Returns the Hermite series c raised to the power pow. The argument c is a sequence of coefficients ordered from low to high. i.e., [1,2,3] is the series P_0 + 2*P_1 + 3*P_2. Parameters ---------- c : array_like 1-D array of Hermite series coefficients ordered from low to high. pow : integer Power to which the series will be raised maxpower : integer, optional Maximum power allowed. This is mainly to limit growth of the series to unmanageable size. Default is 16 Returns ------- coef : ndarray Hermite series of power. See Also -------- hermeadd, hermesub, hermemul, hermediv Examples -------- >>> from numpy.polynomial.hermite_e import hermepow >>> hermepow([1, 2, 3], 2) array([ 23., 28., 46., 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 = hermemul(prd, c) return prd def hermeder(c, m=1, scl=1, axis=0) : """ Differentiate a Hermite_e series. Returns the series coefficients c differentiated m times along axis. At each iteration the result is multiplied by scl (the scaling factor is for use in a linear change of variable). The argument c is an array of coefficients from low to high degree along each axis, e.g., [1,2,3] represents the series 1*He_0 + 2*He_1 + 3*He_2 while [[1,2],[1,2]] represents 1*He_0(x)*He_0(y) + 1*He_1(x)*He_0(y) + 2*He_0(x)*He_1(y) + 2*He_1(x)*He_1(y) if axis=0 is x and axis=1 is y. Parameters ---------- c : array_like Array of Hermite_e series coefficients. If c is multidimensional the different axis correspond to different variables with the degree in each axis given by the corresponding index. m : int, optional Number of derivatives taken, must be non-negative. (Default: 1) scl : scalar, optional Each differentiation is multiplied by scl. The end result is multiplication by scl**m. This is for use in a linear change of variable. (Default: 1) axis : int, optional Axis over which the derivative is taken. (Default: 0). .. versionadded:: 1.7.0 Returns ------- der : ndarray Hermite series of the derivative. See Also -------- hermeint Notes ----- In general, the result of differentiating a Hermite series does not resemble the same operation on a power series. Thus the result of this function may be "unintuitive," albeit correct; see Examples section below. Examples -------- >>> from numpy.polynomial.hermite_e import hermeder >>> hermeder([ 1., 1., 1., 1.]) array([ 1., 2., 3.]) >>> hermeder([-0.25, 1., 1./2., 1./3., 1./4 ], m=2) array([ 1., 2., 3.]) """ c = np.array(c, ndmin=1, copy=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: return c[:1]*0 else : for i in range(cnt): n = n - 1 c *= scl der = np.empty((n,) + c.shape[1:], dtype=c.dtype) for j in range(n, 0, -1): der[j - 1] = j*c[j] c = der c = np.rollaxis(c, 0, iaxis + 1) return c def hermeint(c, m=1, k=[], lbnd=0, scl=1, axis=0): """ Integrate a Hermite_e series. Returns the Hermite_e series coefficients c integrated m times from lbnd along axis. At each iteration the resulting series is **multiplied** by scl and an integration constant, k, is added. The scaling factor is for use in a linear change of variable. ("Buyer beware": note that, depending on what one is doing, one may want scl to be the reciprocal of what one might expect; for more information, see the Notes section below.) The argument c is an array of coefficients from low to high degree along each axis, e.g., [1,2,3] represents the series H_0 + 2*H_1 + 3*H_2 while [[1,2],[1,2]] represents 1*H_0(x)*H_0(y) + 1*H_1(x)*H_0(y) + 2*H_0(x)*H_1(y) + 2*H_1(x)*H_1(y) if axis=0 is x and axis=1 is y. Parameters ---------- c : array_like Array of Hermite_e series coefficients. If c is multidimensional the different axis correspond to different variables with the degree in each axis given by the corresponding index. m : int, optional Order of integration, must be positive. (Default: 1) k : {[], list, scalar}, optional Integration constant(s). The value of the first integral at lbnd is the first value in the list, the value of the second integral at lbnd is the second value, etc. If k == [] (the default), all constants are set to zero. If m == 1, a single scalar can be given instead of a list. lbnd : scalar, optional The lower bound of the integral. (Default: 0) scl : scalar, optional Following each integration the result is *multiplied* by scl before the integration constant is added. (Default: 1) axis : int, optional Axis over which the integral is taken. (Default: 0). .. versionadded:: 1.7.0 Returns ------- S : ndarray Hermite_e series coefficients of the integral. Raises ------ ValueError If m < 0, len(k) > m, np.isscalar(lbnd) == False, or np.isscalar(scl) == False. See Also -------- hermeder Notes ----- Note that the result of each integration is *multiplied* by scl. Why is this important to note? Say one is making a linear change of variable :math:u = ax + b in an integral relative to x. Then .. math::dx = du/a, so one will need to set scl equal to :math:1/a - perhaps not what one would have first thought. Also note that, in general, the result of integrating a C-series needs to be "reprojected" onto the C-series basis set. Thus, typically, the result of this function is "unintuitive," albeit correct; see Examples section below. Examples -------- >>> from numpy.polynomial.hermite_e import hermeint >>> hermeint([1, 2, 3]) # integrate once, value 0 at 0. array([ 1., 1., 1., 1.]) >>> hermeint([1, 2, 3], m=2) # integrate twice, value & deriv 0 at 0 array([-0.25 , 1. , 0.5 , 0.33333333, 0.25 ]) >>> hermeint([1, 2, 3], k=1) # integrate once, value 1 at 0. array([ 2., 1., 1., 1.]) >>> hermeint([1, 2, 3], lbnd=-1) # integrate once, value 0 at -1 array([-1., 1., 1., 1.]) >>> hermeint([1, 2, 3], m=2, k=[1, 2], lbnd=-1) array([ 1.83333333, 0. , 0.5 , 0.33333333, 0.25 ]) """ 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] for j in range(1, n): tmp[j + 1] = c[j]/(j + 1) tmp[0] += k[i] - hermeval(lbnd, tmp) c = tmp c = np.rollaxis(c, 0, iaxis + 1) return c def hermeval(x, c, tensor=True): """ Evaluate an HermiteE series at points x. If c is of length n + 1, this function returns the value: .. math:: p(x) = c_0 * He_0(x) + c_1 * He_1(x) + ... + c_n * He_n(x) The parameter x is converted to an array only if it is a tuple or a list, otherwise it is treated as a scalar. In either case, either x or its elements must support multiplication and addition both with themselves and with the elements of c. If c is a 1-D array, then p(x) will have the same shape as x. If c is multidimensional, then the shape of the result depends on the value of tensor. If tensor is true the shape will be c.shape[1:] + x.shape. If tensor is false the shape will be c.shape[1:]. Note that scalars have shape (,). Trailing zeros in the coefficients will be used in the evaluation, so they should be avoided if efficiency is a concern. Parameters ---------- x : array_like, compatible object If x is a list or tuple, it is converted to an ndarray, otherwise it is left unchanged and treated as a scalar. In either case, x or its elements must support addition and multiplication with with themselves and with the elements of c. c : array_like Array of coefficients ordered so that the coefficients for terms of degree n are contained in c[n]. If c is multidimensional the remaining indices enumerate multiple polynomials. In the two dimensional case the coefficients may be thought of as stored in the columns of c. tensor : boolean, optional If True, the shape of the coefficient array is extended with ones on the right, one for each dimension of x. Scalars have dimension 0 for this action. The result is that every column of coefficients in c is evaluated for every element of x. If False, x is broadcast over the columns of c for the evaluation. This keyword is useful when c is multidimensional. The default value is True. .. versionadded:: 1.7.0 Returns ------- values : ndarray, algebra_like The shape of the return value is described above. See Also -------- hermeval2d, hermegrid2d, hermeval3d, hermegrid3d Notes ----- The evaluation uses Clenshaw recursion, aka synthetic division. Examples -------- >>> from numpy.polynomial.hermite_e import hermeval >>> coef = [1,2,3] >>> hermeval(1, coef) 3.0 >>> hermeval([[1,2],[3,4]], coef) array([[ 3., 14.], [ 31., 54.]]) """ c = np.array(c, ndmin=1, copy=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) if len(c) == 1 : c0 = c[0] c1 = 0 elif len(c) == 2 : c0 = c[0] c1 = c[1] else : nd = len(c) c0 = c[-2] c1 = c[-1] for i in range(3, len(c) + 1) : tmp = c0 nd = nd - 1 c0 = c[-i] - c1*(nd - 1) c1 = tmp + c1*x return c0 + c1*x def hermeval2d(x, y, c): """ Evaluate a 2-D HermiteE series at points (x, y). This function returns the values: .. math:: p(x,y) = \\sum_{i,j} c_{i,j} * He_i(x) * He_j(y) The parameters x and y are converted to arrays only if they are tuples or a lists, otherwise they are treated as a scalars and they must have the same shape after conversion. In either case, either x and y or their elements must support multiplication and addition both with themselves and with the elements of c. If c is a 1-D array a one is implicitly appended to its shape to make it 2-D. The shape of the result will be c.shape[2:] + x.shape. Parameters ---------- x, y : array_like, compatible objects The two dimensional series is evaluated at the points (x, y), where x and y must have the same shape. If x or y is a list or tuple, it is first converted to an ndarray, otherwise it is left unchanged and if it isn't an ndarray it is treated as a scalar. c : array_like Array of coefficients ordered so that the coefficient of the term of multi-degree i,j is contained in c[i,j]. If c has dimension greater than two the remaining indices enumerate multiple sets of coefficients. Returns ------- values : ndarray, compatible object The values of the two dimensional polynomial at points formed with pairs of corresponding values from x and y. See Also -------- hermeval, hermegrid2d, hermeval3d, hermegrid3d Notes ----- .. versionadded::1.7.0 """ try: x, y = np.array((x, y), copy=0) except: raise ValueError('x, y are incompatible') c = hermeval(x, c) c = hermeval(y, c, tensor=False) return c def hermegrid2d(x, y, c): """ Evaluate a 2-D HermiteE series on the Cartesian product of x and y. This function returns the values: .. math:: p(a,b) = \sum_{i,j} c_{i,j} * H_i(a) * H_j(b) where the points (a, b) consist of all pairs formed by taking a from x and b from y. The resulting points form a grid with x in the first dimension and y in the second. The parameters x and y are converted to arrays only if they are tuples or a lists, otherwise they are treated as a scalars. In either case, either x and y or their elements must support multiplication and addition both with themselves and with the elements of c. If c has fewer than two dimensions, ones are implicitly appended to its shape to make it 2-D. The shape of the result will be c.shape[2:] + x.shape. Parameters ---------- x, y : array_like, compatible objects The two dimensional series is evaluated at the points in the Cartesian product of x and y. If x or y is a list or tuple, it is first converted to an ndarray, otherwise it is left unchanged and, if it isn't an ndarray, it is treated as a scalar. c : array_like Array of coefficients ordered so that the coefficients for terms of degree i,j are contained in c[i,j]. If c has dimension greater than two the remaining indices enumerate multiple sets of coefficients. Returns ------- values : ndarray, compatible object The values of the two dimensional polynomial at points in the Cartesian product of x and y. See Also -------- hermeval, hermeval2d, hermeval3d, hermegrid3d Notes ----- .. versionadded::1.7.0 """ c = hermeval(x, c) c = hermeval(y, c) return c def hermeval3d(x, y, z, c): """ Evaluate a 3-D Hermite_e series at points (x, y, z). This function returns the values: .. math:: p(x,y,z) = \\sum_{i,j,k} c_{i,j,k} * He_i(x) * He_j(y) * He_k(z) The parameters x, y, and z are converted to arrays only if they are tuples or a lists, otherwise they are treated as a scalars and they must have the same shape after conversion. In either case, either x, y, and z or their elements must support multiplication and addition both with themselves and with the elements of c. If c has fewer than 3 dimensions, ones are implicitly appended to its shape to make it 3-D. The shape of the result will be c.shape[3:] + x.shape. Parameters ---------- x, y, z : array_like, compatible object The three dimensional series is evaluated at the points (x, y, z), where x, y, and z must have the same shape. If any of x, y, or z is a list or tuple, it is first converted to an ndarray, otherwise it is left unchanged and if it isn't an ndarray it is treated as a scalar. c : array_like Array of coefficients ordered so that the coefficient of the term of multi-degree i,j,k is contained in c[i,j,k]. If c has dimension greater than 3 the remaining indices enumerate multiple sets of coefficients. Returns ------- values : ndarray, compatible object The values of the multidimensional polynomial on points formed with triples of corresponding values from x, y, and z. See Also -------- hermeval, hermeval2d, hermegrid2d, hermegrid3d Notes ----- .. versionadded::1.7.0 """ try: x, y, z = np.array((x, y, z), copy=0) except: raise ValueError('x, y, z are incompatible') c = hermeval(x, c) c = hermeval(y, c, tensor=False) c = hermeval(z, c, tensor=False) return c def hermegrid3d(x, y, z, c): """ Evaluate a 3-D HermiteE series on the Cartesian product of x, y, and z. This function returns the values: .. math:: p(a,b,c) = \\sum_{i,j,k} c_{i,j,k} * He_i(a) * He_j(b) * He_k(c) where the points (a, b, c) consist of all triples formed by taking a from x, b from y, and c from z. The resulting points form a grid with x in the first dimension, y in the second, and z in the third. The parameters x, y, and z are converted to arrays only if they are tuples or a lists, otherwise they are treated as a scalars. In either case, either x, y, and z or their elements must support multiplication and addition both with themselves and with the elements of c. If c has fewer than three dimensions, ones are implicitly appended to its shape to make it 3-D. The shape of the result will be c.shape[3:] + x.shape + y.shape + z.shape. Parameters ---------- x, y, z : array_like, compatible objects The three dimensional series is evaluated at the points in the Cartesian product of x, y, and z. If x,y, or z is a list or tuple, it is first converted to an ndarray, otherwise it is left unchanged and, if it isn't an ndarray, it is treated as a scalar. c : array_like Array of coefficients ordered so that the coefficients for terms of degree i,j are contained in c[i,j]. If c has dimension greater than two the remaining indices enumerate multiple sets of coefficients. Returns ------- values : ndarray, compatible object The values of the two dimensional polynomial at points in the Cartesian product of x and y. See Also -------- hermeval, hermeval2d, hermegrid2d, hermeval3d Notes ----- .. versionadded::1.7.0 """ c = hermeval(x, c) c = hermeval(y, c) c = hermeval(z, c) return c def hermevander(x, deg) : """Pseudo-Vandermonde matrix of given degree. Returns the pseudo-Vandermonde matrix of degree deg and sample points x. The pseudo-Vandermonde matrix is defined by .. math:: V[..., i] = He_i(x), where 0 <= i <= deg. The leading indices of V index the elements of x and the last index is the degree of the HermiteE polynomial. If c is a 1-D array of coefficients of length n + 1 and V is the array V = hermevander(x, n), then np.dot(V, c) and hermeval(x, c) are the same up to roundoff. This equivalence is useful both for least squares fitting and for the evaluation of a large number of HermiteE series of the same degree and sample points. Parameters ---------- x : array_like Array of points. The dtype is converted to float64 or complex128 depending on whether any of the elements are complex. If x is scalar it is converted to a 1-D array. deg : int Degree of the resulting matrix. Returns ------- vander : ndarray The pseudo-Vandermonde matrix. The shape of the returned matrix is x.shape + (deg + 1,), where The last index is the degree of the corresponding HermiteE polynomial. The dtype will be the same as the converted x. Examples -------- >>> from numpy.polynomial.hermite_e import hermevander >>> x = np.array([-1, 0, 1]) >>> hermevander(x, 3) array([[ 1., -1., 0., 2.], [ 1., 0., -1., -0.], [ 1., 1., 0., -2.]]) """ ideg = 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 : v[1] = x for i in range(2, ideg + 1) : v[i] = (v[i-1]*x - v[i-2]*(i - 1)) return np.rollaxis(v, 0, v.ndim) def hermevander2d(x, y, deg) : """Pseudo-Vandermonde matrix of given degrees. Returns the pseudo-Vandermonde matrix of degrees deg and sample points (x, y). The pseudo-Vandermonde matrix is defined by .. math:: V[..., deg[1]*i + j] = He_i(x) * He_j(y), where 0 <= i <= deg[0] and 0 <= j <= deg[1]. The leading indices of V index the points (x, y) and the last index encodes the degrees of the HermiteE polynomials. If V = hermevander2d(x, y, [xdeg, ydeg]), then the columns of V correspond to the elements of a 2-D coefficient array c of shape (xdeg + 1, ydeg + 1) in the order .. math:: c_{00}, c_{01}, c_{02} ... , c_{10}, c_{11}, c_{12} ... and np.dot(V, c.flat) and hermeval2d(x, y, c) will be the same up to roundoff. This equivalence is useful both for least squares fitting and for the evaluation of a large number of 2-D HermiteE series of the same degrees and sample points. Parameters ---------- x, y : array_like Arrays of point coordinates, all of the same shape. The dtypes will be converted to either float64 or complex128 depending on whether any of the elements are complex. Scalars are converted to 1-D arrays. deg : list of ints List of maximum degrees of the form [x_deg, y_deg]. Returns ------- vander2d : ndarray The shape of the returned matrix is x.shape + (order,), where :math:order = (deg[0]+1)*(deg([1]+1). The dtype will be the same as the converted x and y. See Also -------- hermevander, hermevander3d. hermeval2d, hermeval3d Notes ----- .. versionadded::1.7.0 """ 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 = hermevander(x, degx) vy = hermevander(y, degy) v = vx[..., None]*vy[..., None,:] return v.reshape(v.shape[:-2] + (-1,)) def hermevander3d(x, y, z, deg) : """Pseudo-Vandermonde matrix of given degrees. Returns the pseudo-Vandermonde matrix of degrees deg and sample points (x, y, z). If l, m, n are the given degrees in x, y, z, then Hehe pseudo-Vandermonde matrix is defined by .. math:: V[..., (m+1)(n+1)i + (n+1)j + k] = He_i(x)*He_j(y)*He_k(z), where 0 <= i <= l, 0 <= j <= m, and 0 <= j <= n. The leading indices of V index the points (x, y, z) and the last index encodes the degrees of the HermiteE polynomials. If V = hermevander3d(x, y, z, [xdeg, ydeg, zdeg]), then the columns of V correspond to the elements of a 3-D coefficient array c of shape (xdeg + 1, ydeg + 1, zdeg + 1) in the order .. math:: c_{000}, c_{001}, c_{002},... , c_{010}, c_{011}, c_{012},... and np.dot(V, c.flat) and hermeval3d(x, y, z, c) will be the same up to roundoff. This equivalence is useful both for least squares fitting and for the evaluation of a large number of 3-D HermiteE series of the same degrees and sample points. Parameters ---------- x, y, z : array_like Arrays of point coordinates, all of the same shape. The dtypes will be converted to either float64 or complex128 depending on whether any of the elements are complex. Scalars are converted to 1-D arrays. deg : list of ints List of maximum degrees of the form [x_deg, y_deg, z_deg]. Returns ------- vander3d : ndarray The shape of the returned matrix is x.shape + (order,), where :math:order = (deg[0]+1)*(deg([1]+1)*(deg[2]+1). The dtype will be the same as the converted x, y, and z. See Also -------- hermevander, hermevander3d. hermeval2d, hermeval3d Notes ----- .. versionadded::1.7.0 """ 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 = hermevander(x, degx) vy = hermevander(y, degy) vz = hermevander(z, degz) v = vx[..., None, None]*vy[..., None,:, None]*vz[..., None, None,:] return v.reshape(v.shape[:-3] + (-1,)) def hermefit(x, y, deg, rcond=None, full=False, w=None): """ Least squares fit of Hermite series to data. Return the coefficients of a HermiteE series of degree deg that is the least squares fit to the data values y given at points x. If y is 1-D the returned coefficients will also be 1-D. If y is 2-D multiple fits are done, one for each column of y, and the resulting coefficients are stored in the corresponding columns of a 2-D return. The fitted polynomial(s) are in the form .. math:: p(x) = c_0 + c_1 * He_1(x) + ... + c_n * He_n(x), where n is deg. Since numpy version 1.7.0, hermefit also supports NA. If any of the elements of x, y, or w are NA, then the corresponding rows of the linear least squares problem (see Notes) are set to 0. If y is 2-D, then an NA in any row of y invalidates that whole row. 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] : present when full = True Residuals of the least-squares fit, the effective rank of the scaled Vandermonde matrix and its singular values, and the specified 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, polyfit, hermfit, polyfit hermeval : Evaluates a Hermite series. hermevander : pseudo Vandermonde matrix of Hermite series. hermeweight : HermiteE 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 HermiteE series p that minimizes the sum of the weighted squared errors .. math:: E = \\sum_j w_j^2 * |y_j - p(x_j)|^2, where the :math:w_j are the weights. This problem is solved by setting up the (typically) overdetermined matrix equation .. math:: V(x) * c = w * y, where V is the pseudo Vandermonde matrix of x, the elements of c are the coefficients to be solved for, and the elements of y are the observed values. This equation is then solved using the singular value decomposition of V. If some of the singular values of V are so small that they are neglected, then a RankWarning will be issued. This means that the coefficient values may be poorly determined. Using a lower order fit will usually get rid of the warning. The rcond parameter can also be set to a value smaller than its default, but the resulting fit may be spurious and have large contributions from roundoff error. Fits using HermiteE series are probably most useful when the data can be approximated by sqrt(w(x)) * p(x), where w(x) is the HermiteE weight. In that case the weight sqrt(w(x[i]) should be used together with data values y[i]/sqrt(w(x[i]). The weight function is available as hermeweight. References ---------- .. [1] Wikipedia, "Curve fitting", http://en.wikipedia.org/wiki/Curve_fitting Examples -------- >>> from numpy.polynomial.hermite_e import hermefik, hermeval >>> x = np.linspace(-10, 10) >>> err = np.random.randn(len(x))/10 >>> y = hermeval(x, [1, 2, 3]) + err >>> hermefit(x, y, 2) array([ 1.01690445, 1.99951418, 2.99948696]) """ 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 = hermevander(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 hermecompanion(c): """ Return the scaled companion matrix of c. The basis polynomials are scaled so that the companion matrix is symmetric when c is an HermiteE basis polynomial. This provides better eigenvalue estimates than the unscaled case and for basis polynomials the eigenvalues are guaranteed to be real if numpy.linalg.eigvalsh is used to obtain them. Parameters ---------- c : array_like 1-D array of HermiteE series coefficients ordered from low to high degree. Returns ------- mat : ndarray Scaled companion matrix of dimensions (deg, deg). Notes ----- .. versionadded::1.7.0 """ accprod = np.multiply.accumulate # c is a trimmed copy [c] = pu.as_series([c]) if len(c) < 2: raise ValueError('Series must have maximum degree of at least 1.') if len(c) == 2: return np.array([[-c[0]/c[1]]]) n = len(c) - 1 mat = np.zeros((n, n), dtype=c.dtype) scl = np.hstack((1., np.sqrt(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(np.arange(1, n)) bot[...] = top mat[:, -1] -= (c[:-1]/c[-1])*(scl/scl[-1]) return mat def hermeroots(c): """ Compute the roots of a HermiteE series. Return the roots (a.k.a. "zeros") of the polynomial .. math:: p(x) = \\sum_i c[i] * He_i(x). Parameters ---------- c : 1-D array_like 1-D array of coefficients. Returns ------- out : ndarray Array of the roots of the series. If all the roots are real, then out is also real, otherwise it is complex. See Also -------- polyroots, legroots, lagroots, hermroots, chebroots Notes ----- The root estimates are obtained as the eigenvalues of the companion matrix, Roots far from the origin of the complex plane may have large errors due to the numerical instability of the series for such values. Roots with multiplicity greater than 1 will also show larger errors as the value of the series near such points is relatively insensitive to errors in the roots. Isolated roots near the origin can be improved by a few iterations of Newton's method. The HermiteE series basis polynomials aren't powers of x so the results of this function may seem unintuitive. Examples -------- >>> from numpy.polynomial.hermite_e import hermeroots, hermefromroots >>> coef = hermefromroots([-1, 0, 1]) >>> coef array([ 0., 2., 0., 1.]) >>> hermeroots(coef) array([-1., 0., 1.]) """ # c is a trimmed copy [c] = pu.as_series([c]) if len(c) <= 1 : return np.array([], dtype=c.dtype) if len(c) == 2 : return np.array([-c[0]/c[1]]) m = hermecompanion(c) r = la.eigvals(m) r.sort() return r def hermegauss(deg): """ Gauss-HermiteE quadrature. Computes the sample points and weights for Gauss-HermiteE quadrature. These sample points and weights will correctly integrate polynomials of degree :math:2*deg - 1 or less over the interval :math:[-\inf, \inf] with the weight function :math:f(x) = \exp(-x^2/2). Parameters ---------- deg : int Number of sample points and weights. It must be >= 1. Returns ------- x : ndarray 1-D ndarray containing the sample points. y : ndarray 1-D ndarray containing the weights. Notes ----- .. versionadded::1.7.0 The results have only been tested up to degree 100, higher degrees may be problematic. The weights are determined by using the fact that .. math:: w_k = c / (He'_n(x_k) * He_{n-1}(x_k)) where :math:c is a constant independent of :math:k and :math:x_k is the k'th root of :math:He_n, and then scaling the results to get the right value when integrating 1. """ ideg = 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 = hermecompanion(c) x = la.eigvals(m) x.sort() # improve roots by one application of Newton dy = hermeval(x, c) df = hermeval(x, hermeder(c)) x -= dy/df # compute the weights. We scale the factor to avoid possible numerical # overflow. fm = hermeval(x, c[1:]) fm /= np.abs(fm).max() df /= np.abs(df).max() w = 1/(fm * df) # for Hermite_e we can also symmetrize w = (w + w[::-1])/2 x = (x - x[::-1])/2 # scale w to get the right value w *= np.sqrt(2*np.pi) / w.sum() return x, w def hermeweight(x): """Weight function of the Hermite_e polynomials. The weight function is :math:\exp(-x^2/2) and the interval of integration is :math:[-\inf, \inf]. the HermiteE polynomials are orthogonal, but not normalized, with respect to this weight function. Parameters ---------- x : array_like Values at which the weight function will be computed. Returns ------- w : ndarray The weight function at x. Notes ----- .. versionadded::1.7.0 """ w = np.exp(-.5*x**2) return w # # HermiteE series class # exec(polytemplate.substitute(name='HermiteE', nick='herme', domain='[-1,1]'))
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