^Mh!dZddlZddlmZmZmZmZmZmZm Z m Z m Z m Z m Z ddlmZddlmZddlmZddlmZejZgd Zd d d d ddddddddddddZeeezZGddejZdZdDdZdDdZ dDd Z!dDd!Z"dDd"Z#dDd#Z$dDd$Z%dDd%Z&dDd&Z'dEd(Z(d)Z)d*Z*d+Z+d,Z,dEd-Z-d.Z.dDd/Z/dDd0Z0dDd1Z1dDd2Z2dDd3Z3dDd4Z4dDd5Z5dDd6Z6dDd7Z7dDd8Z8dDd9Z9dDd:Z:dDd;Z;dDd<ZZ>dDd?Z?dDd@Z@dDdAZAdDdBZBdDdCZCeDZEeFD]%\ZGZHeEeGeEeH<eIeH&dS)FaI A collection of functions to find the weights and abscissas for Gaussian Quadrature. These calculations are done by finding the eigenvalues of a tridiagonal matrix whose entries are dependent on the coefficients in the recursion formula for the orthogonal polynomials with the corresponding weighting function over the interval. Many recursion relations for orthogonal polynomials are given: .. math:: a1n f_{n+1} (x) = (a2n + a3n x ) f_n (x) - a4n f_{n-1} (x) The recursion relation of interest is .. math:: P_{n+1} (x) = (x - A_n) P_n (x) - B_n P_{n-1} (x) where :math:`P` has a different normalization than :math:`f`. The coefficients can be found as: .. math:: A_n = -a2n / a3n \qquad B_n = ( a4n / a3n \sqrt{h_n-1 / h_n})^2 where .. math:: h_n = \int_a^b w(x) f_n(x)^2 assume: .. math:: P_0 (x) = 1 \qquad P_{-1} (x) == 0 For the mathematical background, see [golub.welsch-1969-mathcomp]_ and [abramowitz.stegun-1965]_. References ---------- .. [golub.welsch-1969-mathcomp] Golub, Gene H, and John H Welsch. 1969. Calculation of Gauss Quadrature Rules. *Mathematics of Computation* 23, 221-230+s1--s10. .. [abramowitz.stegun-1965] Abramowitz, Milton, and Irene A Stegun. (1965) *Handbook of Mathematical Functions: with Formulas, Graphs, and Mathematical Tables*. Gaithersburg, MD: National Bureau of Standards. http://www.math.sfu.ca/~cbm/aands/ .. [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`. N) expinfpisqrtfloorsincosaroundhstackarccosarange)linalg)airy)_specfun)_ufuncs)legendrechebytchebyuchebycchebysjacobilaguerre genlaguerrehermite hermitenorm gegenbauer sh_legendre sh_chebyt sh_chebyu sh_jacobip_rootst_rootsu_rootsc_rootss_rootsj_rootsl_rootsla_rootsh_rootshe_rootscg_rootsps_rootsts_rootsus_rootsjs_roots)roots_legendre roots_chebyt roots_chebyu roots_chebyc roots_chebys roots_jacobiroots_laguerreroots_genlaguerre roots_hermiteroots_hermitenormroots_gegenbauerroots_sh_legendreroots_sh_chebytroots_sh_chebyuroots_sh_jacobic&eZdZ ddZdZdZdS) orthopoly1dN?Fc  fdttD} t|} |r | r| fd}| t|z } d}t jd} tj|| jt|zt j tt| |_ |_ ||_| |_||_dS)NcDg|]}||z SrE).0krootsweightswfuncs Y/var/www/html/test/jupyter/venv/lib/python3.11/site-packages/scipy/special/_orthogonal.py z(orthopoly1d.__init__..ws>111!eeE!Hoo5111c |z SNrE)xevfknns rK eval_funcz'orthopoly1d.__init__..eval_func~s3q66C<'rMrBT)r)rangelenrabsnppoly1d__init__coeffsfloatarraylistziprI weight_funclimitsnormcoef _eval_func)selfrHrIhnknrJramonicrS equiv_weightsmupolyrQrRs `` ` @@rKrZzorthopoly1d.__init__us!111111#CJJ//111 "XX  C (((((((c"ggBBy$''' 4uRyy!8999xS%G%G H HII    $rMc|jr/t|tjs||Stj||SrO)rc isinstancerXrY__call__)rdvs rKrmzorthopoly1d.__call__sI ? /:a#;#; /??1%% %9%%dA.. .rMcdkrdS|xjzc_|jr fd|_|xjzc_dS)NrBc |zSrOrE)rPrQps rKz$orthopoly1d._scale..sA rM)_coeffsrcrb)rdrqrQs `@rK_scalezorthopoly1d._scalesX 88 F  o  322222DO  rM)NrBrBNNFN)__name__ __module__ __qualname__rZrmrtrErMrKrArAssLBF59$$$$4/// rMrAcztj|d}tjd|f} ||dd| dddf<||| dddf<tj| d} ||| } ||| } | | | z z} ||dz | } tjtj| }tjtj| }| tj|| zd z z} | tj|| zd z z} d | | zz }|r"||ddd zdz }| | ddd z dz } ||| z z}|r| ||fS| |fS) a[x,w] = gen_roots_and_weights(n,an_func,sqrt_bn_func,mu) Returns the roots (x) of an nth order orthogonal polynomial, and weights (w) to use in appropriate Gaussian quadrature with that orthogonal polynomial. The polynomials have the recurrence relation P_n+1(x) = (x - A_n) P_n(x) - B_n P_n-1(x) an_func(n) should return A_n sqrt_bn_func(n) should return sqrt(B_n) mu ( = h_0 ) is the integral of the weight over the orthogonal interval d)dtyperNrT)overwrite_a_band@rB) rXr zerosreigvals_bandedlogrWrmaxminsum)nmu0an_funcbn_funcfdf symmetrizerirGcrPydyfmlog_fmlog_dyws rK_gen_roots_and_weightsrs !3A !QAgaennAadG WQZZAacFa$777A !QA AqB2IA 1Q3B VBF2JJ  F VBF2JJ  F"&&**,,-3 4 44B"&&**,,-3 4 44B rBwA 44R4[A  44R4[A quuwwA !Sy!t rMFc x t|}|dks||krtd|dks|dkrtd|dkr|dkrt||S||krt||dz|S||zdkr(d||zdzzt j|dz|dzz}nKt j||zdzt jdzt j |dz|dzz}| | zdkr fd }n fd } fd } fd } fd } t|||||| d|S)aOGauss-Jacobi quadrature. Compute the sample points and weights for Gauss-Jacobi quadrature. The sample points are the roots of the nth degree Jacobi polynomial, :math:`P^{\alpha, \beta}_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:`w(x) = (1 - x)^{\alpha} (1 + x)^{\beta}`. See 22.2.1 in [AS]_ for details. Parameters ---------- n : int quadrature order alpha : float alpha must be > -1 beta : float beta 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.fixed_quad References ---------- .. [AS] Milton Abramowitz and Irene A. Stegun, eds. Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. New York: Dover, 1972. rn must be a positive integer.r~z'alpha and beta must be greater than -1.?ir}cPtj|dkz dzzz dS)Nrr{rrXwhererGabs rKrzroots_jacobi..an_funcs,8AFQUq1uqy$93?? ?rMctj|dkz dzzz zzz d|zzzd|zzzdzzz S)Nrr{r}rrs rKrzroots_jacobi..an_funcsh8QQ1q519%QQC!GaK!Oa! a!8K#LM rMc dd|zzzz tj|z|zzd|zzzdzz ztj|dkdtj||zzzd|zzzdz z zS)Nr}r{rrB)rXrrrs rKrzroots_jacobi..bn_funcs 37Q;? #gq1uQ'1q519q=1+<=>> ?hqAvsBGAQOsQw{QQR?R,S$T$TUU V rMc2tj||SrOr eval_jacobirrPrrs rKrzroots_jacobi..f s"1aA...rMcbd|zzdzztj|dz dzdz|zS)Nrrrrs rKrzroots_jacobi..df"s=a!eai!m$w':1q5!a%QPQ'R'RRRrMF) int ValueErrorr1r;rbetarXrrbetalnr) ralpharrimrrrrrrrs @@rKr6r6sT AA1uuQ8999 {{dbjjBCCC || a$$$ }}59b111  E$JqL!GLq$q&$A$AAfedlQ&"&++5~eAgtAv66788 A A1u|| @ @ @ @ @ @ @            //////SSSSSS !!S'7Ar5" M MMrMc Jdkrtdfd}dkr!tggdd|d|tjSt d\}}}zdz}d |zd z|zz t zd zz} | t zdzt d zz t |zz z} t d z|zd zz t d zz t |zz } t||| | |d|fd } | S) aJacobi 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`. References ---------- .. [AS] Milton Abramowitz and Irene A. Stegun, eds. Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. New York: Dover, 1972. Examples -------- The Jacobi polynomials satisfy the recurrence relation: .. math:: P_n^{(\alpha, \beta-1)}(x) - P_n^{(\alpha-1, \beta)}(x) = P_{n-1}^{(\alpha, \beta)}(x) This can be verified, for example, for :math:`\alpha = \beta = 2` and :math:`n = 1` over the interval :math:`[-1, 1]`: >>> import numpy as np >>> from scipy.special import jacobi >>> x = np.arange(-1.0, 1.0, 0.01) >>> np.allclose(jacobi(0, 2, 2)(x), ... jacobi(1, 2, 1)(x) - jacobi(1, 1, 2)(x)) True Plot of the Jacobi polynomial :math:`P_5^{(\alpha, -0.5)}` for different values of :math:`\alpha`: >>> import matplotlib.pyplot as plt >>> x = np.arange(-1.0, 1.0, 0.01) >>> fig, ax = plt.subplots() >>> ax.set_ylim(-2.0, 2.0) >>> ax.set_title(r'Jacobi polynomials $P_5^{(\alpha, -0.5)}$') >>> for alpha in np.arange(0, 4, 1): ... ax.plot(x, jacobi(5, alpha, -0.5)(x), label=rf'$\alpha={alpha}$') >>> plt.legend(loc='best') >>> plt.show() rn must be nonnegative.c&d|z zd|zzzSNrrE)rPrrs rKrJzjacobi..wfuncusA%1q5T/11rMrBr~rrSTrir{rr}c2tj|SrOr)rPrrrs rKrrzjacobi..sg1!UD!DDrM)rrArX ones_liker6_gam) rrrrgrJrPrriab1rerfrqs ``` rKrr'spV 1uu1222222222Avv2r3UGU%'\333 3Audt444HAq" $, C C1q53; $q5y1}"5"5 5B$q4x#~  a!e ,tAG}} < -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.fixed_quad References ---------- .. [AS] Milton Abramowitz and Irene A. Stegun, eds. Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. New York: Dover, 1972. r~rz>(p - q) must be greater than -1, and q must be greater than 0.rTr{r})rr6) rp1q1rimessagerPrrscales rKr?r?sT 2"}}aR!!!1beRT400GAq! Q! A GEJAJA !Qw!t rMc dkrtdfd}dkr!tggdd|d|tjS}t |\}}t dzt zzt zzt zz dzz}|dzzt dzzdzzz}d} t|||| |d |fd  } | S) aShifted 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}`. rrc,d|z z z|dz zzSNrBrE)rPrqqs rKrJzsh_jacobi..wfuncs#aQU#aAGn44rMrBrrrr{rrc2tj|SrO)reval_sh_jacobi)rPrrqrs rKrrzsh_jacobi..s)?1a)K)KrMrJrargrS)rrArXrr?r) rrqrrgrJn1rPrrerfpps ``` rKr!r!s<H 1uu1222555555Avv2r3UGU%'\333 3 B 2q! $ $DAq a!etAE{{ "T!a%[[ 04A A 3F3F FB1q519a!eai!+ ,,B B Q2rvUKKKKKK M M MB IrMc t|}|dks||krtddkrtdtjdz}|dkr:t jdzgd}t j|gd}|r|||fS||fSfd}fd}fd } fd } t ||||| | d |S) a<Gauss-generalized Laguerre quadrature. Compute the sample points and weights for Gauss-generalized Laguerre quadrature. The sample points are the roots of the nth 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:`w(x) = x^{\alpha} e^{-x}`. See 22.3.9 in [AS]_ for details. 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.fixed_quad References ---------- .. [AS] Milton Abramowitz and Irene A. Stegun, eds. Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. New York: Dover, 1972. rrr~zalpha must be greater than -1.rBrycd|zzdzSNr{rrErGrs rKrz"roots_genlaguerre..an_func0s1uu}q  rMc:tj||zz SrOrXrrs rKrz"roots_genlaguerre..bn_func2sQY((((rMc0tj||SrOreval_genlaguerrerrPrs rKrzroots_genlaguerre..f4s'5!444rMc||tj||z|ztj|dz |zz |z Srrrs rKrzroots_genlaguerre..df6sNG,Qq999u9 8Qq I IIJMNO OrMF)rrrgammarXr]r) rrrirrrPrrrrrs ` rKr8r8s3P AA1uuQ8999 rzz9::: - " "CAvv HeCi[# & & HcUC   a9 a4K!!!!!)))))55555OOOOO "!S'7Ar5" M MMrMc dkrtddkrtddkrdz}n}t|\}}fd}dkrgg}}tzdztdzz }dztdzz }t|||||dtf|fd} | S)aO Generalized (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. See Also -------- laguerre : Laguerre polynomial. hyp1f1 : confluent hypergeometric function 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`. References ---------- .. [AS] Milton Abramowitz and Irene A. Stegun, eds. Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. New York: Dover, 1972. Examples -------- The generalized Laguerre polynomials are closely related to the confluent hypergeometric function :math:`{}_1F_1`: .. math:: L_n^{(\alpha)} = \binom{n + \alpha}{n} {}_1F_1(-n, \alpha +1, x) This can be verified, for example, for :math:`n = \alpha = 3` over the interval :math:`[-1, 1]`: >>> import numpy as np >>> from scipy.special import binom >>> from scipy.special import genlaguerre >>> from scipy.special import hyp1f1 >>> x = np.arange(-1.0, 1.0, 0.01) >>> np.allclose(genlaguerre(3, 3)(x), binom(6, 3) * hyp1f1(-3, 4, x)) True This is the plot of the generalized Laguerre polynomials :math:`L_3^{(\alpha)}` for some values of :math:`\alpha`: >>> import matplotlib.pyplot as plt >>> x = np.arange(-4.0, 12.0, 0.01) >>> fig, ax = plt.subplots() >>> ax.set_ylim(-5.0, 10.0) >>> ax.set_title(r'Generalized Laguerre polynomials $L_3^{\alpha}$') >>> for alpha in np.arange(0, 5): ... ax.plot(x, genlaguerre(3, alpha)(x), label=rf'$L_3^{(alpha)}$') >>> plt.legend(loc='best') >>> plt.show() r~zalpha must be > -1rrrc0t| |zzSrOr)rPrs rKrJzgenlaguerre..wfuncsA2wwe##rMc0tj|SrOrrPrrs rKrrzgenlaguerre..sg6q%CCrM)rr8rrAr) rrrgrrPrrJrerfrqs `` rKrr<sb {{-...1uu1222Avv U  R ' 'DAq$$$$$Avv21 a%i!m  tAE{{ *B q4A;; BAq"b%!S5CCCCC E EA HrMc&t|d|S)aGauss-Laguerre quadrature. Compute the sample points and weights for Gauss-Laguerre quadrature. The sample points are the roots of the nth 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:`w(x) = e^{-x}`. See 22.2.13 in [AS]_ for details. 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.fixed_quad numpy.polynomial.laguerre.laggauss References ---------- .. [AS] Milton Abramowitz and Irene A. Stegun, eds. Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. New York: Dover, 1972. rr)r8)rris rKr7r7sL Q + + ++rMc dkrtddkrdz}n}t|\}}dkrgg}}d}dztdzz }t||||ddtf|fd}|S)aRLaguerre 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. See Also -------- genlaguerre : Generalized (associated) Laguerre polynomial. Notes ----- The polynomials :math:`L_n` are orthogonal over :math:`[0, \infty)` with weight function :math:`e^{-x}`. References ---------- .. [AS] Milton Abramowitz and Irene A. Stegun, eds. Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. New York: Dover, 1972. Examples -------- The Laguerre polynomials :math:`L_n` are the special case :math:`\alpha = 0` of the generalized Laguerre polynomials :math:`L_n^{(\alpha)}`. Let's verify it on the interval :math:`[-1, 1]`: >>> import numpy as np >>> from scipy.special import genlaguerre >>> from scipy.special import laguerre >>> x = np.arange(-1.0, 1.0, 0.01) >>> np.allclose(genlaguerre(3, 0)(x), laguerre(3)(x)) True The polynomials :math:`L_n` also satisfy the recurrence relation: .. math:: (n + 1)L_{n+1}(x) = (2n +1 -x)L_n(x) - nL_{n-1}(x) This can be easily checked on :math:`[0, 1]` for :math:`n = 3`: >>> x = np.arange(0.0, 1.0, 0.01) >>> np.allclose(4 * laguerre(4)(x), ... (7 - x) * laguerre(3)(x) - 3 * laguerre(2)(x)) True This is the plot of the first few Laguerre polynomials :math:`L_n`: >>> import matplotlib.pyplot as plt >>> x = np.arange(-1.0, 5.0, 0.01) >>> fig, ax = plt.subplots() >>> ax.set_ylim(-5.0, 5.0) >>> ax.set_title(r'Laguerre polynomials $L_n$') >>> for n in np.arange(0, 5): ... ax.plot(x, laguerre(n)(x), label=rf'$L_{n}$') >>> plt.legend(loc='best') >>> plt.show() rrrrBr~c"t| SrOrrPs rKrrzlaguerre..&sCGGrMc.tj|SrO)r eval_laguerrerPrs rKrrzlaguerre..'sg3Aq99rM)rr7rrArrrgrrPrrerfrqs` rKrrsZ 1uu1222Avv U  "  DAqAvv21 B q4A;; BAq"b"3"3aXu9999 ; ;A HrMc .t|}|dks||krtdtjtj}|dkr+d}d}t j}d}t||||||d|St|\}} |r|| |fS|| fS)aGauss-Hermite (physicist's) quadrature. Compute the sample points and weights for Gauss-Hermite quadrature. The sample points are the roots of the nth 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:`w(x) = e^{-x^2}`. See 22.2.14 in [AS]_ for details. 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.fixed_quad numpy.polynomial.hermite.hermgauss roots_hermitenorm 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. 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`. .. [AS] Milton Abramowitz and Irene A. Stegun, eds. Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. New York: Dover, 1972. rrc d|zSNrrErGs rKrzroots_hermite..an_funcq 7NrMc0tj|dz SNr}rrs rKrzroots_hermite..bn_funcss71s7## #rMc>d|ztj|dz |zS)Nr}rr eval_hermiterrPs rKrzroots_hermite..dfvs"7W1!a%;;; ;rMT) rrrXrrrrr_roots_hermite_asy rrirrrrrrnodesrIs rKr9r9-s| AA1uuQ8999 '"%..CCxx    $ $ $   < < <%agw2tRPPP+A..w  "'3& &'> !rMc|dzdz }dt|dz zd|zz dztzdt|dz zd|zzdzz fd}d}dtz}t|D]}|||||z z }|S)aHelper 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 r{r@r}@c.|t|z z SrO)r)rPrs rKrz_compute_tauk..fs3q66zA~rMc&dt|z Sr)r rs rKrz_compute_tauk..dfsSVV|rM)rrrU) rrGmaxitrrrxiirs @rK _compute_taukrs4 A A U1S5\\ CE !C '+s53<z dz }d|dddfzd|d ddfzzd?|d"ddfzz d@|dddfzzd$|zzd%z }dA|d'ddfzdB|d)ddfzzdC|d+ddfzz dD|dddfzz dE|dddfzz dFz d/z }d0|d1ddfzd2|d3ddfzz dG|d5ddfzzdH|dddfzz dI|d ddfzzdJ|d"ddfzzdK|dddfzzdL|zzd;z } t |dz|z\}!}"}#}$dt t z|dMzz|z}%|td+dNd+dz}&||z}'||z|&dOddf|z|zz|&dPddf|z|zz|dzz }(||z|&dOddf|z|zz|&dPddf|z|zz|&dddf|z|zz|&dddf|z|zz|d+zz })| |z|&dOddf|z|zz |dzz }*| |z|&dOddf| z|zz|&dPddf| z|zz|&dddf|z|zz |d"zz }+| |z|&dOddf| z|zz|&dPddf| z|zz|&dddf| z|zz|&dddf| z|zz|&dddf|z|zz |d)zz },|%|!|'|(|dzz z|)|dQzz zz|"|*|+|dzz z|,|dQzz zz|dRzz zz}-t dt z|dSzz|z }.||z|&dOddf|z|zz |z }/||z|&dOddf|z|zz|&dPddf|z|zz|&dddf|z|zz |dzz }0||z|&dOddf|z|zz|&dPddf|z|zz|&dddf|z|zz|&dddf|z|zz|&dddf|z| zz |d zz }1||z}2| |z|&dOddf| z|zz|&dPddf|z|zz|dzz }3| |z|&dOddf| z|zz|&dPddf| z|zz|&dddf| z|zz|&dddf|z|zz|d+zz }4|.|!|/|0|dzz z|1|dQzz zz|dzz |"|2|3|dzz z|4|dQzz zzzz}5|-|5fS)TaAsymptotic 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] https://dlmf.nist.gov/12.10#vii r}rBrrgUUUUUU?r{rg?g98c?gHxi?gd?g_cJ6?g¿grqGgk~X X¿g@9SsMԿg:(,a(rrNg@g8@g"rg o@g b@g@g g@@g@rg؍AgЦAgPAg@@ gAg0Ag8 JAgH;Ag \hAgAg夓AgdjA gm6A g AgP/Agf!Bgݱ;BghdBgAg.@gpt@ga@g`@g Aggj AgAgH$iAgAig.Ag(AgӍAg5e Bgd{>Bl+2VgUUUUUU?rrrgUUUUUU?r)rr r reshaperrr)6rthetastctrietazetaphia0a1a2a3a4a5b0b1b2b3b4b5ctpu0u1u2u3u4u5v0v1v2v3v4v5AiAipBiBipPphipA0A1A2B0B1B2UPdC0C1C2D0D1D2Uds6 rK_pbcfrOs D UB UB QB e)c"fRi C WS[g & &D 52q5=d #C B B B B B B B B B B B B r ""6** *C B c!AAA#h,R 4 'B s1QQQ3x-%AaaaC. (5 0F :B #ac( WS111X- -AaaaC0@ @ S111X   (  ,/7 8B #bd) hs1QQQ3x/ /(3qs82C C s1QQQ3x  "-c!AAA#h"6 79C DGQ RB SAAAY SAAAY!6 6c"QQQ$i9O O QqqqS ! "$0QqqqS$9 :B R%$qs)B,r/ !D111IbLO 3d1QQQ3il2o E 111IbLO #Qw 'B b54!!!9R<? " #dAg -B b54!!!9R<? "T!AAA#Yr\"_ 4tAaaaCy|B F G$PQ' QB b54!!!9R<? "T!AAA#Yr\"_ 4tAaaaCy|B F !AAA#Yr\"_ #AaaaCy|B / 026' :B R22s7 ?RCZ/ 0 BBGObSj0 1BM AB CA c"fW % +B b54!!!9R<? " #d *B b54!!!9R<? "T!AAA#Yr\"_ 4tAaaaCy|B F G$PQ' QB b54!!!9R<? "T!AAA#Yr\"_ 4tAaaaCy|B F !AAA#Yr\"_ #AaaaCy|B / 026' :B BB R%$qs)B,r/ !D111IbLO 3tQw >B R%$qs)B,r/ !D111IbLO 3d1QQQ3il2o E 111IbLO #Qw 'B rR"RW*_r"c'z12R']Bb2b#g:o2s7 234 5B b5LrMctd|zdz}||z }t|}t|D]d}t||\}}|td|zt |z|zz } || z}t t | dkrne|t|z} |dzdkrd| d<t| dz d|dzzz } | | fS)a+Newton iteration for polishing the asymptotic approximation to the zeros of the Hermite polynomials. Parameters ---------- n : int Quadrature order x_initial : ndarray Initial guesses for the roots maxit : int Maximal number of Newton iterations. The default 5 is sufficient, usually only one or two steps are needed. Returns ------- nodes : ndarray Quadrature nodes weights : ndarray Quadrature weights See Also -------- roots_hermite_asy r}rBg+=r{rrr) rr rUrOrrrWr r) r x_initialrritrruuddthetarPrs rK_newtonrVs6 c!eck  BBA 1IIE 5\\a2d3ii"ns5zz1B67 s6{{  e # # E $ SZZA1uzz! QTE c"a%i A a4KrMct|}t||\}}|dzdkr6t|ddd |g}t|ddd|g}n5t|ddd |g}t|ddd|g}|ttt |z z}||fS)a,Gauss-Hermite (physicist's) quadrature for large n. Computes the sample points and weights for Gauss-Hermite quadrature. The sample points are the roots of the nth 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}`. This method relies on asymptotic expansions which work best for n > 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`. r{rNr~)rrVr rrr)rr rrIs rKrrsV   BQ^^NE71uzzttt e,--'$$B$-122r!Bw/00'"Qr'*G455 tBxx#g,,&&G '>rMc <dkrtddkrdz}n}t|\}}d}dkrgg}}dztdzzttz}dz}t |||||t t f|fd}|S)a/Physicist's Hermite polynomial. Defined by .. math:: H_n(x) = (-1)^ne^{x^2}\frac{d^n}{dx^n}e^{-x^2}; :math:`H_n` is a polynomial of degree :math:`n`. Parameters ---------- n : int Degree of the polynomial. monic : bool, optional If `True`, scale the leading coefficient to be 1. Default is `False`. Returns ------- H : orthopoly1d Hermite polynomial. Notes ----- The polynomials :math:`H_n` are orthogonal over :math:`(-\infty, \infty)` with weight function :math:`e^{-x^2}`. Examples -------- >>> from scipy import special >>> import matplotlib.pyplot as plt >>> import numpy as np >>> p_monic = special.hermite(3, monic=True) >>> p_monic poly1d([ 1. , 0. , -1.5, 0. ]) >>> p_monic(1) -0.49999999999999983 >>> x = np.linspace(-3, 3, 400) >>> y = p_monic(x) >>> plt.plot(x, y) >>> plt.title("Monic Hermite polynomial of degree 3") >>> plt.xlabel("x") >>> plt.ylabel("H_3(x)") >>> plt.show() rrrc(t| |zSrOrrs rKrJzhermite..wfunc3sA26{{rMr{c.tj|SrOrrs rKrrzhermite..:sg21a88rM)rr9rrrrAr rrgrrPrrJrerfrqs ` rKrrsb 1uu1222Avv U    DAqAvv21 AQU d2hh &B ABAq"b%3$e8888 : :A HrMc |t|}|dks||krtdtjdtjz}|dkr+d}d}t j}d}t||||||d|St|\}} |td z}| td z} |r|| |fS|| fS) a-Gauss-Hermite (statistician's) quadrature. Compute the sample points and weights for Gauss-Hermite quadrature. The sample points are the roots of the nth 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:`w(x) = e^{-x^2/2}`. See 22.2.15 in [AS]_ for more details. 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.fixed_quad numpy.polynomial.hermite_e.hermegauss 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. References ---------- .. [AS] Milton Abramowitz and Irene A. Stegun, eds. Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. New York: Dover, 1972. rrr}rc d|zSrrErs rKrz"roots_hermitenorm..an_funcyrrMc*tj|SrOrrs rKrz"roots_hermitenorm..bn_func{s71:: rMc8|tj|dz |zSrreval_hermitenormrs rKrzroots_hermitenorm..df~sw/Aq999 9rMTr{) rrrXrrrrarrrs rKr:r:@sf AA1uuQ8999 '#be)  CCxx        $ : : :%agw2tRPPP+A..w a477  "'3& &'> !rMc 2dkrtddkrdz}n}t|\}}d}dkrgg}}tdtzt dzz}d}t |||||t t f|fd}|S) abNormalized (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}`. rrrc.t| |zdz Srrrs rKrJzhermitenorm..wfuncsA26C<   rMr{rBc.tj|SrOr`rs rKrrzhermitenorm..(@A(F(FrMr)rr:rrrrArr[s ` rKrrs> 1uu1222Avv U  R DAq!!!Avv21 a"fQU #B BAq"btSkFFFF H H HA HrMc t|}|dks||krtddkrtddkrt||SdkrOtjtjt jdzzt jdzz }nudz }tjgd }|d }tdt|D]}||z||z}|tjtjz z}d }fd } fd } fd} t|||| | | d|S)a(Gauss-Gegenbauer quadrature. Compute the sample points and weights for Gauss-Gegenbauer quadrature. The sample points are the roots of the nth 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:`w(x) = (1 - x^2)^{\alpha - 1/2}`. See 22.2.3 in [AS]_ for more details. 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.fixed_quad References ---------- .. [AS] Milton Abramowitz and Irene A. Stegun, eds. Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. New York: Dover, 1972. rrz alpha must be greater than -0.5.rrrB)gF90(+?g& {XgkʰDg3Ts?g?grBrc d|zSrrErs rKrz!roots_gegenbauer..an_func QwrMchtj||dzzdz zd|zz|zdz zz S)Nr{rrrrs rKrz!roots_gegenbauer..bn_funcs@wqAE MA-.!q5y/QYQR]2STUUUrMc0tj||SrOreval_gegenbauerrs rKrzroots_gegenbauer..fs&q%333rMc| |ztj||z|dzzdz tj|dz |zzd|dzz z Srrmrs rKrzroots_gegenbauer..df sc BFW,Qq99 91u9}q G$;AE5!$L$LL M aZ rMT) rrr2rXrrrrr]rUrVr) rrrirr inv_alphar[termrrrrs ` rKr;r;sP AA1uuQ8999 t||;<<< # Ar""" ||wru~~ eck : :: eai(() J >>>??Qi!S[[)) 1 1D /F4L0CCBGBEEM***VVVVV44444 "!S'7Ar4 L LLrMctjrdkrtdtdz dz |}|sdkr|St dzzt dzzt dzz t dzzz }||fd|jd<|S) aGegenbauer (ultraspherical) polynomial. Defined to be the solution of .. math:: (1 - x^2)\frac{d^2}{dx^2}C_n^{(\alpha)} - (2\alpha + 1)x\frac{d}{dx}C_n^{(\alpha)} + n(n + 2\alpha)C_n^{(\alpha)} = 0 for :math:`\alpha > -1/2`; :math:`C_n^{(\alpha)}` is a polynomial of degree :math:`n`. Parameters ---------- n : int Degree of the polynomial. alpha : float Parameter, must be greater than -0.5. monic : bool, optional If `True`, scale the leading coefficient to be 1. Default is `False`. Returns ------- C : orthopoly1d Gegenbauer polynomial. Notes ----- The polynomials :math:`C_n^{(\alpha)}` are orthogonal over :math:`[-1,1]` with weight function :math:`(1 - x^2)^{(\alpha - 1/2)}`. Examples -------- >>> import numpy as np >>> from scipy import special >>> import matplotlib.pyplot as plt We can initialize a variable ``p`` as a Gegenbauer polynomial using the `gegenbauer` function and evaluate at a point ``x = 1``. >>> p = special.gegenbauer(3, 0.5, monic=False) >>> p poly1d([ 2.5, 0. , -1.5, 0. ]) >>> p(1) 1.0 To evaluate ``p`` at various points ``x`` in the interval ``(-3, 3)``, simply pass an array ``x`` to ``p`` as follows: >>> x = np.linspace(-3, 3, 400) >>> y = p(x) We can then visualize ``x, y`` using `matplotlib.pyplot`. >>> fig, ax = plt.subplots() >>> ax.plot(x, y) >>> ax.set_title("Gegenbauer (ultraspherical) polynomial of degree 3") >>> ax.set_xlabel("x") >>> ax.set_ylabel("G_3(x)") >>> plt.show() rgz1`alpha` must be a finite number greater than -1/2rrgrr{cJtjt|SrO)rrnr\rs rKrrzgegenbauer..\s#G,CE!HHDI1-N-NrMrc)rXisfiniterrrrt__dict__)rrrgbasefactors`` rKrrsB ;u  N$LMMM !US[%#+U ; ; ;D Q 1U7Q;$us{"3"331U7mm"53;?334FKK#N#N#N#N#NDM, KrMct|}|dks||krtdtjt j| dz|dd|zz }t j|t|z }|r ||tfS||fS)a.Gauss-Chebyshev (first kind) quadrature. Computes the sample points and weights for Gauss-Chebyshev quadrature. The sample points are the roots of the nth 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:`w(x) = 1/\sqrt{1 - x^2}`. See 22.2.4 in [AS]_ for more details. 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.fixed_quad numpy.polynomial.chebyshev.chebgauss References ---------- .. [AS] Milton Abramowitz and Irene A. Stegun, eds. Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. New York: Dover, 1972. rrr{)rrr_sinpirXr full_liker)rrirrPrs rKr2r2esN AA1uuQ8999ry!aA..!A#677A Q1A !Rx!t rMc dkrtdd}dkrtggtd|d|fdS}t|d\}}}td z }d d z z}t|||||d|fd } | S) ao Chebyshev 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. See Also -------- chebyu : Chebyshev polynomial of the second kind. Notes ----- The polynomials :math:`T_n` are orthogonal over :math:`[-1, 1]` with weight function :math:`(1 - x^2)^{-1/2}`. References ---------- .. [AS] Milton Abramowitz and Irene A. Stegun, eds. Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. New York: Dover, 1972. Examples -------- Chebyshev polynomials of the first kind of order :math:`n` can be obtained as the determinant of specific :math:`n \times n` matrices. As an example we can check how the points obtained from the determinant of the following :math:`3 \times 3` matrix lay exactly on :math:`T_3`: >>> import numpy as np >>> import matplotlib.pyplot as plt >>> from scipy.linalg import det >>> from scipy.special import chebyt >>> x = np.arange(-1.0, 1.0, 0.01) >>> fig, ax = plt.subplots() >>> ax.set_ylim(-2.0, 2.0) >>> ax.set_title(r'Chebyshev polynomial $T_3$') >>> ax.plot(x, chebyt(3)(x), label=rf'$T_3$') >>> for p in np.arange(-1.0, 1.0, 0.1): ... ax.plot(p, ... det(np.array([[p, 1, 0], [1, 2*p, 1], [0, 1, 2*p]])), ... 'rx') >>> plt.legend(loc='best') >>> plt.show() They are also related to the Jacobi Polynomials :math:`P_n^{(-0.5, -0.5)}` through the relation: .. math:: P_n^{(-0.5, -0.5)}(x) = \frac{1}{4^n} \binom{2n}{n} T_n(x) Let's verify it for :math:`n = 3`: >>> from scipy.special import binom >>> from scipy.special import jacobi >>> x = np.arange(-1.0, 1.0, 0.01) >>> np.allclose(jacobi(3, -0.5, -0.5)(x), ... 1/64 * binom(6, 3) * chebyt(3)(x)) True We can plot the Chebyshev polynomials :math:`T_n` for some values of :math:`n`: >>> x = np.arange(-1.5, 1.5, 0.01) >>> fig, ax = plt.subplots() >>> ax.set_ylim(-4.0, 4.0) >>> ax.set_title(r'Chebyshev polynomials $T_n$') >>> for n in np.arange(2,5): ... ax.plot(x, chebyt(n)(x), label=rf'$T_n={n}$') >>> plt.legend(loc='best') >>> plt.show() rrc2dtd||zz z S)NrBrrrs rKrJzchebyt..wfuncsT!a!e)__$$rMrBrc.tj|SrOr eval_chebytrs rKrrzchebyt..sW%8A%>%>rMTrr{rc.tj|SrOrrs rKrrzchebyt..sg1!Q77rM)rrArr2) rrgrJrrPrrirerfrqs ` rKrrst 1uu1222%%%Avv2r2sE7E>>>>@@ @ BB4(((HAq" aB QUBAq"b%%7777 9 9A HrMc6t|}|dks||krtdtj|ddtz|dzz }tj|}ttj|dzz|dzz }|r ||tdz fS||fS)aGauss-Chebyshev (second kind) quadrature. Computes the sample points and weights for Gauss-Chebyshev quadrature. The sample points are the roots of the nth 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:`w(x) = \sqrt{1 - x^2}`. See 22.2.5 in [AS]_ for details. 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.fixed_quad References ---------- .. [AS] Milton Abramowitz and Irene A. Stegun, eds. Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. New York: Dover, 1972. rrrr~r{)rrrXr rr r)rrirrRrPrs rKr3r3sL AA1uuQ8999 !Qb AE*A q A RVAYY\QU#A !R!V|!t rMct|dd|}|r|Sttdz t|dzzt|dzz }|||S)a) Chebyshev 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. See Also -------- chebyt : Chebyshev polynomial of the first kind. Notes ----- The polynomials :math:`U_n` are orthogonal over :math:`[-1, 1]` with weight function :math:`(1 - x^2)^{1/2}`. References ---------- .. [AS] Milton Abramowitz and Irene A. Stegun, eds. Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. New York: Dover, 1972. Examples -------- Chebyshev polynomials of the second kind of order :math:`n` can be obtained as the determinant of specific :math:`n \times n` matrices. As an example we can check how the points obtained from the determinant of the following :math:`3 \times 3` matrix lay exactly on :math:`U_3`: >>> import numpy as np >>> import matplotlib.pyplot as plt >>> from scipy.linalg import det >>> from scipy.special import chebyu >>> x = np.arange(-1.0, 1.0, 0.01) >>> fig, ax = plt.subplots() >>> ax.set_ylim(-2.0, 2.0) >>> ax.set_title(r'Chebyshev polynomial $U_3$') >>> ax.plot(x, chebyu(3)(x), label=rf'$U_3$') >>> for p in np.arange(-1.0, 1.0, 0.1): ... ax.plot(p, ... det(np.array([[2*p, 1, 0], [1, 2*p, 1], [0, 1, 2*p]])), ... 'rx') >>> plt.legend(loc='best') >>> plt.show() They satisfy the recurrence relation: .. math:: U_{2n-1}(x) = 2 T_n(x)U_{n-1}(x) where the :math:`T_n` are the Chebyshev polynomial of the first kind. Let's verify it for :math:`n = 2`: >>> from scipy.special import chebyt >>> x = np.arange(-1.0, 1.0, 0.01) >>> np.allclose(chebyu(3)(x), 2 * chebyt(2)(x) * chebyu(1)(x)) True We can plot the Chebyshev polynomials :math:`U_n` for some values of :math:`n`: >>> x = np.arange(-1.0, 1.0, 0.01) >>> fig, ax = plt.subplots() >>> ax.set_ylim(-1.5, 1.5) >>> ax.set_title(r'Chebyshev polynomials $U_n$') >>> for n in np.arange(1,5): ... ax.plot(x, chebyu(n)(x), label=rf'$U_n={n}$') >>> plt.legend(loc='best') >>> plt.show() rrsr}r{?)rrrrrtrrgrwrxs rKrr7slr !S#U + + +D  "XX^d1q5kk )DSMM 9FKK KrMc^t|d\}}}|dz}|dz}|dz}|r|||fS||fS)a Gauss-Chebyshev (first kind) quadrature. Compute the sample points and weights for Gauss-Chebyshev quadrature. The sample points are the roots of the nth 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:`w(x) = 1 / \sqrt{1 - (x/2)^2}`. See 22.2.6 in [AS]_ for more details. 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.fixed_quad References ---------- .. [AS] Milton Abramowitz and Irene A. Stegun, eds. Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. New York: Dover, 1972. Tr{r2rrirPrrs rKr4r4QL1d##GAq!FAFAFA !Qw!t rMc @dkrtddkrdz}n}t|\}}dkrgg}}dtzdkdzz}d}t||||dd|}|s.|d |d z fd |jd <|S) agChebyshev polynomial of the first kind on :math:`[-2, 2]`. Defined as :math:`C_n(x) = 2T_n(x/2)`, where :math:`T_n` is the nth Chebychev polynomial of the first kind. 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 Chebyshev polynomial of the first kind on :math:`[-2, 2]`. See Also -------- chebyt : Chebyshev polynomial of the first kind. Notes ----- The polynomials :math:`C_n(x)` are orthogonal over :math:`[-2, 2]` with weight function :math:`1/\sqrt{1 - (x/2)^2}`. References ---------- .. [1] Abramowitz and Stegun, "Handbook of Mathematical Functions" Section 22. National Bureau of Standards, 1972. rrrrrBc8dtd||zdz z z S)NrBrrr~rs rKrrzchebyc..sC$q1q53;*?*?$?rMr{rJrargr}r{c.tj|SrO)r eval_chebycrs rKrrzchebyc..W-@A-F-FrMrc)rr4rrArtrvrs` rKrrsD 1uu1222Avv U    DAqAvv21 RAFa< B BAq"b??"% 1 1 1A G qqtt#F#F#F#F < HrMc^t|d\}}}|dz}|dz}|dz}|r|||fS||fS)aGauss-Chebyshev (second kind) quadrature. Compute the sample points and weights for Gauss-Chebyshev quadrature. The sample points are the roots of the nth degree Chebyshev polynomial of the second kind, :math:`S_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:`w(x) = \sqrt{1 - (x/2)^2}`. See 22.2.7 in [AS]_ for more details. 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.fixed_quad References ---------- .. [AS] Milton Abramowitz and Irene A. Stegun, eds. Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. New York: Dover, 1972. Tr{)r3rs rKr5r5rrMc 0dkrtddkrdz}n}t|\}}dkrgg}}t}d}t||||dd|}|s3dz|dz }||fd |jd <|S) afChebyshev polynomial of the second kind on :math:`[-2, 2]`. Defined as :math:`S_n(x) = U_n(x/2)` where :math:`U_n` is the nth Chebychev polynomial of the second kind. Parameters ---------- n : int Degree of the polynomial. monic : bool, optional If `True`, scale the leading coefficient to be 1. Default is `False`. Returns ------- S : orthopoly1d Chebyshev polynomial of the second kind on :math:`[-2, 2]`. See Also -------- chebyu : Chebyshev polynomial of the second kind Notes ----- The polynomials :math:`S_n(x)` are orthogonal over :math:`[-2, 2]` with weight function :math:`\sqrt{1 - (x/2)}^2`. References ---------- .. [1] Abramowitz and Stegun, "Handbook of Mathematical Functions" Section 22. National Bureau of Standards, 1972. rrrrBc2td||zdz z S)Nrrr~rs rKrrzchebys..bsDQUS[$9$9rMrrr{c.tj|SrO)r eval_chebysrs rKrrzchebys..grrMrc)rr5rrArtrv) rrgrrPrrerfrqrxs ` rKrr3sD 1uu1222Avv U    DAqAvv21 B BAq"b99"% 1 1 1A Gc'QQqTT! #F#F#F#F < HrMcVt||}|ddzdz f|ddzS)aGauss-Chebyshev (first kind, shifted) quadrature. Compute the sample points and weights for Gauss-Chebyshev quadrature. The sample points are the roots of the nth degree shifted 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:`[0, 1]` with weight function :math:`w(x) = 1/\sqrt{x - x^2}`. See 22.2.8 in [AS]_ for more details. 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.fixed_quad References ---------- .. [AS] Milton Abramowitz and Irene A. Stegun, eds. Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. New York: Dover, 1972. rrr{Nr)rrixws rKr=r=ms6L a  B UQY!O 122 &&rMct|dd|}|r|S|dkr d|zdz }nd}|||S)aXShifted Chebyshev polynomial of the first kind. Defined as :math:`T^*_n(x) = T_n(2x - 1)` for :math:`T_n` the nth Chebyshev polynomial of the first kind. 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 Shifted Chebyshev polynomial of the first kind. Notes ----- The polynomials :math:`T^*_n` are orthogonal over :math:`[0, 1]` with weight function :math:`(x - x^2)^{-1/2}`. rrrsrrr}rBr!rtrs rKrrs[2 QS . . .D  1uuAKK KrMct|d\}}}|dzdz }tjdd}|||z z}|r|||fS||fS)aGauss-Chebyshev (second kind, shifted) quadrature. Computes the sample points and weights for Gauss-Chebyshev quadrature. The sample points are the roots of the nth degree shifted 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:`[0, 1]` with weight function :math:`w(x) = \sqrt{x - x^2}`. See 22.2.9 in [AS]_ for more details. 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.fixed_quad References ---------- .. [AS] Milton Abramowitz and Irene A. Stegun, eds. Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. New York: Dover, 1972. Trr{r)r3rr)rrirPrrm_uss rKr>r>sbL1d##GAq! Q! A <S ! !DMA !Tz!t rMcht|dd|}|r|Sd|z}|||S)aZShifted Chebyshev polynomial of the second kind. Defined as :math:`U^*_n(x) = U_n(2x - 1)` for :math:`U_n` the nth Chebyshev polynomial of the second kind. 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 Shifted Chebyshev polynomial of the second kind. Notes ----- The polynomials :math:`U^*_n` are orthogonal over :math:`[0, 1]` with weight function :math:`(x - x^2)^{1/2}`. r}rrsrrrs rKr r sG2 QS . . .D  TFKK KrMc t|}|dks||krtdd}d}d}tj}d}t ||||||d|S)a Gauss-Legendre quadrature. Compute the sample points and weights for Gauss-Legendre quadrature [GL]_. The sample points are the roots of the nth degree 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:`[-1, 1]` with weight function :math:`w(x) = 1`. See 2.2.10 in [AS]_ for more details. 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.fixed_quad numpy.polynomial.legendre.leggauss References ---------- .. [AS] Milton Abramowitz and Irene A. Stegun, eds. Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. New York: Dover, 1972. .. [GL] Gauss-Legendre quadrature, Wikipedia, https://en.wikipedia.org/wiki/Gauss%E2%80%93Legendre_quadrature Examples -------- >>> import numpy as np >>> from scipy.special import roots_legendre, eval_legendre >>> roots, weights = roots_legendre(9) ``roots`` holds the roots, and ``weights`` holds the weights for Gauss-Legendre quadrature. >>> roots array([-0.96816024, -0.83603111, -0.61337143, -0.32425342, 0. , 0.32425342, 0.61337143, 0.83603111, 0.96816024]) >>> weights array([0.08127439, 0.18064816, 0.2606107 , 0.31234708, 0.33023936, 0.31234708, 0.2606107 , 0.18064816, 0.08127439]) Verify that we have the roots by evaluating the degree 9 Legendre polynomial at ``roots``. All the values are approximately zero: >>> eval_legendre(9, roots) array([-8.88178420e-16, -2.22044605e-16, 1.11022302e-16, 1.11022302e-16, 0.00000000e+00, -5.55111512e-17, -1.94289029e-16, 1.38777878e-16, -8.32667268e-17]) Here we'll show how the above values can be used to estimate the integral from 1 to 2 of f(t) = t + 1/t with Gauss-Legendre quadrature [GL]_. First define the function and the integration limits. >>> def f(t): ... return t + 1/t ... >>> a = 1 >>> b = 2 We'll use ``integral(f(t), t=a, t=b)`` to denote the definite integral of f from t=a to t=b. The sample points in ``roots`` are from the interval [-1, 1], so we'll rewrite the integral with the simple change of variable:: x = 2/(b - a) * t - (a + b)/(b - a) with inverse:: t = (b - a)/2 * x + (a + b)/2 Then:: integral(f(t), a, b) = (b - a)/2 * integral(f((b-a)/2*x + (a+b)/2), x=-1, x=1) We can approximate the latter integral with the values returned by `roots_legendre`. Map the roots computed above from [-1, 1] to [a, b]. >>> t = (b - a)/2 * roots + (a + b)/2 Approximate the integral as the weighted sum of the function values. >>> (b - a)/2 * f(t).dot(weights) 2.1931471805599276 Compare that to the exact result, which is 3/2 + log(2): >>> 1.5 + np.log(2) 2.1931471805599454 rrr}c d|zSrrErs rKrzroots_legendre..an_func rjrMcH|tjdd|z|zdz z zS)NrBrrrrs rKrzroots_legendre..bn_func s(273!a%!)a-01111rMc| |ztj||z|tj|dz |zzd|dzz z S)Nrr{r eval_legendrers rKrzroots_legendre..df sRQ.q!444g+AE15556:;a1f*F FrMT)rrrrr)rrirrrrrrs rKr1r1 sX AA1uuQ8999 C222AFFF "!S'7Ar4 L LLrMc 4dkrtddkrdz}n}t|\}}dkrgg}}ddzdzz }tdzdztdzdzz dzz }t||||dd|fd }|S) a*Legendre polynomial. Defined to be the solution of .. math:: \frac{d}{dx}\left[(1 - x^2)\frac{d}{dx}P_n(x)\right] + n(n + 1)P_n(x) = 0; :math:`P_n(x)` 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 ------- P : orthopoly1d Legendre polynomial. Notes ----- The polynomials :math:`P_n` are orthogonal over :math:`[-1, 1]` with weight function 1. Examples -------- Generate the 3rd-order Legendre polynomial 1/2*(5x^3 + 0x^2 - 3x + 0): >>> from scipy.special import legendre >>> legendre(3) poly1d([ 2.5, 0. , -1.5, 0. ]) rrrr}r{cdSrrErs rKrrzlegendre.. s#rMrc.tj|SrOrrs rKrrzlegendre.. s(=a(C(CrMr)rr1rrArs` rKrr sL 1uu1222Avv U  "  DAqAvv21 A B a!eai4A;;> )CF 2BAq"b gCCCC E E EA HrMcVt|\}}|dzdz }|dz}|r||dfS||fS)aGauss-Legendre (shifted) quadrature. Compute the sample points and weights for Gauss-Legendre quadrature. The sample points are the roots of the nth 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:`w(x) = 1.0`. See 2.2.11 in [AS]_ for details. 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.fixed_quad References ---------- .. [AS] Milton Abramowitz and Irene A. Stegun, eds. Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. New York: Dover, 1972. rr{rB)r1)rrirPrs rKr<r< sIJ !  DAq Q! AFA !Sy!t rMc :dkrtdd}dkrtggdd|d|fdSt\}}ddzdzz }tdzdztdzdzz }t|||||d|fd  }|S) aShifted 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. 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