a a @sdZddlZddlmZmZmZmZmZmZm Z m Z m Z m Z m Z ddlmZddlmZddlmZddlmZejZddlmZgd Zd d d d ddddddddddddZgdZeeeeddgZGdddejZdd Z dnd"d#Z!dod$d%Z"dpd&d'Z#dqd(d)Z$drd*d+Z%dsd,d-Z&dtd.d/Z'dud0d1Z(dvd2d3Z)dwd5d6Z*d7d8Z+d9d:Z,d;d<Z-d=d>Z.dxd?d@Z/dAdBZ0dydCdDZ1dzdEdFZ2d{dGdHZ3d|dIdJZ4d}dKdLZ5d~dMdNZ6ddOdPZ7ddQdRZ8ddSdTZ9ddUdVZ:ddWdXZ;ddYdZZdd_d`Z?ddadbZ@ddcddZAddedfZBddgdhZCddidjZDddkdlZEejFZFddmlmGZGmHZHmIZImJZJmKZKmLZLmMZMmNZNmOZOmPZPmQZQmRZRmSZSmTZTmUZUmVZVeWZXeYD] \ZZZ[eXeZeXe[<e\e[qdS)aI 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)_ufuncs)specfun)legendrechebytchebyuchebycchebysjacobilaguerre genlaguerrehermite hermitenorm gegenbauer sh_legendre sh_chebyt sh_chebyu sh_jacobiZp_rootsZt_rootsZu_rootsZc_rootsZs_rootsZj_rootsZl_rootsZla_rootsZh_rootsZhe_rootsZcg_rootsZps_rootsZts_rootsZus_rootsZjs_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_jacobi) eval_legendre eval_chebyt eval_chebyu eval_chebyc eval_chebys eval_jacobi eval_laguerreeval_genlaguerre eval_hermiteeval_hermitenormeval_gegenbauereval_sh_legendreeval_sh_chebyteval_sh_chebyueval_sh_jacobipochbinomc@s&eZdZd ddZddZdd ZdS) orthopoly1dN?Fc sfddttD} t|} |rT|rD|fdd}| t|} d}tjdd} tj|| jt|t t t | |_ |_ ||_| |_||_dS)Ncs g|]}||qSrC).0k)rootsweightswfuncrCh/Users/vegardjervell/Documents/master/model/venv/lib/python3.9/site-packages/scipy/special/orthogonal.py ~sz(orthopoly1d.__init__..cs |SNrCx)evfknnrCrIz&orthopoly1d.__init__..rBT)r)rangelenrabsnppoly1d__init__ZcoeffsfloatarraylistziprGZ weight_funclimitsnormcoef _eval_func) selfrFrGhnknrHr]monic eval_funcZ equiv_weightsmuZpolyrC)rNrOrFrGrHrIrX|s$  zorthopoly1d.__init__cCs.|jrt|tjs||Stj||SdSrK)r_ isinstancerVrW__call__)r`vrCrCrIrgs zorthopoly1d.__call__csFdkr dS|j9_|jr4fdd|_|j9_dS)NrBcs |SrKrCrLrNprCrIrPrQz$orthopoly1d._scale..)Z_coeffsr_r^)r`rjrCrirI_scaleszorthopoly1d._scale)NrBrBNNFN)__name__ __module__ __qualname__rXrgrkrCrCrCrIrAzs  rAcCstj|dd}td|f} ||dd| dddf<||| dddf<tj| dd} ||| } ||| } | | | 8} ||d| } | t| } | t| } d | | }|r||ddd d}| | ddd d} |||9}|r| ||fS| |fSdS) 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)ZdtyperNrT)Zoverwrite_a_bandrB)rVr Zzerosr Zeigvals_bandedrUmaxsum)nmu0an_funcbn_funcfdfZ symmetrizererEcrMyZdyZfmwrCrCrI_gen_roots_and_weightss&     r}Fc st|}|dks||kr td|dks0|dkr8td|dkrR|dkrRt||S||krjt||d|Sd||dt|d|d}||dkrfdd }nfd d }fd d }fd d }fd d } t|||||| d|S)anGauss-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.quadrature 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.rqz'alpha and beta must be greater than -1.?@cs"t|dkddS)NrrprrVwhererEabrCrIrP rQzroots_jacobi..csRt|dkdd|d|dS)NrrprrrrrCrIrPs2c svdd|t||d|dt|dkdt||d|dS)NrrprrB)rVrrrrrCrIrPs<6cst||SrKcephesr5rtrMrrCrIrPrQcs0d|dt|ddd|S)NrrrrrrCrIrPsF)int ValueErrorr!r+rbetar}) rtalpharremrurvrwrxryrCrrIr&s&+ $ r&c sdkrtdfdd}dkr@tggdd|d|tjdStdd \}}}d}d |d |td } | tdtd t|9} td |d td t|} t||| | |d|fd d} | 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`. rn must be nonnegative.csd|d|SNrrCrL)rrrCrIrPDrQzjacobi..rBrqrrdTrerprrcst|SrK)r5rLrrrtrCrIrPOrQ)rrArV ones_liker&_gam) rtrrrcrHrMr|reZab1rarbrjrCrrIrs & $,0rcCsv||dks|dkrtdt||||dd\}}}|dd}d|}||}||}|rj|||fS||fSdS) apGauss-Jacobi (shifted) quadrature. Compute the sample points and weights for Gauss-Jacobi (shifted) quadrature. The sample points are the roots of the nth 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:`w(x) = (1 - x)^{p-q} x^{q-1}`. See 22.2.2 in [AS]_ for details. 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 References ---------- .. [AS] Milton Abramowitz and Irene A. Stegun, eds. Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. New York: Dover, 1972. rqrz>(p - q) must be greater than -1, and q must be greater than 0.rTrprN)rr&)rtp1Zq1rerMr|rZscalerCrCrIr/Us+  r/c sdkrtdfdd}dkr@tggdd|d|tjdS}t|\}}tdtttd}|d td d }d} t|||| |d |fd dd } | 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}`. rrcsd||dSNrBrCrL)rjqrCrIrPrQzsh_jacobi..rBrrrrprrcst|SrK)r>rLrtrjrrCrIrPrQrHr]rcrd)rrArVrr/r) rtrjrrcrHn1rMr|rarbpprCrrIr s $8$r c st|}|dks||kr tddkr0tdtd}|dkr|tdgd}t|gd}|rt|||fS||fSfdd}fd d}fd d} fd d} 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.quadrature 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~rqzalpha must be greater than -1.rBrocsd|dSNrprrCrrrCrIrPrQz#roots_genlaguerre..cst|| SrKrVrrrrCrIrPrQcst||SrKrr7rrrCrIrPrQcs0|t|||t|d||SrrrrrCrIrPsF)rrrgammarVrZr}) rtrrerrurMr|rvrwrxryrCrrIr(s")     r(c sdkrtddkr tddkr2d}n}t|\}}fdd}dkrbgg}}tdtd}dtd}t|||||dtf|fdd} | S) aGeneralized (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. rqzalpha must be > -1rrrcst| |SrKrrLrrCrIrP9rQzgenlaguerre..cs t|SrK)r7rLrrtrCrIrP?rQ)rr(rrAr) rtrrcrrMr|rHrarbrjrCrrIrs"*    rcCst|d|dS)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.quadrature 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)r()rtrerCrCrIr'Es'r'c sdkrtddkr"d}n}t|\}}dkrDgg}}d}dtd}t||||dddtf|fdd}|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. Notes ----- The polynomials :math:`L_n` are orthogonal over :math:`[0, \infty)` with weight function :math:`e^{-x}`. rrrrBrqcSs t| SrKrrLrCrCrIrPrQzlaguerre..cs t|SrK)r6rLrtrCrIrPrQ)rr'rrArrtrcrrMr|rarbrjrCrrIros    rc Cst|}|dks||kr tdttj}|dkrhdd}dd}tj}dd}t||||||d|St|\}} |r|| |fS|| fSd S) a=Gauss-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 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`. .. [AS] Milton Abramowitz and Irene A. Stegun, eds. Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. New York: Dover, 1972. rr~cSsd|SNrrCrrCrCrIrPrQzroots_hermite..cSst|dSNrrrrCrCrIrPrQcSsd|t|d|S)Nrr)rr8rrCrCrIrPrQTN) rrrVrrrr8r}_roots_hermite_asy rtrerrurvrwrxrynodesrGrCrCrIr)s?   r)cs|dd}dt|dd|dtdt|dd|dfdd}dd}dt}t|D]}|||||}ql|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 rpr@r@cs|t|SrK)rrLrzrCrIrP rQz_compute_tauk..cSs dt|Sr)rrLrCrCrIrPrQ)rrrS)rtrEmaxitrrxryxiirCrrI _compute_tauks <  rcCs~t||}td|d}|dd}dt|dd|d}||dd|ddd|ddd|d}|S) aeTricomi 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 rrprrrBrg@?)rrr)rtrEZtaukZsigkrnuxksqrCrCrI_initial_nodes_as   8rcCs|dd}dt|dd|d}t|ddddd}|d ||d d |d|d d d|d|dd|d|dd |dd|dd|dd|d}|S)abGatteschi initial guesses Computes an initial approximation to the square of the kth (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 rprrrrrNrqgd}d|dddfd|d ddfd?|d"ddfd@|dddfd$|d%}dA|d'ddfdB|d)ddfdC|d+ddfdD|dddfdE|dddfdFd/}d0|d1ddfd2|d3ddfdG|d5ddfdH|dddfdI|d ddfdJ|d"ddfdK|dddfdL|d;} t|d|\}!}"}#}$dtt|dM|}%|td+dNd+d}&||}'|||&dOddf|||&dPddf|||d}(|||&dOddf|||&dPddf|||&dddf|||&dddf|||d+})| ||&dOddf|| |d}*| ||&dOddf| ||&dPddf| ||&dddf|| |d"}+| ||&dOddf| ||&dPddf| ||&dddf| ||&dddf| ||&dddf|| |d)},|%|!|'|(|d|)|dQ|"|*|+|d|,|dQ|dR}-tdt|dS|}.|||&dOddf|| |}/|||&dOddf|||&dPddf|||&dddf|| |d}0|||&dOddf|||&dPddf|||&dddf|||&dddf|||&dddf||  |d }1||}2| ||&dOddf| ||&dPddf|||d}3| ||&dOddf| ||&dPddf| ||&dddf| ||&dddf|||d+}4|.|!|/|0|d|1|dQ|d|"|2|3|d|4|dQ}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 rrBrrgUUUUUU?rprg?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)rrr Zreshaperrr)6rtthetastctreetazetaphiZa0Za1Za2a3Za4Za5Zb0b1b2Zb3Zb4Zb5ZctpZu0u1u2u3Zu4Zu5Zv0v1v2Zv3Zv4Zv5ZAiZAipZBiZBipPZphipZA0A1A2B0ZB1ZB2UZPdZC0ZC1ZC2ZD0ZD1ZD2ZUdrCrCrI_pbcfs"  0\lb 0\lb@p*Z &&Z@p(rc Cstd|d}||}t|}t|D]J}t||\}}|td|t||} || }tt| dkr(qtq(|t|} |ddkrd| d<t| d d|d} | | 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 rrBg+=rprrr) rr rSrrrrrUrr) rtZ x_initialrretrruZudZdthetarMr|rCrCrI_newtons   rcCst|}t||\}}|ddkrRt|ddd |g}t|ddd|g}n.t|ddd |g}t|ddd|g}|ttt|9}||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`. rprNrq)rrr rrrs)rtrrrGrCrCrIr$s+ rc sdkrtddkr"d}n}t|\}}dd}dkrLgg}}dtdtt}d}t|||||t tf|fdd}|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() rrrcSst| |SrKrrLrCrCrIrPrQzhermite..rpcs t|SrK)r8rLrrCrIrPrQ)rr)rrrrAr rtrcrrMr|rHrarbrjrCrrIr]s1    rc Cst|}|dks||kr tdtdtj}|dkrldd}dd}tj}dd}t||||||d |St|\}} |td 9}| td 9} |r|| |fS|| fSd S) aLGauss-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 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 References ---------- .. [AS] Milton Abramowitz and Irene A. Stegun, eds. Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. New York: Dover, 1972. rr~rrcSsd|SrrCrrCrCrIrPrQz#roots_hermitenorm..cSs t|SrKrrrCrCrIrPrQcSs|t|d|Sr)rr9rrCrCrIrPrQTrpN) rrrVrrrr9r}rrrCrCrIr*s 4    r*c sdkrtddkr"d}n}t|\}}dd}dkrLgg}}tdttd}d}t|||||t tf|fddd }|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}`. rrrcSst| |dSrrrLrCrCrIrPrQzhermitenorm..rprBcs t|SrK)r9rLrrCrIrPrQr)rr*rrrrArrrCrrIrs    rc st|}|dks||kr tddkr2tdndkrDt||Sttjtdtd}dd}fd d}fd d}fd d}t||||||d |S) aGGauss-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.quadrature 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~gz alpha must be greater than -0.5.rrcSsd|SrrCrrCrCrIrPWrQz"roots_gegenbauer..cs2t||ddd||dS)NrprrrrrrCrIrPXscst||SrKrr:rrrCrIrPZrQcsF| |t|||ddt|d|d|dSrrrrrCrIrP[s  T) rrr"rVrrrrr}) rtrrerrurvrwrxryrCrrIr+!s)  (   r+csztdd|d}|r |Stdtdtdtd}||fdd|jd<|S)amGegenbauer (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 -------- >>> 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() rrcrpcstt|SrK)r:rYrLrrCrIrPrQzgegenbauer..r_)rrrk__dict__)rtrrcbasefactorrCrrIr`s@  rcCslt|}|dks||kr tdtt| d|dd|}t|t|}|r`||tfS||fSdS)aMGauss-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.quadrature 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. rr~rpN)rrrZ_sinpirVr Z full_liker)rtrerrMr|rCrCrIr"s(" r"c sdkrtddd}dkr>tggtd|d|fddS}t|dd \}}}td }d d }t|||||d|fd d} | S) aChebyshev 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. rrcSsdtd||S)NrBrrrLrCrCrIrPrQzchebyt..rBrcs t|SrKr1rLrrCrIrP rQTrrprcs t|SrKrrLrrCrIrPrQ)rrArr") rtrcrHrrMr|rerarbrjrCrrIrs!   rcCs|t|}|dks||kr tdt|ddt|d}t|}tt|d|d}|rp||tdfS||fSdS)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.quadrature 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~rrqrpN)rrrVr rrr)rtrerrrMr|rCrCrIr#s' r#cCsJt|dd|d}|r|Sttdt|dt|d}|||S)aChebyshev 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. rrrrp?)rrrrrkrtrcrrrCrCrIrIs "$ rcCsBt|d\}}}|d9}|d9}|d9}|r6|||fS||fSdS)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.quadrature 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. TrpNr"rtrerMr|rrCrCrIr$us' r$c sdkrtddkr"d}n}t|\}}dkrDgg}}dtdkd}d}t||||ddd|d }|s|d |d fd d|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]`. Notes ----- The polynomials :math:`C_n(x)` are orthogonal over :math:`[-2, 2]` with weight function :math:`1/\sqrt{1 - (x/2)^2}`. See Also -------- chebyt : Chebyshev polynomial of the first kind. References ---------- .. [1] Abramowitz and Stegun, "Handbook of Mathematical Functions" Section 22. National Bureau of Standards, 1972. rrrrrBcSsdtd||dS)NrBrrrrLrCrCrIrPrQzchebyc..rprHr]rcrrpcs t|SrK)r3rLrrCrIrPrQr_)rr$rrArkrrrCrrIrs$"    rcCsBt|d\}}}|d9}|d9}|d9}|r6|||fS||fSdS)a'Gauss-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.quadrature 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. TrpN)r#rrCrCrIr%s' r%c sdkrtddkr"d}n}t|\}}dkrDgg}}t}d}t||||ddd|d}|sd|d }||fd d|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]`. Notes ----- The polynomials :math:`S_n(x)` are orthogonal over :math:`[-2, 2]` with weight function :math:`\sqrt{1 - (x/2)}^2`. See Also -------- chebyu : Chebyshev polynomial of the second kind References ---------- .. [1] Abramowitz and Stegun, "Handbook of Mathematical Functions" Section 22. National Bureau of Standards, 1972. rrrrBcSstd||dS)NrrrrLrCrCrIrP?rQzchebys..rrrpcs t|SrK)r4rLrrCrIrPDrQr_)rr%rrArkr) rtrcrrMr|rarbrjrrCrrIrs&"     rcCs(t||}|dddf|ddS)a3Gauss-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.quadrature 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. rrrpNr)rtreZxwrCrCrIr-Js' r-cCs@t|dd|d}|r|S|dkr.d|d}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}`. rrrrrrrBr rkrrCrCrIrus rcCsNt|d\}}}|dd}tdd}|||9}|rB|||fS||fSdS)a4Gauss-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.quadrature 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. TrrprN)r#rr)rtrerMr|rZm_usrCrCrIr.s'    r.cCs.t|dd|d}|r|Sd|}|||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}`. rrrrrrrCrCrIrs  rc CsXt|}|dks||kr tdd}dd}dd}tj}dd}t||||||d|S) aGauss-Legendre quadrature. Compute the sample points and weights for Gauss-Legendre quadrature. 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) = 1.0`. 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.quadrature 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. rr~rcSsd|SrrCrrCrCrIrPrQz roots_legendre..cSs|tdd||dS)NrBrrrrrCrCrIrPrQcSs6| |t|||t|d|d|dS)Nrrp)rr0rrCrCrIrPs T)rrrr0r})rtrerrurvrwrxryrCrCrIr!s'r!c sdkrtddkr"d}n}t|\}}dkrDgg}}ddd}tddtddd}t||||ddd|fd dd }|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. ]) rrrrrpcSsdSrrCrLrCrCrIrPTrQzlegendre..rcs t|SrK)r0rLrrCrIrPUrQr)rr!rrArrCrrIr"s&   ( rcCs:t|\}}|dd}|d}|r.||dfS||fSdS)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.quadrature 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. rrprBN)r!)rtrerMr|rCrCrIr,[s &   r,c sdkrtddd}dkr>tggdd|d|fddSt\}}ddd}tdd td d}t|||||d|fd dd }|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. Notes ----- The polynomials :math:`P^*_n` are orthogonal over :math:`[0, 1]` with weight function 1. rrcSs d|dS)NrrBrCrLrCrCrIrPrQzsh_legendre..rBrcs t|SrKr;rLrrCrIrPrQrprcs t|SrKrrLrrCrIrPrQ)r]rcrd)rrAr,r)rtrcrHrMr|rarbrjrCrrIrs    r)r@r5r>r:r1r2r4r3r<r=r0r;r7r6r8r9)F)F)F)F)F)F)F)F)F)r)r)F)F)F)F)F)F)F)F)F)F)F)F)F)F)F)F)F)F)F)F)F)]__doc__ZnumpyrVrrrrrrrr r r r Zscipyr Z scipy.specialrrrrrrZ _polyfunsZ _rootfuns_mapZ _evalfunsr[keys__all__rWrAr}r&rr/r r(rr'rr)rrrrrrrrr*rr+rr"rr#rr$rr%rr-rr.rr!rr,rr?r@r5r>r:r1r2r4r3r<r=r0r;r7r6r8r9globalsZ _modattrsitemsZnewfunZoldfunappendrCrCrCrIsM4     ,- H : 7 8 A @ * 0 R $#('n /9 E J 5 ? O 3 4 3 , 1 9 1 : + % 1 # 4 9 . .H