(ph!SrSSKrSSKJrJrJrJrJrJrJ r J r J r J r J r SSKJr SSKJr SSKJr SSKJr \R*r/S QrS S S S SSSSSSSSSSSS.r\\"\R555-r"SS\R85rSrSBSjrSBSjr SBSjr!SBS jr"SBS!jr#SBS"jr$SBS#jr%SBS$jr&SBS%jr'SCS&jr(S'r)S(r*S)r+S*r,SCS+jr-S,r.SBS-jr/SBS.jr0SBS/jr1SBS0jr2SBS1jr3SBS2jr4SBS3jr5SBS4jr6SBS5jr7SBS6jr8SBS7jr9SBS8jr:SBS9jr;SBS:jrSBS=jr?SBS>jr@SBS?jrASBS@jrBSBSAjrC\D"5rE\R5HurGrH\E\G\E\H'\R\H5 M! g)DaI 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.\rSrSrSSjrSrSrSrg) orthopoly1dsNc ^ ^[[U55V s/sHn X)U"X5- PM n n [U5n U(a"Um T (a UmU U4SjnU [U5- n Sn[R "USS9n [R R X R[U5-5 [R"[[XU 555Ul XPl X`lXlXlgs sn f)Nc>T"U5T- $N)xevfknns L/var/www/html/venv/lib/python3.13/site-packages/scipy/special/_orthogonal.py eval_func'orthopoly1d.__init__..eval_func~sq6C<'?T)r)rangelenrabsnppoly1d__init__coeffsfloatarraylistzipweights weight_funclimitsnormcoef _eval_func)selfrootsr[hnknwfuncr]monicrKk equiv_weightsmupolyrHrIs @@rJrUorthopoly1d.__init__us$CJ/1/!eEHo5/ 1 "X C(c"gBByy$' 4uRy!89xxS%G HI    $-1sC9cUR(a0[U[R5(dURU5$[RR X5$rE)r_ isinstancerSrT__call__)r`vs rJrmorthopoly1d.__call__s= ??:a#;#;??1% %99%%d. .rMc^^TS:XagU=RT-slURmT(a UU4SjUlU=RT-slg)NrNc>T"U5T-$rErF)rGrHps rJ$orthopoly1d._scale..s A rM)_coeffsr_r^)r`rrrHs `@rJ_scaleorthopoly1d._scales< 8   oo 2DO  rM)r_r]r^r\r[)NrNrNNNFN)__name__ __module__ __qualname____firstlineno__rUrmrv__static_attributes__rFrMrJrArAssBF59$4/ rMrAc@[R"USS9n[R"SU45n U"USS5U SSS24'U"U5U SSS24'[R"U SS9n U"X 5n U"X 5n XU - -n U"US- U 5n [R "[R "U 55n[R "[R "U 55nU [R"UR5UR5-S - 5-n U [R"UR5UR5-S - 5-n S X-- nU(aUUSSS 2-S- nXSSS 2- S- n UUUR5- -nU(aU UU4$U U4$) 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@rN) rSr zerosreigvals_bandedlogrRrmaxminsum)nmu0an_funcbn_funcfdf symmetrizerhrfcrGydyfmlog_fmlog_dyws rJ_gen_roots_and_weightsrst !3A !QAaenAadG QZAacFa$7A !A AB2IA 1Q3B VVBFF2J F VVBFF2J F"&&&**,-3 44B"&&&**,-3 44B rwA 4R4[A  4R4[A quuwA !Sy!t rMc N^ ^ [U5nUS:dX:wa [S5eUS::dUS::a [S5eUS:XaUS:Xa [XC5$X:Xa[XAS-U5$X-S::a)SX-S--[R "US-US-5-nOP[ R"X-S-[ R"S5-[R"US-US-5-5nUm Um T T -S:XaU U 4S jnOU U 4S jnU U 4S jnU U 4S jnU U 4S jn [XEXgXSU5$)aGauss-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.rz'alpha and beta must be greater than -1.?ircR>[R"US:HTT- ST-T-- S5$)NrrrrSwhererfabs rJrroots_jacobi..an_funcs+88AFQUq1uqy$93? ?rMc>[R"US:HTT- ST-T-- TT-TT-- SU-T-T-SU-T-T-S--- 5$)Nrrrrrs rJrrse88QQ1q519%QQC!GaK!Oa! a!8K#LM rMc  >SSU-T-T-- [R"UT-UT--SU-T-T-S-- 5-[R"US:HS[R"XT-T--SU-T-T-S- - 55-$)NrrrrN)rSrrrs rJrroots_jacobi..bn_funcs 37Q;? #ggq1uQ'1q519q=1+<=> ?hhqAvsBGGAQOsQw{QQR?R,S$TU V rMc6>[R"UTTU5$rEr eval_jacobirrGrrs rJrroots_jacobi..f s""1aA..rMcf>SUT-T-S--[R"US- TS-TS-U5-$)Nrrrrs rJrroots_jacobi..df"s=a!eai!m$w':':1q5!a%QPQ'RRRrMF) int ValueErrorr1r;rbetarSrrbetalnr) ralpharrhmrrrrrrrs @@rJr6r6s#T AA1u899 {dbjBCC | a$$ }9b11 EJqL!GLLq$q&$AAffelQ&"&&+5~~eAgtAv678 A A1u| @  /S !!'A5" MMrMc ^^^TS:a [S5eUU4SjnTS:Xa[//SSUSU[RS9$[ TTTSS9upVnTT-S-nS U-S T-U-- [ TT-S -5-n U [ TT-S-5[ TS -5- [ TU-5- -n [ S T-U-5S T-- [ TS -5- [ TU-5- n [XVXUSUUUU4S j5n U $) 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&>SU- T-SU-T--$NrrF)rGrrs rJrdjacobi..wfuncusA%1q5T/11rMrNrrrKTrhrrrc6>[R"TTTU5$rEr)rGrrrs rJrsjacobi..sg11!UD!DrM)rrArS ones_liker6_gam) rrrrerdrGrrhab1rbrcrrs ``` rJrr'sV 1u1222Av2r3UGU%'\\3 3Audt4HA" $, C C1q53; $q5y1}"5 5B$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. rrz>(p - q) must be greater than -1, and q must be greater than 0.rTrr)rr6) rp1q1rhmessagerGrrscales rJr?r?suT "}aR!!1eRT40GA! Q! A GEJAJA Qwt rMc ^^^TS:a [S5eUU4SjnTS:Xa[//SSUSU[RS9$Tn[ UTT5upg[ TS-5[ TT-5-[ TT-5-[ TT-T- S-5-nUST-T-[ ST-T-5S---nSn [XgXUS UUUU4S jS 9n U $) aShifted Jacobi polynomial. Defined by .. math:: G_n^{(p, q)}(x) = \binom{2n + p - 1}{n}^{-1}P_n^{(p - q, q - 1)}(2x - 1), where :math:`P_n^{(\cdot, \cdot)}` is the nth Jacobi polynomial. Parameters ---------- n : int Degree of the polynomial. p : float Parameter, must have :math:`p > q - 1`. q : float Parameter, must be greater than 0. monic : bool, optional If `True`, scale the leading coefficient to be 1. Default is `False`. Returns ------- G : orthopoly1d Shifted Jacobi polynomial. Notes ----- For fixed :math:`p, q`, the polynomials :math:`G_n^{(p, q)}` are orthogonal over :math:`[0, 1]` with weight function :math:`(1 - x)^{p - q}x^{q - 1}`. rrc,>SU- TT- -UTS- --$NrNrF)rGrrqs rJrdsh_jacobi..wfuncs#aQU#aAGn44rMrNrrrrrrc6>[R"TTTU5$rE)reval_sh_jacobi)rGrrrrs rJrssh_jacobi..s)?)?1a)KrMrdr]rerK)rrArSrr?r) rrrrrerdn1rGrrbrcpps ``` rJr!r!sH 1u1225Av2r3UGU%'\\3 3 B 2q! $DA a!etAE{ "T!a%[ 04A A 3F FB1q519a!eai!+ ,,B B Q2vUK MB IrMc v^[U5nUS:dX:wa [S5eTS:a [S5e[R"TS-5nUS:XaA[R "TS-/S5n[R "U/S5nU(aXVU4$XV4$U4SjnU4SjnU4S jn U4S jn [ X4XxXS U5$) aGauss-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. rrrzalpha must be greater than -1.rNr~c>SU-T-S-$NrrrFrfrs rJr"roots_genlaguerre..an_func0s1uu}q  rMc<>[R"XT--5*$rErSrrs rJr"roots_genlaguerre..bn_func2sY(((rMc4>[R"UTU5$rEreval_genlaguerrerrGrs rJrroots_genlaguerre..f4s''5!44rMc>U[R"UTU5-UT-[R"US- TU5-- U- $rrrs rJrroots_genlaguerre..df6sNG,,Qq99u9 8 8Qq IIJMNO OrMF)rrrgammarSrXr) rrrhrrrGrrrrrs ` rJr8r8sP AA1u899 rz9:: -- "CAv HHeCi[# & HHcUC  9 4K!)5O "!'A5" MMrMc B^^TS::a [S5eTS:a [S5eTS:XaTS-nOTn[UT5upEU4SjnTS:Xa//pT[TT-S-5[TS-5- nST-[TS-5- n[XEXxUS[4UUU4Sj5n U $)aSGeneralized (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() rzalpha must be > -1rrrc(>[U*5UT--$rEr)rGrs rJrdgenlaguerre..wfuncsA2we##rMc4>[R"TTU5$rErrGrrs rJrsgenlaguerre..sg66q%CrM)rr8rrAr) rrrerrGrrdrbrcrrs `` rJrr<sb {-..1u122Av U  R 'DA$Av21 a%i!m tAE{ *B q4A; BA"%!S5C EA HrMc[USUS9$)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)rrhs rJr7r7sL Q ++rMc ^TS:a [S5eTS:XaTS-nOTn[U5up4TS:Xa//pCSnST-[TS-5- n[X4XVSS[4UU4Sj5nU$)ajLaguerre 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() rrrrNrc[U*5$rErrGs rJrslaguerre..&s CGrMc2>[R"TU5$rE)r eval_laguerrerGrs rJrsr'sg33Aq9rM)rr7rrArrrerrGrrbrcrrs` rJrrsZ 1u122Av U  " DAAv21 B q4A; BA""3aXu9 ;A HrMc [U5nUS:dX:wa [S5e[R"[R5nUS::a(SnSn[ R nSn[X#XEXgSU5$[U5upU(aXU4$X4$)aNGauss-Hermite (physicist's) quadrature. Compute the sample points and weights for Gauss-Hermite quadrature. The sample points are the roots of the nth degree Hermite polynomial, :math:`H_n(x)`. These sample points and weights correctly integrate polynomials of degree :math:`2n - 1` or less over the interval :math:`[-\infty, \infty]` with weight function :math:`w(x) = e^{-x^2}`. See 22.2.14 in [AS]_ for details. Parameters ---------- n : int quadrature order mu : bool, optional If True, return the sum of the weights, optional. Returns ------- x : ndarray Sample points w : ndarray Weights mu : float Sum of the weights See Also -------- scipy.integrate.fixed_quad numpy.polynomial.hermite.hermgauss roots_hermitenorm Notes ----- For small n up to 150 a modified version of the Golub-Welsch algorithm is used. Nodes are computed from the eigenvalue problem and improved by one step of a Newton iteration. The weights are computed from the well-known analytical formula. For n larger than 150 an optimal asymptotic algorithm is applied which computes nodes and weights in a numerically stable manner. The algorithm has linear runtime making computation for very large n (several thousand or more) feasible. References ---------- .. [townsend.trogdon.olver-2014] Townsend, A. and Trogdon, T. and Olver, S. (2014) *Fast computation of Gauss quadrature nodes and weights on the whole real line*. :arXiv:`1410.5286`. .. [townsend.trogdon.olver-2015] Townsend, A. and Trogdon, T. and Olver, S. (2015) *Fast computation of Gauss quadrature nodes and weights on the whole real line*. IMA Journal of Numerical Analysis :doi:`10.1093/imanum/drv002`. .. [AS] Milton Abramowitz and Irene A. Stegun, eds. Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. New York: Dover, 1972. rrc SU-$NrrFrfs rJrroots_hermite..an_funcq 7NrMc4[R"US- 5$Nrrrs rJrroots_hermite..bn_funcss771s7# #rMcBSU-[R"US- U5-$)Nrrr eval_hermiterrGs rJrroots_hermite..dfvs"7W11!a%;; ;rMT) rrrSrrrrr_roots_hermite_asy rrhrrrrrrnodesr[s rJr9r9-s| AA1u899 ''"%%.CCx  $   <%agtRPP+A. 3& &> !rMc^US-S- nS[US- 5-SU-- S-[-S[US- 5-SU--S-- mU4SjnSnS[-n[U5HnXd"U5U"U5- - nM U$)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 rr@r@c&>U[U5- T- $rE)r)rGrs rJr_compute_tauk..fs3q6zA~rMcS[U5- $r)r rs rJr_compute_tauk..dfsSV|rM)rrrP) rrfmaxitrrrxiirs @rJ _compute_taukrs4 A A U1S5\ CE !C '+s53- S- nSUSSS24-SUS SS24--S?US"SS24-- S@USSS24--S$U--S%- nSAUS'SS24-SBUS)SS24--SCUS+SS24-- SDUSSS24-- SEUSSS24-- SF- S/- nS0US1SS24-S2US3SS24-- SGUS5SS24--SHUSSS24-- SIUS SS24--SJUS"SS24--SKUSSS24--SLU--S;- n [ US-U-5un!n"n#n$S[ [ 5-USM--U-n%U[S+SNS+5RS5-n&UU-n'UU-U&SOSS24U-U--U&SPSS24U-U--US-- n(UU-U&SOSS24U-U--U&SPSS24U-U--U&SSS24U-U--U&SSS24U-U--US+-- n)U U-U&SOSS24U-U--*US-- n*U U-U&SOSS24U -U--U&SPSS24U -U--U&SSS24U-U--*US"-- n+U U-U&SOSS24U -U--U&SPSS24U -U--U&SSS24U -U--U&SSS24U -U--U&SSS24U-U--*US)-- n,U%U!U'U(US-- -U)USQ-- --U"U*U+US-- -U,USQ-- --USR-- --n-[ S[ -5USS--U- n.UU-U&SOSS24U-U--*U- n/UU-U&SOSS24U-U--U&SPSS24U-U--U&SSS24U-U--*US-- n0UU-U&SOSS24U-U--U&SPSS24U-U--U&SSS24U-U--U&SSS24U-U--U&SSS24U-U --*US -- n1UU-n2U U-U&SOSS24U -U--U&SPSS24U-U--US-- n3U U-U&SOSS24U -U--U&SPSS24U -U--U&SSS24U -U--U&SSS24U-U--US+-- n4U.U!U/U0US-- -U1USQ-- --US-- U"U2U3US-- -U4USQ-- ----n5U-U54$)Ta\Asymptotic 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 rrNrrgUUUUUU?rrg?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$iAgAig.Ag(AgӍAg5e Bgd{>Bl+2VgUUUUUU?rrrgUUUUUU?r)rr r reshaperrr)6rthetastctrhetazetaphia0a1a2a3a4a5b0b1b2b3b4b5ctpu0u1u2u3u4u5v0v1v2v3v4v5AiAipBiBipPphipA0A1A2B0B1B2UPdC0C1C2D0D1D2Uds6 rJ_pbcfrhsD UB UB QB e)c"fRi C WS[g & &D 52q5=d #C B B B B B B B B B B B B r ""6* *C B c!A#h,R 4 'B s1Q3x-%AaC. (5 0F :B #ac( WS1X- -AaC0@ @ S1X   (  ,/7 8B #bd) hs1Q3x/ /(3qs82C C s1Q3x  "-c!A#h"6 79C DGQ RB SAY SAY!6 6c"Q$i9O O QqS ! "$0QqS$9 :B R%$qs)B,r/ !D1IbLO 3d1Q3il2o E 1IbLO #Qw 'B b54!9R<? " #dAg -B b54!9R<? "T!A#Yr\"_ 4tAaCy|B F G$PQ' QB b54!9R<? "T!A#Yr\"_ 4tAaCy|B F !A#Yr\"_ #AaCy|B / 026' :B R22s7 ?RCZ/ 0 BBGObSj0 1BM AB CA c"fW % +B b54!9R<? " #d *B b54!9R<? "T!A#Yr\"_ 4tAaCy|B F G$PQ' QB b54!9R<? "T!A#Yr\"_ 4tAaCy|B F !A#Yr\"_ #AaCy|B / 026' :B BB R%$qs)B,r/ !D1IbLO 3tQw >B R%$qs)B,r/ !D1IbLO 3d1Q3il2o E 1IbLO #Qw 'B rR"RW*_r"c'z12R']Bb2b#g:o2s7 234 5B b5LrMcj[SU-S-5nX- n[U5n[U5HNn[X5upxU[S5U-[ U5-U-- n XY-n[ [ U 55S:dMN O U[U5-n US-S:XaSU S'[U S-*5SWS--- n X4$)aNewton iteration for polishing the asymptotic approximation to the zeros of the Hermite polynomials. Parameters ---------- n : int Quadrature order x_initial : ndarray Initial guesses for the roots maxit : int Maximal number of Newton iterations. The default 5 is sufficient, usually only one or two steps are needed. Returns ------- nodes : ndarray Quadrature nodes weights : ndarray Quadrature weights See Also -------- roots_hermite_asy rrNg+=rrrr) rr rPrhrrrRr r) r x_initialr rhtr4r uuddthetarGrs rJ_newtonros6 c!eck BA 1IE 5\ad3i"ns5z1B67 s6{ e #   SZA1uz! QTE c"a%i A 4KrMc$[U5n[X5up#US-S:Xa([USSS2*U/5n[USSS2U/5nO'[USSS2*U/5n[USSS2U/5nU[[5[ U5- -nX#4$)aGauss-Hermite (physicist's) quadrature for large n. Computes the sample points and weights for Gauss-Hermite quadrature. The sample points are the roots of the nth degree Hermite polynomial, :math:`H_n(x)`. These sample points and weights correctly integrate polynomials of degree :math:`2n - 1` or less over the interval :math:`[-\infty, \infty]` with weight function :math:`f(x) = e^{-x^2}`. This method relies on asymptotic expansions which work best for n > 150. The algorithm has linear runtime making computation for very large n feasible. Parameters ---------- n : int quadrature order Returns ------- nodes : ndarray Quadrature nodes weights : ndarray Quadrature weights See Also -------- roots_hermite References ---------- .. [townsend.trogdon.olver-2014] Townsend, A. and Trogdon, T. and Olver, S. (2014) *Fast computation of Gauss quadrature nodes and weights on the whole real line*. :arXiv:`1410.5286`. .. [townsend.trogdon.olver-2015] Townsend, A. and Trogdon, T. and Olver, S. (2015) *Fast computation of Gauss quadrature nodes and weights on the whole real line*. IMA Journal of Numerical Analysis :doi:`10.1093/imanum/drv002`. rrNr)r'ror rrr)rr&rr[s rJrrsV  BQ^NE1uztt e,-'$B$-12r!Bw/0'"Qr'*G45 tBx#g,&&G >rMc ^TS:a [S5eTS:XaTS-nOTn[U5up4SnTS:Xa//pCST-[TS-5-[[5-nST-n[ X4XgU[ *[ 4UU4Sj5nU$)aPhysicist'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 [U*U-5$rErrs rJrdhermite..wfunc3sA26{rMrc2>[R"TU5$rErrs rJrshermite..:sg221a8rM)rr9rrrrAr rrerrGrrdrbrcrrs ` rJrrsb 1u122Av U   DAAv21 AQU d2h &B ABA"%3$e8 :A HrMc \[U5nUS:dX:wa [S5e[R"S[R-5nUS::a(SnSn[ R nSn[X#XEXgSU5$[U5upU[S 5-nU [S 5-n U(aXU4$X4$) aGauss-Hermite (statistician's) quadrature. Compute the sample points and weights for Gauss-Hermite quadrature. The sample points are the roots of the nth degree Hermite polynomial, :math:`He_n(x)`. These sample points and weights correctly integrate polynomials of degree :math:`2n - 1` or less over the interval :math:`[-\infty, \infty]` with weight function :math:`w(x) = e^{-x^2/2}`. See 22.2.15 in [AS]_ for more details. Parameters ---------- n : int quadrature order mu : bool, optional If True, return the sum of the weights, optional. Returns ------- x : ndarray Sample points w : ndarray Weights mu : float Sum of the weights See Also -------- scipy.integrate.fixed_quad numpy.polynomial.hermite_e.hermegauss Notes ----- For small n up to 150 a modified version of the Golub-Welsch algorithm is used. Nodes are computed from the eigenvalue problem and improved by one step of a Newton iteration. The weights are computed from the well-known analytical formula. For n larger than 150 an optimal asymptotic algorithm is used which computes nodes and weights in a numerical stable manner. The algorithm has linear runtime making computation for very large n (several thousand or more) feasible. References ---------- .. [AS] Milton Abramowitz and Irene A. Stegun, eds. Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. New York: Dover, 1972. rrrrc SU-$rrFrs rJr"roots_hermitenorm..an_funcyrrMc.[R"U5$rErrs rJr"roots_hermitenorm..bn_func{s771: rMc<U[R"US- U5-$rreval_hermitenormrs rJrroots_hermitenorm..df~sw//Aq99 9rMTr) rrrSrrrr~rrrs rJr:r:@sf AA1u899 ''#bee) CCx    $ $ :%agtRPP+A. a47 3& &> !rMc ^TS:a [S5eTS:XaTS-nOTn[U5up4SnTS:Xa//pC[S[-5[ TS-5-nSn[ X4XgU[ *[ 4UU4SjS9nU$) aNormalized (probabilist's) Hermite polynomial. Defined by .. math:: He_n(x) = (-1)^ne^{x^2/2}\frac{d^n}{dx^n}e^{-x^2/2}; :math:`He_n` is a polynomial of degree :math:`n`. Parameters ---------- n : int Degree of the polynomial. monic : bool, optional If `True`, scale the leading coefficient to be 1. Default is `False`. Returns ------- He : orthopoly1d Hermite polynomial. Notes ----- The polynomials :math:`He_n` are orthogonal over :math:`(-\infty, \infty)` with weight function :math:`e^{-x^2/2}`. rrrc&[U*U-S- 5$rrrs rJrdhermitenorm..wfuncsA26C<  rMrrNc2>[R"TU5$rEr}rs rJrshermitenorm..(@(@A(FrMr)rr:rrrrArrvs ` rJrrs> 1u122Av U  R DA!Av21 a"fQU #B BA"tSkF HA HrMc r^[U5nUS:dX:wa [S5eTS:a [S5eTS:Xa [X5$TS::aY[R"[R 5[ R"TS-5-[ R"TS-5- nOqST- n[R"/S Q5nUS n[S[U55H nXE-Xg-nM U[R"[R T- 5-nS nU4S jn U4S jn U4Sjn [X4XXSU5$)aGauss-Gegenbauer quadrature. Compute the sample points and weights for Gauss-Gegenbauer quadrature. The sample points are the roots of the nth degree Gegenbauer polynomial, :math:`C^{\alpha}_n(x)`. These sample points and weights correctly integrate polynomials of degree :math:`2n - 1` or less over the interval :math:`[-1, 1]` with weight function :math:`w(x) = (1 - x^2)^{\alpha - 1/2}`. See 22.2.3 in [AS]_ for more details. Parameters ---------- n : int quadrature order alpha : float alpha must be > -0.5 mu : bool, optional If True, return the sum of the weights, optional. Returns ------- x : ndarray Sample points w : ndarray Weights mu : float Sum of the weights See Also -------- scipy.integrate.fixed_quad References ---------- .. [AS] Milton Abramowitz and Irene A. Stegun, eds. Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. New York: Dover, 1972. rrz alpha must be greater than -0.5.rrrN)gF90(+?g& {XgkʰDg3Ts?g?grNrc SU-$rrFrs rJr!roots_gegenbauer..an_func QwrMcj>[R"XST--S- -SUT--UT-S- -- 5$)Nrrrrrs rJr!roots_gegenbauer..bn_funcs=wwqE MA-.!q5y/QYQR]2STUUrMc4>[R"UTU5$rEreval_gegenbauerrs rJrroots_gegenbauer..fs&&q%33rMc>U*U-[R"UTU5-UST--S- [R"US- TU5--SUS-- - $rrrs rJrroots_gegenbauer..df sc BFW,,Qq9 91u9}q G$;$;AE5!$LL M aZ rMT) rrr2rSrrrrrXrPrQr) rrrhrr inv_alpharVtermrrrrs ` rJr;r;sP AA1u899 t|;<< # A"" |wwruu~ eck :: eai() J >?Qi!S[)D/FL0C*BGGBEEEM**V4 "!'A4 LLrMcp^^[R"T5(aTS::a [S5e[TTS- TS- US9nU(dTS:XaU$[ ST-T-5[ TS-5-[ ST-5- [ TS-T-5- nUR U5 UU4SjUR S'U$) 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() rz1`alpha` must be a finite number greater than -1/2rrerrcF>[R"[T5TU5$rE)rrrWrs rJrsgegenbauer..\sG,C,CE!HDI1-NrMr_)rSisfiniterrrrv__dict__)rrrebasefactors`` rJrrsB ;;u  $LMM !US[%#+U ;D Q 1U7Q;$us{"331U7m"53;?34FKK#NDMM, KrMc[U5nUS:dX:wa [S5e[R"[R "U*S-US5SU-- 5n[R "U[U- 5nU(aX4[4$X44$)aGauss-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_sinpirSr full_liker)rrhrrGrs rJr2r2esvN AA1u899ryy!aA.!A#67A Q1A Rxt rMc ^TS:a [S5eSnTS:Xa[//[SUSUU4Sj5$Tn[USS9upEn[S - nS TS - -n[XEXxUSUU4S j5n U $) aS 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() rrc(S[SX-- 5- $)NrNrrrs rJrdchebyt..wfuncsT!ae)_$$rMrNrc2>[R"TU5$rEr eval_chebytrs rJrschebyt..sW%8%8A%>rMTrrrc2>[R"TU5$rErrs rJrsrsg11!Q7rM)rrArr2) rrerdrrGrrhrbrcrrs ` rJrrst 1u122%Av2r2sE7E>@ @ BB4(HA" aB QUBA"%%7 9A HrMc0[U5nUS:dX:wa [S5e[R"USS5[-US-- n[R "U5n[[R "U5S--US-- nU(a XE[S- 4$XE4$)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. rrrrr)rrrSr rr r)rrhrrkrGrs rJr3r3sL AA1u899 !Qb AE*A q A RVVAY\QU#A R!V|t rMc[USSUS9nU(aU$[[5S- [US-5-[US-5- nUR U5 U$)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() rrrr?)rrrrrvrrerrs rJrr7sUr !S#U +D  "X^d1q5k )DSM 9FKK KrMcZ[US5up#nUS-nUS-nUS-nU(aX#U4$X#4$)aGauss-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. Trr2rrhrGrrs rJr4r4CL1d#GA!FAFAFA Qwt rMc ^TS:a [S5eTS:XaTS-nOTn[U5up4TS:Xa//pCS[-TS:HS--nSn[X4XVSSUS9nU(d-UR S U"S 5- 5 U4S jUR S 'U$) aChebyshev 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. rrrrrNc.S[SX-S- - 5- $)NrNrrrrs rJrschebyc..sC$q153;*?$?rMrrdr]rerrc2>[R"TU5$rE)r eval_chebycrs rJrsrW-@-@A-FrMr_)rr4rrArvrrs` rJrrsD 1u122Av U   DAAv21 RAFa< B BA"?"% 1A  qt#F < HrMcZ[US5up#nUS-nUS-nUS-nU(aX#U4$X#4$)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 rJr5r5rrMc ^TS:a [S5eTS:XaTS-nOTn[U5up4TS:Xa//pC[nSn[X4XVSSUS9nU(d2TS-U"S5- nUR U5 U4S jUR S 'U$) aChebyshev polynomial of the second kind on :math:`[-2, 2]`. Defined as :math:`S_n(x) = U_n(x/2)` where :math:`U_n` is the nth Chebychev polynomial of the second kind. Parameters ---------- n : int Degree of the polynomial. monic : bool, optional If `True`, scale the leading coefficient to be 1. Default is `False`. Returns ------- S : orthopoly1d Chebyshev polynomial of the second kind on :math:`[-2, 2]`. See Also -------- chebyu : Chebyshev polynomial of the second kind Notes ----- The polynomials :math:`S_n(x)` are orthogonal over :math:`[-2, 2]` with weight function :math:`\sqrt{1 - (x/2)}^2`. References ---------- .. [1] Abramowitz and Stegun, "Handbook of Mathematical Functions" Section 22. National Bureau of Standards, 1972. rrrrNc([SX-S- - 5$)Nrrrrs rJrschebys..bsDQUS[$9rMrrrc2>[R"TU5$rE)r eval_chebysrs rJrsrgrrMr_)rr5rrArvr) rrerrGrrbrcrrrs ` rJrr3sD 1u122Av U   DAAv21 B BA"9"% 1A c'QqT! #F < HrMc<[X5nUSS-S- 4USS-$)aGauss-Chebyshev (first kind, shifted) quadrature. Compute the sample points and weights for Gauss-Chebyshev quadrature. The sample points are the roots of the nth degree shifted Chebyshev polynomial of the first kind, :math:`T_n(x)`. These sample points and weights correctly integrate polynomials of degree :math:`2n - 1` or less over the interval :math:`[0, 1]` with weight function :math:`w(x) = 1/\sqrt{x - x^2}`. See 22.2.8 in [AS]_ for more details. Parameters ---------- n : int quadrature order mu : bool, optional If True, return the sum of the weights, optional. Returns ------- x : ndarray Sample points w : ndarray Weights mu : float Sum of the weights See Also -------- scipy.integrate.fixed_quad References ---------- .. [AS] Milton Abramowitz and Irene A. Stegun, eds. Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. New York: Dover, 1972. rrrNr)rrhxws rJr=r=ms/L a B UQY!O 12 &&rMct[USSUS9nU(aU$US:a SU-S- nOSnURU5 U$)aShifted Chebyshev polynomial of the first kind. Defined as :math:`T^*_n(x) = T_n(2x - 1)` for :math:`T_n` the nth Chebyshev polynomial of the first kind. Parameters ---------- n : int Degree of the polynomial. monic : bool, optional If `True`, scale the leading coefficient to be 1. Default is `False`. Returns ------- T : orthopoly1d Shifted Chebyshev polynomial of the first kind. Notes ----- The polynomials :math:`T^*_n` are orthogonal over :math:`[0, 1]` with weight function :math:`(x - x^2)^{-1/2}`. rrrrrrrNr!rvrs rJrrsF2 QS .D  1uAKK KrMc[US5up#nUS-S- n[R"SS5nX5U- -nU(aX#U4$X#4$)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. Trrr)r3rr)rrhrGrrm_uss rJr>r>sQL1d#GA! Q! A <<S !DMA Tzt rMc\[USSUS9nU(aU$SU-nURU5 U$)aShifted 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}`. rrrrrrs rJr r s62 QS .D  TFKK KrMc [U5nUS:dX:wa [S5eSnSnSn[RnSn[ X#XEXgSU5$)ay 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 rrrc SU-$rrFrs rJrroots_legendre..an_func rrMcLU[R"SSU-U-S- - 5-$)NrNrrrrs rJrroots_legendre..bn_func s'2773!a%!)a-0111rMcU*U-[R"X5-U[R"US- U5--SUS-- - $)Nrrr eval_legendrers rJrroots_legendre..df sPQ..q44g++AE1556:;a1f*F FrMT)rrrrr)rrhrrrrrrs rJr1r1 sZX AA1u899 C2AF "!'A4 LLrMc ^TS:a [S5eTS:XaTS-nOTn[U5up4TS:Xa//pCSST-S-- n[ST-S-5[TS-5S-- ST-- n[X4XVSSUU4SjS 9nU$) aLegendre 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. ]) rrrrrcgrrFrs rJrslegendre.. s#rMrc2>[R"TU5$rErrs rJrsr s(=(=a(CrMr)rr1rrArs` rJrr sL 1u122Av U  " DAAv21 A B a!eai4A;> )CF 2BA" gC EA HrMcR[U5up#US-S- nUS-nU(aX#S4$X#4$)ajGauss-Legendre (shifted) quadrature. Compute the sample points and weights for Gauss-Legendre quadrature. The sample points are the roots of the nth degree shifted Legendre polynomial :math:`P^*_n(x)`. These sample points and weights correctly integrate polynomials of degree :math:`2n - 1` or less over the interval :math:`[0, 1]` with weight function :math:`w(x) = 1.0`. See 2.2.11 in [AS]_ for details. Parameters ---------- n : int quadrature order mu : bool, optional If True, return the sum of the weights, optional. Returns ------- x : ndarray Sample points w : ndarray Weights mu : float Sum of the weights See Also -------- scipy.integrate.fixed_quad References ---------- .. [AS] Milton Abramowitz and Irene A. Stegun, eds. Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. New York: Dover, 1972. rrrN)r1)rrhrGrs rJr<r< s;J ! DA Q! AFA Syt rMc ^TS:a [S5eSnTS:Xa[//SSUSUU4Sj5$[T5up4SST-S-- n[ST-S-5[TS-5S-- n[X4XVUSUU4S jS 9nU$) aShifted Legendre polynomial. Defined as :math:`P^*_n(x) = P_n(2x - 1)` for :math:`P_n` the nth Legendre polynomial. Parameters ---------- n : int Degree of the polynomial. monic : bool, optional If `True`, scale the leading coefficient to be 1. Default is `False`. Returns ------- P : orthopoly1d Shifted Legendre polynomial. Notes ----- The polynomials :math:`P^*_n` are orthogonal over :math:`[0, 1]` with weight function 1. rrcSU-S-$)NrrNrFrs rJrdsh_legendre..wfunc sQw}rMrNrc2>[R"TU5$rEreval_sh_legendrers rJrssh_legendre.. sW%=%=a%CrMrrc2>[R"TU5$rErrs rJrsr rrM)r]rerK)rrAr<r)rrerdrGrrbrcrrs` rJrr s2 1u122Av2r3UFECE E Q DA A B a!eai4A;> )BA"%eF HA HrM)F)r)J__doc__numpyrSrrrrrrr r r r r scipyr scipy.specialrrrrr _polyfuns _rootfuns_maprYkeys__all__rTrArr6rr?r!r8rr7rr9rrrr'rhrorrr:rr;rr2rr3rr4rr5rr=rr>r r1rr<rglobals _modattrsitemsnewfunoldfunappendrFrMrJrsGZ++++}} ' $-!*!*!*!*!*#,&0"+&0%/&0$.$.$.0 d=--/0 0*"))*Z,bSNl[ @5p4rANHc P&,R[ @Q"h#L F&R$Nup,^6rA LI"X/ lMM`Lf/dh \/d^F-`4 r-`5 t''T!J-`FyMx5 t+\& T I #))+NFF!&)If NN6,rM