ó ʽ÷Xc@`sIdZddlmZmZmZddlZddlmZmZm Z m Z m Z m Z m Z mZmZmZmZmZddlmZddlmZddlmZejZdd lmZd d d d dddddddddddgZidd6dd6dd6dd 6d!d"6d#d$6d%d&6d'd(6d)d*6d+d,6d-d.6d/d06d1d26d3d46d5d66Zd7d8d9d:d;d<d=d>d?d@dAdBdCdDdEgZee ej!ƒƒedFdGgZ"dHej#fdI„ƒYZ$dJ„Z%e&dK„Z'e&dL„Z(e&dM„Z)e&dN„Z*e&dO„Z+e&dP„Z,e&dQ„Z-e&dR„Z.e&dS„Z/dTdU„Z0dV„Z1dW„Z2dX„Z3dY„Z4dTdZ„Z5d[„Z6e&d\„Z7e&d]„Z8e&d^„Z9e&d_„Z:e&d`„Z;e&da„Z<e&db„Z=e&dc„Z>e&dd„Z?e&de„Z@e&df„ZAe&dg„ZBe&dh„ZCe&di„ZDe&dj„ZEe&dk„ZFe&dl„ZGe&dm„ZHe&dn„ZIe&do„ZJe&dp„ZKejLZLddqlmMZMmNZNmOZOmPZPmQZQmRZRmSZSmTZTmUZUmVZVmWZWmXZXmYZYmZZZm[Z[m\Z\e]ƒZ^x5ej_ƒD]'\Z`Zae^e`e^eaŠsgð?tnormcoeftcoeffst _eval_func( tnptpoly1dt__init__trangetlentarraytlisttzipt__dict__Rtabstfloat( tselftrootsRSthnRXtwfuncRUtmonict eval_functkt equiv_weightstmu((RWRXs7/tmp/pip-build-7oUkmx/scipy/scipy/special/orthogonal.pyR_}s"4     cC`s@|jr)t|tjƒ r)|j|ƒStjj||ƒSdS(N(R\t isinstanceR]R^t__call__(Rhtv((s7/tmp/pip-build-7oUkmx/scipy/scipy/special/orthogonal.pyRr“s c`siˆdkrdS|jdcˆ9<|jd‰ˆrR‡‡fd†|jd -1 beta : float beta must be > 0 mu : bool, optional If True, return the sum of the weights, optional. Returns ------- x : ndarray Sample points w : ndarray Weights mu : float Sum of the weights See Also -------- scipy.integrate.quadrature scipy.integrate.fixed_quad isn must be a positive integer.iÿÿÿÿs'alpha and beta must be greater than -1.ggà?g@c`s)tj|dkˆˆdˆˆdƒS(Niig(R]twhere(Rn(tatb(s7/tmp/pip-build-7oUkmx/scipy/scipy/special/orthogonal.pyRYsc`sYtj|dkˆˆdˆˆˆˆˆˆd|ˆˆd|ˆˆdƒS(Niig@(R]R(Rn(RR‘(s7/tmp/pip-build-7oUkmx/scipy/scipy/special/orthogonal.pyRYs"c`sƒdd|ˆˆtj|ˆ|ˆd|ˆˆdƒtj|dkdtj||ˆˆd|ˆˆdƒƒS(Ng@iigð?(R]RR(Rn(RR‘(s7/tmp/pip-build-7oUkmx/scipy/scipy/special/orthogonal.pyRYsc`stj|ˆˆ|ƒS(N(tcephesRE(R‚RV(RR‘(s7/tmp/pip-build-7oUkmx/scipy/scipy/special/orthogonal.pyRY sc`s6d|ˆˆdtj|dˆdˆd|ƒS(Ngà?i(R’RE(R‚RV(RR‘(s7/tmp/pip-build-7oUkmx/scipy/scipy/special/orthogonal.pyRY s(R t ValueErrorR#R7R’tbetaRŽRy( R‚talphaR”RptmRƒR„R…R†R‡((RR‘s7/tmp/pip-build-7oUkmx/scipy/scipy/special/orthogonal.pyR-Ðs&#   *c `sUˆdkrtdƒ‚n‡‡fd†}ˆdkr^tggdd|d |dtjƒStˆˆˆdtƒ\}}}ˆˆd}d |d ˆ|tˆˆdƒ} | tˆˆdƒtˆdƒtˆ|ƒ9} td ˆ|ƒd ˆtˆdƒtˆ|ƒ} t||| | |d |‡‡‡fd †ƒ} | S(sÞJacobi polynomial. Defined to be the solution of .. math:: (1 - x^2)\frac{d^2}{dx^2}P_n^{(\alpha, \beta)} + (\beta - \alpha - (\alpha + \beta + 2)x) \frac{d}{dx}P_n^{(\alpha, \beta)} + n(n + \alpha + \beta + 1)P_n^{(\alpha, \beta)} = 0 for :math:`\alpha, \beta > -1`; :math:`P_n^{(\alpha, \beta)}` is a polynomial of degree :math:`n`. Parameters ---------- n : int Degree of the polynomial. alpha : float Parameter, must be greater than -1. beta : float Parameter, must be greater than -1. monic : bool, optional If `True`, scale the leading coefficient to be 1. Default is `False`. Returns ------- P : orthopoly1d Jacobi polynomial. Notes ----- For fixed :math:`\alpha, \beta`, the polynomials :math:`P_n^{(\alpha, \beta)}` are orthogonal over :math:`[-1, 1]` with weight function :math:`(1 - x)^\alpha(1 + x)^\beta`. isn must be nonnegative.c`sd|ˆd|ˆS(Ni((RV(R•R”(s7/tmp/pip-build-7oUkmx/scipy/scipy/special/orthogonal.pyRY9sgð?iÿÿÿÿiRmRpig@c`stˆˆˆ|ƒS(N(RE(RV(R•R”R‚(s7/tmp/pip-build-7oUkmx/scipy/scipy/special/orthogonal.pyRYDs(iÿÿÿÿi(iÿÿÿÿi(R“RQR]t ones_likeR-Rt_gam( R‚R•R”RlRkRVRRptab1RjRXRt((R•R”R‚s7/tmp/pip-build-7oUkmx/scipy/scipy/special/orthogonal.pyRs&   !(48cC`sž||dks|dkr+tdƒ‚nt||||dtƒ\}}}|dd}d|}||}||}|r|||fS||fSdS(s{Gauss-Jacobi (shifted) quadrature. Computes the sample points and weights for Gauss-Jacobi (shifted) quadrature. The sample points are the roots of the n-th degree shifted Jacobi polynomial, :math:`G^{p,q}_n(x)`. These sample points and weights correctly integrate polynomials of degree :math:`2n - 1` or less over the interval :math:`[0, 1]` with weight function :math:`f(x) = (1 - x)^{p-q} x^{q-1}` Parameters ---------- n : int quadrature order p1 : float (p1 - q1) must be > -1 q1 : float q1 must be > 0 mu : bool, optional If True, return the sum of the weights, optional. Returns ------- x : ndarray Sample points w : ndarray Weights mu : float Sum of the weights See Also -------- scipy.integrate.quadrature scipy.integrate.fixed_quad iÿÿÿÿis>(p - q) must be greater than -1, and q must be greater than 0.iig@N(R“R-R(R‚tp1tq1RpRVRR–tscale((s7/tmp/pip-build-7oUkmx/scipy/scipy/special/orthogonal.pyR?Js#&    c `s5ˆdkrtdƒ‚n‡‡fd†}ˆdkr^tggdd|d|dtjƒSˆ}t|ˆˆdtƒ\}}}tˆdƒtˆˆƒtˆˆƒtˆˆˆdƒ} | d ˆˆtd ˆˆƒd } d} t||| | d |d dd |d‡‡‡fd †ƒ} | S(s Shifted Jacobi polynomial. Defined by .. math:: G_n^{(p, q)}(x) = \binom{2n + p - 1}{n}^{-1}P_n^{(p - q, q - 1)}(2x - 1), where :math:`P_n^{(\cdot, \cdot)}` is the nth Jacobi polynomial. Parameters ---------- n : int Degree of the polynomial. p : float Parameter, must have :math:`p > q - 1`. q : float Parameter, must be greater than 0. monic : bool, optional If `True`, scale the leading coefficient to be 1. Default is `False`. Returns ------- G : orthopoly1d Shifted Jacobi polynomial. Notes ----- For fixed :math:`p, q`, the polynomials :math:`G_n^{(p, q)}` are orthogonal over :math:`[0, 1]` with weight function :math:`(1 - x)^{p - q}x^{q - 1}`. isn must be nonnegative.c`sd|ˆˆ|ˆdS(Ngð?((RV(Rttq(s7/tmp/pip-build-7oUkmx/scipy/scipy/special/orthogonal.pyRY sgð?iÿÿÿÿiRmRpiRkRURlc`stˆˆˆ|ƒS(N(RN(RV(R‚RtR(s7/tmp/pip-build-7oUkmx/scipy/scipy/special/orthogonal.pyRY«s(iÿÿÿÿi(ii(R“RQR]R—R?RR˜( R‚RtRRlRktn1RVRRƒRjRXtpp((R‚RtRs7/tmp/pip-build-7oUkmx/scipy/scipy/special/orthogonal.pyR!ys$   !B($c `st|ƒ}|dks$||kr3tdƒ‚nˆdkrNtdƒ‚ntjˆdƒ}|dkr»tjˆdgdƒ}tj|gdƒ}|r®|||fS||fSn‡fd†}‡fd†}‡fd †} ‡fd †} t||||| | t|ƒS( sgGauss-generalized Laguerre quadrature. Computes the sample points and weights for Gauss-generalized Laguerre quadrature. The sample points are the roots of the n-th degree generalized Laguerre polynomial, :math:`L^{\alpha}_n(x)`. These sample points and weights correctly integrate polynomials of degree :math:`2n - 1` or less over the interval :math:`[0, \infty]` with weight function :math:`f(x) = x^{\alpha} e^{-x}`. Parameters ---------- n : int quadrature order alpha : float alpha must be > -1 mu : bool, optional If True, return the sum of the weights, optional. Returns ------- x : ndarray Sample points w : ndarray Weights mu : float Sum of the weights See Also -------- scipy.integrate.quadrature scipy.integrate.fixed_quad isn must be a positive integer.iÿÿÿÿsalpha must be greater than -1.gð?R{c`sd|ˆdS(Nii((Rn(R•(s7/tmp/pip-build-7oUkmx/scipy/scipy/special/orthogonal.pyRYâsc`stj||ˆƒ S(N(R]R(Rn(R•(s7/tmp/pip-build-7oUkmx/scipy/scipy/special/orthogonal.pyRYãsc`stj|ˆ|ƒS(N(R’RG(R‚RV(R•(s7/tmp/pip-build-7oUkmx/scipy/scipy/special/orthogonal.pyRYäsc`s:|tj|ˆ|ƒ|ˆtj|dˆ|ƒ|S(Ni(R’RG(R‚RV(R•(s7/tmp/pip-build-7oUkmx/scipy/scipy/special/orthogonal.pyRYås(R R“R’tgammaR]RbRŽRy( R‚R•RpR–RƒRVRR„R…R†R‡((R•s7/tmp/pip-build-7oUkmx/scipy/scipy/special/orthogonal.pyR1±s"!     c `sˆdkrtdƒ‚nˆdkr6tdƒ‚nˆdkrOˆd}nˆ}t|ˆdtƒ\}}}‡fd†}ˆdkržgg}}ntˆˆdƒtˆdƒ}dˆtˆdƒ} t|||| |dtf|‡‡fd†ƒ} | S( sßGeneralized (associated) Laguerre polynomial. Defined to be the solution of .. math:: x\frac{d^2}{dx^2}L_n^{(\alpha)} + (\alpha + 1 - x)\frac{d}{dx}L_n^{(\alpha)} + nL_n^{(\alpha)} = 0, where :math:`\alpha > -1`; :math:`L_n^{(\alpha)}` is a polynomial of degree :math:`n`. Parameters ---------- n : int Degree of the polynomial. alpha : float Parameter, must be greater than -1. monic : bool, optional If `True`, scale the leading coefficient to be 1. Default is `False`. Returns ------- L : orthopoly1d Generalized Laguerre polynomial. Notes ----- For fixed :math:`\alpha`, the polynomials :math:`L_n^{(\alpha)}` are orthogonal over :math:`[0, \infty)` with weight function :math:`e^{-x}x^\alpha`. The Laguerre polynomials are the special case where :math:`\alpha = 0`. See Also -------- laguerre : Laguerre polynomial. iÿÿÿÿsalpha must be > -1isn must be nonnegative.iRpc`st| ƒ|ˆS(N(R(RV(R•(s7/tmp/pip-build-7oUkmx/scipy/scipy/special/orthogonal.pyRYsc`stˆˆ|ƒS(N(RG(RV(R•R‚(s7/tmp/pip-build-7oUkmx/scipy/scipy/special/orthogonal.pyRY$s(R“R1RR˜RQR( R‚R•RlRžRVRRƒRkRjRXRt((R•R‚s7/tmp/pip-build-7oUkmx/scipy/scipy/special/orthogonal.pyRês *     "cC`st|dd|ƒS(s%Gauss-Laguerre quadrature. Computes the sample points and weights for Gauss-Laguerre quadrature. The sample points are the roots of the n-th degree Laguerre polynomial, :math:`L_n(x)`. These sample points and weights correctly integrate polynomials of degree :math:`2n - 1` or less over the interval :math:`[0, \infty]` with weight function :math:`f(x) = e^{-x}`. Parameters ---------- n : int quadrature order mu : bool, optional If True, return the sum of the weights, optional. Returns ------- x : ndarray Sample points w : ndarray Weights mu : float Sum of the weights See Also -------- scipy.integrate.quadrature scipy.integrate.fixed_quad numpy.polynomial.laguerre.laggauss gRp(R1(R‚Rp((s7/tmp/pip-build-7oUkmx/scipy/scipy/special/orthogonal.pyR/*sc `sƈdkrtdƒ‚nˆdkr4ˆd}nˆ}t|dtƒ\}}}ˆdkrqgg}}nd}dˆtˆdƒ}t||||d„dtf|‡fd†ƒ}|S( sRLaguerre polynomial. Defined to be the solution of .. math:: x\frac{d^2}{dx^2}L_n + (1 - x)\frac{d}{dx}L_n + nL_n = 0; :math:`L_n` is a polynomial of degree :math:`n`. Parameters ---------- n : int Degree of the polynomial. monic : bool, optional If `True`, scale the leading coefficient to be 1. Default is `False`. Returns ------- L : orthopoly1d Laguerre Polynomial. Notes ----- The polynomials :math:`L_n` are orthogonal over :math:`[0, \infty)` with weight function :math:`e^{-x}`. isn must be nonnegative.iRpgð?iÿÿÿÿcS`s t| ƒS(N(R(RV((s7/tmp/pip-build-7oUkmx/scipy/scipy/special/orthogonal.pyRYusc`s tˆ|ƒS(N(RF(RV(R‚(s7/tmp/pip-build-7oUkmx/scipy/scipy/special/orthogonal.pyRYvs(R“R/RR˜RQR( R‚RlRžRVRRƒRjRXRt((R‚s7/tmp/pip-build-7oUkmx/scipy/scipy/special/orthogonal.pyRLs    !c C`sÇt|ƒ}|dks$||kr3tdƒ‚ntjtjƒ}|dkr”d„}d„}tj}d„}t||||||t|ƒSt |ƒ\}} |r¹|| |fS|| fSdS(s[Gauss-Hermite (physicst's) quadrature. Computes the sample points and weights for Gauss-Hermite quadrature. The sample points are the roots of the n-th degree Hermite polynomial, :math:`H_n(x)`. These sample points and weights correctly integrate polynomials of degree :math:`2n - 1` or less over the interval :math:`[-\infty, \infty]` with weight function :math:`f(x) = e^{-x^2}`. Parameters ---------- n : int quadrature order mu : bool, optional If True, return the sum of the weights, optional. Returns ------- x : ndarray Sample points w : ndarray Weights mu : float Sum of the weights Notes ----- For small n up to 150 a modified version of the Golub-Welsch algorithm is used. Nodes are computed from the eigenvalue problem and improved by one step of a Newton iteration. The weights are computed from the well-known analytical formula. For n larger than 150 an optimal asymptotic algorithm is applied which computes nodes and weights in a numerically stable manner. The algorithm has linear runtime making computation for very large n (several thousand or more) feasible. See Also -------- scipy.integrate.quadrature scipy.integrate.fixed_quad numpy.polynomial.hermite.hermgauss roots_hermitenorm References ---------- .. [townsend.trogdon.olver-2014] Townsend, A. and Trogdon, T. and Olver, S. (2014) *Fast computation of Gauss quadrature nodes and weights on the whole real line*. :arXiv:`1410.5286`. .. [townsend.trogdon.olver-2015] Townsend, A. and Trogdon, T. and Olver, S. (2015) *Fast computation of Gauss quadrature nodes and weights on the whole real line*. IMA Journal of Numerical Analysis :doi:`10.1093/imanum/drv002`. isn must be a positive integer.i–cS`sd|S(Ng((Rn((s7/tmp/pip-build-7oUkmx/scipy/scipy/special/orthogonal.pyRY¼scS`stj|dƒS(Ng@(R]R(Rn((s7/tmp/pip-build-7oUkmx/scipy/scipy/special/orthogonal.pyRY½scS`sd|tj|d|ƒS(Ng@i(R’RH(R‚RV((s7/tmp/pip-build-7oUkmx/scipy/scipy/special/orthogonal.pyRY¿sN( R R“R]RRR’RHRŽRt_roots_hermite_asy( R‚RpR–RƒR„R…R†R‡tnodesRS((s7/tmp/pip-build-7oUkmx/scipy/scipy/special/orthogonal.pyR3|s:       ic`s§|dd}dt|dƒd|dtdt|dƒd|d‰‡fd†}d„}dt}x.t|ƒD] }|||ƒ||ƒ}qW|S(sHelper function for Tricomi initial guesses For details, see formula 3.1 in lemma 3.1 in the original paper. Parameters ---------- n : int Quadrature order k : ndarray of type int Index of roots :math:` au_k` to compute maxit : int Number of Newton maxit performed, the default value of 5 is sufficient. Returns ------- tauk : ndarray Roots of equation 3.1 See Also -------- initial_nodes_a roots_hermite_asy igà?g@g@g@c`s|t|ƒˆS(N(R(RV(R‰(s7/tmp/pip-build-7oUkmx/scipy/scipy/special/orthogonal.pyRYåscS`sdt|ƒS(Ngð?(R (RV((s7/tmp/pip-build-7oUkmx/scipy/scipy/special/orthogonal.pyRYæs(RRR`(R‚RntmaxitRR†R‡txiti((R‰s7/tmp/pip-build-7oUkmx/scipy/scipy/special/orthogonal.pyt _compute_taukÉsB  cC`st||ƒ}td|ƒd}|dd}dt|dƒd|d}||dd|ddd|ddd|d}|S( seTricomi initial guesses Computes an initial approximation to the square of the `k`-th (positive) root :math:`x_k` of the Hermite polynomial :math:`H_n` of order :math:`n`. The formula is the one from lemma 3.1 in the original paper. The guesses are accurate except in the region near :math:`\sqrt{2n + 1}`. Parameters ---------- n : int Quadrature order k : ndarray of type int Index of roots to compute Returns ------- xksq : ndarray Square of the approximate roots See Also -------- initial_nodes roots_hermite_asy gà?ig@g@gð?g@g@gÐ?(R¦R R(R‚RnttauktsigkRtnutxksq((s7/tmp/pip-build-7oUkmx/scipy/scipy/special/orthogonal.pyt_initial_nodes_aís  :cC`sõ|dd}dt|dƒd|d}tj|jƒdƒdddd…}|dd||dd dd!|d|d"d#d$|d|d d%|d&|ddd'|d(d)|dd*|ddd+|d,}|S(-seGatteschi initial guesses Computes an initial approximation to the square of the `k`-th (positive) root :math:`x_k` of the Hermite polynomial :math:`H_n` of order :math:`n`. The formula is the one from lemma 3.2 in the original paper. The guesses are accurate in the region just below :math:`\sqrt{2n + 1}`. Parameters ---------- n : int Quadrature order k : ndarray of type int Index of roots to compute Returns ------- xksq : ndarray Square of the approximate root See Also -------- initial_nodes roots_hermite_asy igà?g@g@iiNiÿÿÿÿg@gð?g@gð¿g"@g€a@g(@gàe@ig0@gœ˜@gW@gþ@igÀg˜Í@g€¡!GAig‘@g°›ý@gÀgUUUUUUå?gUUUUUUÕ?gš™™™™™É?gUUUUUUõ?gUUUUUUÕ¿gPuPu°?gE'«å±?gH³g®΄?g`A÷îí‡?gUUUUUUå?g«ªªªªªú¿g ^PTxt?gÅôJ±‘_‚?gUUUUUUÕ?g«ªªªªªÀ(RRtairyzoR€(R‚RnRR©takRª((s7/tmp/pip-build-7oUkmx/scipy/scipy/special/orthogonal.pyt_initial_nodes_bs  )o+cC`sÈd|d}t|ƒjtƒ}tdtt|dƒdƒƒ}|ddd…}t|||d ƒ}t|||dƒ}tt||gƒƒ}|ddkrÄtd|gƒ}n|S( sÐInitial guesses for the Hermite roots Computes an initial approximation to the non-negative roots :math:`x_k` of the Hermite polynomial :math:`H_n` of order :math:`n`. The Tricomi and Gatteschi initial guesses are used in the region where they are accurate. Parameters ---------- n : int Quadrature order Returns ------- xk : ndarray Approximate roots See Also -------- roots_hermite_asy g_—Dj˜iß?g•ê Ï®ƒ@igà?Niÿÿÿÿig( R tastypeR RRR«R®RR (R‚tfittturnovertiatibtxasqtxbsqtiv((s7/tmp/pip-build-7oUkmx/scipy/scipy/special/orthogonal.pyt_initial_nodes8s#c6C`s t|ƒ}t|ƒ}d|d}d|d||}d|ddQ }| |dd}d}d} d} d } d } d } d}d }d }d}d}d}|tdƒjdRƒ}d}d|ddd…fd|d}d|ddd…fd|ddd…fdd}d|ddd…fd|d dd…fd!|d"dd…fd#|ddd…fd$|d%}d&|d'dd…fd(|d)dd…fd*|d+dd…fd,|ddd…fd-|ddd…fd.d/}d0|d1dd…fd2|d3dd…fd4|d5dd…fd6|ddd…fd7|d dd…fd8|d"dd…fd9|ddd…fd:|d;}d}d|ddd…fd|d}d<|ddd…fd=|ddd…fd>d}d|ddd…fd|d dd…fd?|d"dd…fd@|ddd…fd$|d%}dA|d'dd…fdB|d)dd…fdC|d+dd…fdD|ddd…fdE|ddd…fdFd/}d0|d1dd…fd2|d3dd…fdG|d5dd…fdH|ddd…fdI|d dd…fdJ|d"dd…fdK|ddd…fdL|d;} t|dS|ƒ\}!}"}#}$dttƒ|dT|}%|td+dNd+ƒjdUƒ}&||}'|||&dOdd…f|||&ddd…f|||d}(|||&dOdd…f|||&ddd…f|||&ddd…f|||&ddd…f|||d+})| ||&dOdd…f|| |d}*| ||&dOdd…f| ||&ddd…f| ||&ddd…f|| |d"}+| ||&dOdd…f| ||&ddd…f| ||&ddd…f| ||&ddd…f| ||&ddd…f|| |d)},|%|!|'|(|d|)|dM|"|*|+|d|,|dM|dV}-tdtƒ|dW|}.|||&dOdd…f|| |}/|||&dOdd…f|||&ddd…f|||&ddd…f|| |d}0|||&dOdd…f|||&ddd…f|||&ddd…f|||&ddd…f|||&ddd…f||  |d }1||}2| ||&dOdd…f| ||&ddd…f|||d}3| ||&dOdd…f| ||&ddd…f| ||&ddd…f| ||&ddd…f|||d+}4|.|!|/|0|d|1|dM|dX|"|2|3|d|4|dM}5|-|5fS(YsÇAsymptotic series expansion of parabolic cylinder function The implementation is based on sections 3.2 and 3.3 from the original paper. Compared to the published version this code adds one more term to the asymptotic series. The detailed formulas can be found at [parabolic-asymptotics]_. The evaluation is done in a transformed variable :math:`\theta := \arccos(t)` where :math:`t := x / \mu` and :math:`\mu := \sqrt{2n + 1}`. Parameters ---------- n : int Quadrature order theta : ndarray Transformed position variable Returns ------- U : ndarray Value of the parabolic cylinder function :math:`U(a, \theta)`. Ud : ndarray Value of the derivative :math:`U^{\prime}(a, \theta)` of the parabolic cylinder function. See Also -------- roots_hermite_asy References ---------- .. [parabolic-asymptotics] http://dlmf.nist.gov/12.10#vii g@gð?gà?g@igÐ?g«ªªªªªº?g9Žã8Žcµ?g±HxºiÀ?gdŠ§­Ò?g_ÙcJ6ì?g«ªªªªªÂ¿grÇqG¹¿gk~X¤ X¿g@ñ9SsMÔ¿g:¼(,a(î¿iiÿÿÿÿiiNg@g8@g"Àig o@g b@g’@g”¯Ài g@ÃÑ@igÀŸÛ@igØAgЦAgPAg@Ãñ@i gìœAig0ëAig8 JˆAgH;Ag ß\hAgü‚AgÀ夓Aigd—jÂAi gÀm6ºÝAi gâï ëAgP”/„ðAgìâf!Bg…ݱ×;Bgh¬d¬BgÀêøAg.@gpt@gàa@g`Äá@gÈ AgÀšýÀgjÖ Ag°ŒAgH$i‹Ag¸¦ŸAiÇê¸gÀÁÇ.ÞAg€(òëAg ÓìAgÈ5¿e Bgãëd{>BI+ëg@iig @gUUUUUUå?(iÿÿÿÿigUUUUUUå?gUUUUUUÅ?(iÿÿÿÿigUUUUUUõ?gUUUUUUÕ?gUUUUUUå?(RR RtreshapeRRR(6R‚tthetatsttctRptetatzetatphita0ta1ta2ta3ta4ta5tb0tb1tb2tb3tb4tb5tctptu0tu1tu2tu3tu4tu5tv0tv1tv2tv3tv4tv5tAitAiptBitBiptPtphiptA0tA1tA2tB0tB1tB2tUtPdtC0tC1tC2tD0tD1tD2tUd((s7/tmp/pip-build-7oUkmx/scipy/scipy/special/orthogonal.pyt_pbcf_sh"  &:n‚¶&:n‚¶  J‚/gŸ",+gŸ J‚*$c C`sòtd|dƒ}||}t|ƒ}xqt|ƒD]c}t||ƒ\}}|tdƒ|t|ƒ|} || }tt| ƒƒdkr7Pq7q7W|t|ƒ} |ddkrËd| d 150. The algorithm has linear runtime making computation for very large n feasible. Parameters ---------- n : int quadrature order Returns ------- nodes : ndarray Quadrature nodes weights : ndarray Quadrature weights See Also -------- roots_hermite References ---------- .. [townsend.trogdon.olver-2014] Townsend, A. and Trogdon, T. and Olver, S. (2014) *Fast computation of Gauss quadrature nodes and weights on the whole real line*. :arXiv:`1410.5286`. .. [townsend.trogdon.olver-2015] Townsend, A. and Trogdon, T. and Olver, S. (2015) *Fast computation of Gauss quadrature nodes and weights on the whole real line*. IMA Journal of Numerical Analysis :doi:`10.1093/imanum/drv002`. iiNiÿÿÿÿ(R·RóR RRR(R‚R¶R¢RS((s7/tmp/pip-build-7oUkmx/scipy/scipy/special/orthogonal.pyR¡üs+  " c `sÛˆdkrtdƒ‚nˆdkr4ˆd}nˆ}t|dtƒ\}}}d„}ˆdkrzgg}}ndˆtˆdƒttƒ}dˆ}t|||||t tf|‡fd†ƒ} | S(sHPhysicist's Hermite polynomial. Defined by .. math:: H_n(x) = (-1)^ne^{x^2}\frac{d^n}{dx^n}e^{-x^2}; :math:`H_n` is a polynomial of degree :math:`n`. Parameters ---------- n : int Degree of the polynomial. monic : bool, optional If `True`, scale the leading coefficient to be 1. Default is `False`. Returns ------- H : orthopoly1d Hermite polynomial. Notes ----- The polynomials :math:`H_n` are orthogonal over :math:`(-\infty, \infty)` with weight function :math:`e^{-x^2}`. isn must be nonnegative.iRpcS`st| |ƒS(N(R(RV((s7/tmp/pip-build-7oUkmx/scipy/scipy/special/orthogonal.pyRY[sic`s tˆ|ƒS(N(RH(RV(R‚(s7/tmp/pip-build-7oUkmx/scipy/scipy/special/orthogonal.pyRYas(R“R3RR˜RRRQR( R‚RlRžRVRRƒRkRjRXRt((R‚s7/tmp/pip-build-7oUkmx/scipy/scipy/special/orthogonal.pyR5s     " c C`sët|ƒ}|dks$||kr3tdƒ‚ntjdtjƒ}|dkr˜d„}d„}tj}d„}t||||||t|ƒSt |ƒ\}} |tdƒ9}| tdƒ9} |rÝ|| |fS|| fSd S( sNGauss-Hermite (statistician's) quadrature. Computes the sample points and weights for Gauss-Hermite quadrature. The sample points are the roots of the n-th degree Hermite polynomial, :math:`He_n(x)`. These sample points and weights correctly integrate polynomials of degree :math:`2n - 1` or less over the interval :math:`[-\infty, \infty]` with weight function :math:`f(x) = e^{-x^2/2}`. Parameters ---------- n : int quadrature order mu : bool, optional If True, return the sum of the weights, optional. Returns ------- x : ndarray Sample points w : ndarray Weights mu : float Sum of the weights Notes ----- For small n up to 150 a modified version of the Golub-Welsch algorithm is used. Nodes are computed from the eigenvalue problem and improved by one step of a Newton iteration. The weights are computed from the well-known analytical formula. For n larger than 150 an optimal asymptotic algorithm is used which computes nodes and weights in a numerical stable manner. The algorithm has linear runtime making computation for very large n (several thousand or more) feasible. See Also -------- scipy.integrate.quadrature scipy.integrate.fixed_quad numpy.polynomial.hermite_e.hermegauss isn must be a positive integer.g@i–cS`sd|S(Ng((Rn((s7/tmp/pip-build-7oUkmx/scipy/scipy/special/orthogonal.pyRY˜scS`s tj|ƒS(N(R]R(Rn((s7/tmp/pip-build-7oUkmx/scipy/scipy/special/orthogonal.pyRY™scS`s|tj|d|ƒS(Ni(R’RI(R‚RV((s7/tmp/pip-build-7oUkmx/scipy/scipy/special/orthogonal.pyRY›siN( R R“R]RRR’RIRŽRR¡( R‚RpR–RƒR„R…R†R‡R¢RS((s7/tmp/pip-build-7oUkmx/scipy/scipy/special/orthogonal.pyR5gs +       c `s߈dkrtdƒ‚nˆdkr4ˆd}nˆ}t|dtƒ\}}}d„}ˆdkrzgg}}ntdtƒtˆdƒ}d}t||||d|d t tfd |d ‡fd †ƒ} | S( sbNormalized (probabilist's) Hermite polynomial. Defined by .. math:: He_n(x) = (-1)^ne^{x^2/2}\frac{d^n}{dx^n}e^{-x^2/2}; :math:`He_n` is a polynomial of degree :math:`n`. Parameters ---------- n : int Degree of the polynomial. monic : bool, optional If `True`, scale the leading coefficient to be 1. Default is `False`. Returns ------- He : orthopoly1d Hermite polynomial. Notes ----- The polynomials :math:`He_n` are orthogonal over :math:`(-\infty, \infty)` with weight function :math:`e^{-x^2/2}`. isn must be nonnegative.iRpcS`st| |dƒS(Ng@(R(RV((s7/tmp/pip-build-7oUkmx/scipy/scipy/special/orthogonal.pyRYÏsigð?RkRURlRmc`s tˆ|ƒS(N(RI(RV(R‚(s7/tmp/pip-build-7oUkmx/scipy/scipy/special/orthogonal.pyRYÕs(R“R5RRRR˜RQR( R‚RlRžRVRRƒRkRjRXRt((R‚s7/tmp/pip-build-7oUkmx/scipy/scipy/special/orthogonal.pyR¨s     +c `sðt|ƒ}|dks$||kr3tdƒ‚nˆdkrNtdƒ‚nˆdkrgt||ƒStjtjƒtjˆdƒtjˆdƒ}d„}‡fd†}‡fd †}‡fd †}t||||||t |ƒS( sNGauss-Gegenbauer quadrature. Computes the sample points and weights for Gauss-Gegenbauer quadrature. The sample points are the roots of the n-th degree Gegenbauer polynomial, :math:`C^{\alpha}_n(x)`. These sample points and weights correctly integrate polynomials of degree :math:`2n - 1` or less over the interval :math:`[-1, 1]` with weight function :math:`f(x) = (1 - x^2)^{\alpha - 1/2}`. Parameters ---------- n : int quadrature order alpha : float alpha must be > -0.5 mu : bool, optional If True, return the sum of the weights, optional. Returns ------- x : ndarray Sample points w : ndarray Weights mu : float Sum of the weights See Also -------- scipy.integrate.quadrature scipy.integrate.fixed_quad isn must be a positive integer.gà¿s alpha must be greater than -0.5.ggà?cS`sd|S(Ng((Rn((s7/tmp/pip-build-7oUkmx/scipy/scipy/special/orthogonal.pyRY sc`s5tj||dˆdd|ˆ|ˆdƒS(Niii(R]R(Rn(R•(s7/tmp/pip-build-7oUkmx/scipy/scipy/special/orthogonal.pyRY sc`stj|ˆ|ƒS(N(R’RJ(R‚RV(R•(s7/tmp/pip-build-7oUkmx/scipy/scipy/special/orthogonal.pyRYsc`sO| |tj|ˆ|ƒ|dˆdtj|dˆ|ƒd|dS(Nii(R’RJ(R‚RV(R•(s7/tmp/pip-build-7oUkmx/scipy/scipy/special/orthogonal.pyRYs( R R“R%R]RRR’R RŽR( R‚R•RpR–RƒR„R…R†R‡((R•s7/tmp/pip-build-7oUkmx/scipy/scipy/special/orthogonal.pyR7Ýs!    4 c`s–tˆˆdˆdd|ƒ}|r*|Stdˆˆƒtˆdƒtdˆƒtˆdˆƒ}|j|ƒ‡‡fd†|jd<|S(sÿGegenbauer (ultraspherical) polynomial. Defined to be the solution of .. math:: (1 - x^2)\frac{d^2}{dx^2}C_n^{(\alpha)} - (2\alpha + 1)x\frac{d}{dx}C_n^{(\alpha)} + n(n + 2\alpha)C_n^{(\alpha)} = 0 for :math:`\alpha > -1/2`; :math:`C_n^{(\alpha)}` is a polynomial of degree :math:`n`. Parameters ---------- n : int Degree of the polynomial. monic : bool, optional If `True`, scale the leading coefficient to be 1. Default is `False`. Returns ------- C : orthopoly1d Gegenbauer polynomial. Notes ----- The polynomials :math:`C_n^{(\alpha)}` are orthogonal over :math:`[-1,1]` with weight function :math:`(1 - x^2)^{(\alpha - 1/2)}`. gà?Rlic`sttˆƒˆ|ƒS(N(RJRg(RV(R•R‚(s7/tmp/pip-build-7oUkmx/scipy/scipy/special/orthogonal.pyRY<sR\(RR˜RuRe(R‚R•Rltbasetfactor((R•R‚s7/tmp/pip-build-7oUkmx/scipy/scipy/special/orthogonal.pyRs! B cC`s¦t|ƒ}|dks$||kr3tdƒ‚ntjtjd|dddƒtd|ƒ}tj|ƒ}|jt|ƒ|r˜||tfS||fSdS(sOGauss-Chebyshev (first kind) quadrature. Computes the sample points and weights for Gauss-Chebyshev quadrature. The sample points are the roots of the n-th degree Chebyshev polynomial of the first kind, :math:`T_n(x)`. These sample points and weights correctly integrate polynomials of degree :math:`2n - 1` or less over the interval :math:`[-1, 1]` with weight function :math:`f(x) = 1/\sqrt{1 - x^2}`. Parameters ---------- n : int quadrature order mu : bool, optional If True, return the sum of the weights, optional. Returns ------- x : ndarray Sample points w : ndarray Weights mu : float Sum of the weights See Also -------- scipy.integrate.quadrature scipy.integrate.fixed_quad numpy.polynomial.chebyshev.chebgauss isn must be a positive integer.iiiþÿÿÿN(R R“R]R RRt empty_liketfill(R‚RpR–RVR((s7/tmp/pip-build-7oUkmx/scipy/scipy/special/orthogonal.pyR%Ds 2 c `s¿ˆdkrtdƒ‚nd„}ˆdkrXtggtd|d |‡fd†ƒSˆ}t|dtƒ\}}}td }d ˆd}t|||||d |‡fd †ƒ} | S( sÓChebyshev polynomial of the first kind. Defined to be the solution of .. math:: (1 - x^2)\frac{d^2}{dx^2}T_n - x\frac{d}{dx}T_n + n^2T_n = 0; :math:`T_n` is a polynomial of degree :math:`n`. Parameters ---------- n : int Degree of the polynomial. monic : bool, optional If `True`, scale the leading coefficient to be 1. Default is `False`. Returns ------- T : orthopoly1d Chebyshev polynomial of the first kind. Notes ----- The polynomials :math:`T_n` are orthogonal over :math:`[-1, 1]` with weight function :math:`(1 - x^2)^{-1/2}`. See Also -------- chebyu : Chebyshev polynomial of the second kind. isn must be nonnegative.cS`sdtd||ƒS(Ngð?i(R(RV((s7/tmp/pip-build-7oUkmx/scipy/scipy/special/orthogonal.pyRY“sgð?iÿÿÿÿic`s tˆ|ƒS(N(RA(RV(R‚(s7/tmp/pip-build-7oUkmx/scipy/scipy/special/orthogonal.pyRY–sRpic`s tˆ|ƒS(N(RA(RV(R‚(s7/tmp/pip-build-7oUkmx/scipy/scipy/special/orthogonal.pyRYœs(iÿÿÿÿi(iÿÿÿÿi(R“RQRR%R( R‚RlRkRžRVRRpRjRXRt((R‚s7/tmp/pip-build-7oUkmx/scipy/scipy/special/orthogonal.pyRos!    cC`s§t|ƒ}|dks$||kr3tdƒ‚ntj|ddƒt|d}tj|ƒ}ttj|ƒd|d}|r™||tdfS||fSdS(s&Gauss-Chebyshev (second kind) quadrature. Computes the sample points and weights for Gauss-Chebyshev quadrature. The sample points are the roots of the n-th degree Chebyshev polynomial of the second kind, :math:`U_n(x)`. These sample points and weights correctly integrate polynomials of degree :math:`2n - 1` or less over the interval :math:`[-1, 1]` with weight function :math:`f(x) = \sqrt{1 - x^2}`. Parameters ---------- n : int quadrature order mu : bool, optional If True, return the sum of the weights, optional. Returns ------- x : ndarray Sample points w : ndarray Weights mu : float Sum of the weights See Also -------- scipy.integrate.quadrature scipy.integrate.fixed_quad isn must be a positive integer.iiÿÿÿÿiN(R R“R]RRR R(R‚RpR–RïRVR((s7/tmp/pip-build-7oUkmx/scipy/scipy/special/orthogonal.pyR'£s !cC`s_t|ddd|ƒ}|r"|Sttƒdt|dƒt|dƒ}|j|ƒ|S(sãChebyshev polynomial of the second kind. Defined to be the solution of .. math:: (1 - x^2)\frac{d^2}{dx^2}U_n - 3x\frac{d}{dx}U_n + n(n + 2)U_n = 0; :math:`U_n` is a polynomial of degree :math:`n`. Parameters ---------- n : int Degree of the polynomial. monic : bool, optional If `True`, scale the leading coefficient to be 1. Default is `False`. Returns ------- U : orthopoly1d Chebyshev polynomial of the second kind. Notes ----- The polynomials :math:`U_n` are orthogonal over :math:`[-1, 1]` with weight function :math:`(1 - x^2)^{1/2}`. See Also -------- chebyt : Chebyshev polynomial of the first kind. gà?Rlg@igø?(RRRR˜Ru(R‚RlRôRõ((s7/tmp/pip-build-7oUkmx/scipy/scipy/special/orthogonal.pyRÍs ", cC`sWt|tƒ\}}}|d9}|d9}|d9}|rI|||fS||fSdS(s*Gauss-Chebyshev (first kind) quadrature. Computes the sample points and weights for Gauss-Chebyshev quadrature. The sample points are the roots of the n-th degree Chebyshev polynomial of the first kind, :math:`C_n(x)`. These sample points and weights correctly integrate polynomials of degree :math:`2n - 1` or less over the interval :math:`[-2, 2]` with weight function :math:`f(x) = 1/\sqrt{1 - (x/2)^2}`. Parameters ---------- n : int quadrature order mu : bool, optional If True, return the sum of the weights, optional. Returns ------- x : ndarray Sample points w : ndarray Weights mu : float Sum of the weights See Also -------- scipy.integrate.quadrature scipy.integrate.fixed_quad iN(R%R(R‚RpRVRR–((s7/tmp/pip-build-7oUkmx/scipy/scipy/special/orthogonal.pyR)ùs    c `sóˆdkrtdƒ‚nˆdkr4ˆd}nˆ}t|dtƒ\}}}ˆdkrqgg}}ndtˆdkd}d}t||||dd„d dd |ƒ}|sï|jd |d ƒƒ‡fd†|jd>> from scipy.special import legendre >>> legendre(3) poly1d([ 2.5, 0. , -1.5, 0. ]) isn must be nonnegative.iRpg@iRkcS`sdS(Ngð?((RV((s7/tmp/pip-build-7oUkmx/scipy/scipy/special/orthogonal.pyRY®sRUiÿÿÿÿRlRmc`s tˆ|ƒS(N(R@(RV(R‚(s7/tmp/pip-build-7oUkmx/scipy/scipy/special/orthogonal.pyRY¯s(iÿÿÿÿi(R“R#RR˜RQ( R‚RlRžRVRRƒRjRXRt((R‚s7/tmp/pip-build-7oUkmx/scipy/scipy/special/orthogonal.pyR|s&    .!cC`sKt|ƒ\}}|dd}|d}|r=||dfS||fSdS(s Gauss-Legendre (shifted) quadrature. Computes the sample points and weights for Gauss-Legendre quadrature. The sample points are the roots of the n-th degree shifted Legendre polynomial :math:`P^*_n(x)`. These sample points and weights correctly integrate polynomials of degree :math:`2n - 1` or less over the interval :math:`[0, 1]` with weight function :math:`f(x) = 1.0`. Parameters ---------- n : int quadrature order mu : bool, optional If True, return the sum of the weights, optional. Returns ------- x : ndarray Sample points w : ndarray Weights mu : float Sum of the weights See Also -------- scipy.integrate.quadrature scipy.integrate.fixed_quad iigð?N(R#(R‚RpRVR((s7/tmp/pip-build-7oUkmx/scipy/scipy/special/orthogonal.pyR9µs   c `sâˆdkrtdƒ‚nd„}ˆdkrXtggdd|d |‡fd†ƒStˆdtƒ\}}}ddˆd}tdˆdƒtˆdƒd}t|||||d dd |d ‡fd †ƒ}|S(sShifted Legendre polynomial. Defined as :math:`P^*_n(x) = P_n(2x - 1)` for :math:`P_n` the nth Legendre polynomial. Parameters ---------- n : int Degree of the polynomial. monic : bool, optional If `True`, scale the leading coefficient to be 1. Default is `False`. Returns ------- P : orthopoly1d Shifted Legendre polynomial. Notes ----- The polynomials :math:`P^*_n` are orthogonal over :math:`[0, 1]` with weight function 1. isn must be nonnegative.cS`s d|dS(Nggð?((RV((s7/tmp/pip-build-7oUkmx/scipy/scipy/special/orthogonal.pyRY÷sgð?ic`s tˆ|ƒS(N(RK(RV(R‚(s7/tmp/pip-build-7oUkmx/scipy/scipy/special/orthogonal.pyRYúsRpiRURlRmc`s tˆ|ƒS(N(RK(RV(R‚(s7/tmp/pip-build-7oUkmx/scipy/scipy/special/orthogonal.pyRYÿs(ii(ii(R“RQR9RR˜( R‚RlRkRVRRƒRjRXRt((R‚s7/tmp/pip-build-7oUkmx/scipy/scipy/special/orthogonal.pyRÛs   &!(RPRERNRJRARBRDRCRLRMR@RKRGRFRHRI(ct__doc__t __future__RRRtnumpyR]RRRRRRR R R R R RtscipyRt scipy.specialRtRR’R R˜Rt _polyfunst _rootfuns_mapt _evalfunsRctkeyst__all__R^RQRŽRyR-RR?R!R1RR/RR3R¦R«R®R·RíRóR¡RR5RR7RR%RR'RR)RR+RR;RR=R R#RR9RRORPRERNRJRARBRDRCRLRMR@RKRGRFRHRItglobalst _modattrstitemstnewfuntoldfuntappend(((s7/tmp/pip-build-7oUkmx/scipy/scipy/special/orthogonal.pytHsž R           $( - @ : / 8 9 @ " 0 M $ # ( ' n / 9 2 A 5 7 0 + 4 * , ( 9 ( : # % ) # , 9 & . j