MZdN!"dZddlmZddlmZmZmZmZmZddl m Z m Z ddl m Z ddlmZdZedd Zd Zdd Zd Zd ZeddZeddZdZdZeddZeddZdZeddZdZeddZdZ dZ!ddZ"y)z:Efficient functions for generating orthogonal polynomials.)Dummy)dup_muldup_mul_ground dup_lshiftdup_subdup_add)ZZQQ) named_poly)publicc @|dkr |jgS|jg||z|dz |jz||z |dz g}}td|dzD]B}||||z|zz||z|d|zz|dz z}||z|d|zz|jz ||z||zz z|d|zz }||z|d|zz|jz ||z|d|zz|dz z||z|d|zzz|d|zz } ||z|jz ||z|jz z||z|d|zzz|z } t|||} tt|d|| |} t|| |} |t t | | || |}}E|S)z/Low-level implementation of Jacobi polynomials.)onerangerrrr)nabKm2m1idenf0f1f2p0p1p2s 8/usr/lib/python3/dist-packages/sympy/polys/orthopolys.py dup_jacobir! s1uweeW!QqTzAEE)AaC1:6B 1ac]8dAEAIA!Q1 56!ead1fnquu$1qs 3qtCx @!ead1fnquu$Q1a!A$)> ?1q51Q4PQ6> RVWXYVZ[^V^ _!eaeema!eaeem ,a!ead1fn = C BA & Jr1a0"a 8 BA &WWRQ/Q7B8 INc 0t|tdd|||f|S)aGenerates the Jacobi polynomial `P_n^{(a,b)}(x)`. Parameters ========== n : int Degree of the polynomial. a Lower limit of minimal domain for the list of coefficients. b Upper limit of minimal domain for the list of coefficients. x : optional polys : bool, optional If True, return a Poly, otherwise (default) return an expression. NzJacobi polynomial)r r!)rrrxpolyss r jacobi_polyr&s " aT+>Aq 5 QQr"c|dkr |jgS|jg|d|z|jg}}td|dzD]}tt |d||d||jz z||z |dz|}t||d||jz z||z |jz|}|t |||}}|S)z3Low-level implementation of Gegenbauer polynomials.rrrzerorrrr)rrrrrrrrs r dup_gegenbauerr*,s1uweeWqtAvqvv&B 1ac]( Jr1a0!A$!%%.12E!2La P B!agqt 3aee ;Q ?WRQ'B( Ir"c.t|tdd||f|S)a?Generates the Gegenbauer polynomial `C_n^{(a)}(x)`. Parameters ========== n : int Degree of the polynomial. x : optional a Decides minimal domain for the list of coefficients. polys : bool, optional If True, return a Poly, otherwise (default) return an expression. NzGegenbauer polynomial)r r*)rrr$r%s r gegenbauer_polyr,7s a/FAPU VVr"c |dkr |jgS|jg|j|jg}}td|dzD]-}|tt t |d||d|||}}/|S)zDLow-level implementation of Chebyshev polynomials of the first kind.rrrr)rrrrrrrrrs r dup_chebyshevtr0Gsx1uweeWquuaffoB 1ac]SW^Jr1a,@!A$JBPQRBS Ir"c |dkr |jgS|jg|d|jg}}td|dzD]-}|tt t |d||d|||}}/|S)zELow-level implementation of Chebyshev polynomials of the second kind.rrr.r/s r dup_chebyshevur2Psx1uweeWqtQVVnB 1ac]SW^Jr1a,@!A$JBPQRBS Ir"c4t|ttd|f|S)aGenerates the Chebyshev polynomial of the first kind `T_n(x)`. Parameters ========== n : int Degree of the polynomial. x : optional polys : bool, optional If True, return a Poly, otherwise (default) return an expression. z&Chebyshev polynomial of the first kind)r r0r rr$r%s r chebyshevt_polyr5Ys" a 4qdE CCr"c4t|ttd|f|S)aGenerates the Chebyshev polynomial of the second kind `U_n(x)`. Parameters ========== n : int Degree of the polynomial. x : optional polys : bool, optional If True, return a Poly, otherwise (default) return an expression. z'Chebyshev polynomial of the second kind)r r2r r4s r chebyshevu_polyr7is" a 5tU DDr"c |dkr |jgS|jg|d|jg}}td|dzD]E}t|d|}t |||dz |}|t t ||||d|}}G|S)z0Low-level implementation of Hermite polynomials.rrrr)rrrrrrrrrrrs r dup_hermiter;ys1uweeWqtQVVnB 1ac]? r1a  2q1vq )^GAq!$4adA>B? Ir"c|dkr |jgS|jg|j|jg}}td|dzD]4}t|d|}t |||dz |}|t |||}}6|S)z>Low-level implementation of probabilist's Hermite polynomials.rrr9r:s r dup_hermite_probr=s1uweeWquuaffoB 1ac]& r1a  2q1vq )WQ1%B& Ir"c4t|ttd|f|S)zGenerates the Hermite polynomial `H_n(x)`. Parameters ========== n : int Degree of the polynomial. x : optional polys : bool, optional If True, return a Poly, otherwise (default) return an expression. zHermite polynomial)r r;r r4s r hermite_polyr?s ab*>e LLr"c4t|ttd|f|S)aGenerates the probabilist's Hermite polynomial `He_n(x)`. Parameters ========== n : int Degree of the polynomial. x : optional polys : bool, optional If True, return a Poly, otherwise (default) return an expression. z probabilist's Hermite polynomial)r r=r r4s r hermite_prob_polyrAs! a)2 .e ==r"c2|dkr |jgS|jg|j|jg}}td|dzD]M}tt |d||d|zdz ||}t|||dz ||}|t |||}}O|S)z1Low-level implementation of Legendre polynomials.rrr(r:s r dup_legendrerCs1uweeWquuaffoB 1ac]& :b!Q/1Q3q5!a @ 2q1ay! ,WQ1%B& Ir"c4t|ttd|f|S)zGenerates the Legendre polynomial `P_n(x)`. Parameters ========== n : int Degree of the polynomial. x : optional polys : bool, optional If True, return a Poly, otherwise (default) return an expression. zLegendre polynomial)r rCr r4s r legendre_polyrEs ar+@1$ NNr"c d|jg|jg}}td|dzD]}t||j ||z ||jz ||z |dzg|}t |||jz ||z |jz|}|t |||}}|S)z1Low-level implementation of Laguerre polynomials.rr)r)rrrrr)ralpharrrrrrs r dup_laguerrerHsffXwB 1ac]& B!%%!uQUU{AaD&81Q4&?@! D 2aee QqT1AEE91 =WQ1%B& Ir"c.t|tdd||f|S)aQGenerates the Laguerre polynomial `L_n^{(\alpha)}(x)`. Parameters ========== n : int Degree of the polynomial. x : optional alpha : optional Decides minimal domain for the list of coefficients. polys : bool, optional If True, return a Poly, otherwise (default) return an expression. NzLaguerre polynomial)r rH)rr$rGr%s r laguerre_polyrJs at-BQJPU VVr"c *|dkr|j|jgS|jg|j|jg}}td|dzD]3}|tt t |d||d|zdz |||}}5t |d|S)z%Low-level implementation of fn(n, x).rrr.r/s r dup_spherical_bessel_fnrLs1uqvveeWquuaffoB 1ac]WW^Jr1a,@!AaCE(ANPRTUVBW b!Q r"c |j|jg|jg}}td|dzD]3}|tt t |d||dd|zz |||}}5|S)z&Low-level implementation of fn(-n, x).rrr.r/s r dup_spherical_bessel_fn_minusrOsoeeQVV_qvvhB 1ac]WW^Jr1a,@!AacE(ANPRTUVBW Ir"c | td}|dkrtnt}tt ||t dt d|z f|S)a Coefficients for the spherical Bessel functions. These are only needed in the jn() function. The coefficients are calculated from: fn(0, z) = 1/z fn(1, z) = 1/z**2 fn(n-1, z) + fn(n+1, z) == (2*n+1)/z * fn(n, z) Parameters ========== n : int Degree of the polynomial. x : optional polys : bool, optional If True, return a Poly, otherwise (default) return an expression. Examples ======== >>> from sympy.polys.orthopolys import spherical_bessel_fn as fn >>> from sympy import Symbol >>> z = Symbol("z") >>> fn(1, z) z**(-2) >>> fn(2, z) -1/z + 3/z**3 >>> fn(3, z) -6/z**2 + 15/z**4 >>> fn(4, z) 1/z - 45/z**3 + 105/z**5 r$rr)rrOrLr absr r )rr$r%fs r spherical_bessel_fnrTsEJ y #J)*Q%4KA c!faR"Q%'U ;;r")NF)NrF)#__doc__sympy.core.symbolrsympy.polys.densearithrrrrrsympy.polys.domainsr r sympy.polys.polytoolsr sympy.utilitiesr r!r&r*r,r0r2r5r7r;r=r?rArCrErHrJrLrOrTr"r r\s@#""&," RR$ W  C C D D   M M = =  O OWW  (