RieN!NdZddlmZddlmZmZmZmZmZddl m Z m Z ddl m Z ddlmZdZedd Zd Zdd Zd ZdZeddZeddZdZdZeddZeddZdZeddZdZeddZdZ dZ!ddZ"dS)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]g}||||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 | | || |}}i|S)z/Low-level implementation of Jacobi polynomials.)onerangerrrr)nabKm2m1idenf0f1f2p0p1p2s 6lib/python3.11/site-packages/sympy/polys/orthopolys.py dup_jacobir! s1uuweW!QQqTTzAE)AaC1:6B 1ac]]88addAEAIA!Q1 56!eaadd1fnqu$1qs 3qqttCx @!eaadd1fnqu$Q1a!!A$$)> ?1q511Q44PQ6> RVWVWXYVZVZ[^V^ _!eaema!eaem ,a!eaadd1fn = C BA & & Jr1a00"a 8 8 BA & &WWRQ//Q77B INFc :t|tdd|||f|S)aGenerates the Jacobi polynomial `P_n^{(a,b)}(x)`. Parameters ========== n : int Degree of the polynomial. a Lower limit of minimal domain for the list of coefficients. b Upper limit of minimal domain for the list of coefficients. x : optional polys : bool, optional If True, return a Poly, otherwise (default) return an expression. NzJacobi polynomial)r r!)rrrxpolyss r jacobi_polyr&s#" aT+>Aq 5 Q QQr"c|dkr|jgS|jg|d|z|jg}}td|dzD]}tt |d||d||jz z||z |dz|}t||d||jz z||z |jz|}|t |||}}|S)z3Low-level implementation of Gegenbauer polynomials.rrrzerorrrr)rrrrrrrrs r dup_gegenbauerr*,s1uuweWqqttAvqv&B 1ac]](( Jr1a00!!A$$!%.12E!2La P P B!agqqtt 3ae ;Q ? ?WRQ''B Ir"c8t|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 V 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_chebyshevtr0Gs1uuweWquafoB 1ac]]SSW^Jr1a,@,@!!A$$JJBPQRRB Ir"c |dkr|jgS|jg|d|jg}}td|dzD]<}|tt t |d||d|||}}=|S)zELow-level implementation of Chebyshev polynomials of the second kind.rrr.r/s r dup_chebyshevur2Ps1uuweWqqttQVnB 1ac]]SSW^Jr1a,@,@!!A$$JJBPQRRB Ir"c@t|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 C CCr"c@t|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 D DDr"c 4|dkr|jgS|jg|d|jg}}td|dzD][}t|d|}t |||dz |}|t t ||||d|}}\|S)z0Low-level implementation of Hermite polynomials.rrrr)rrrrrrrrrrrs r dup_hermiter;ys1uuweWqqttQVnB 1ac]]?? r1a  2qq1vvq ) )^GAq!$4$4aaddA>>B Ir"c|dkr|jgS|jg|j|jg}}td|dzD]C}t|d|}t |||dz |}|t |||}}D|S)z>Low-level implementation of probabilist's Hermite polynomials.rrr9r:s r dup_hermite_probr=s1uuweWquafoB 1ac]]&& r1a  2qq1vvq ) )WQ1%%B Ir"c@t|ttd|f|S)zGenerates the Hermite polynomial `H_n(x)`. Parameters ========== n : int Degree of the polynomial. x : optional polys : bool, optional If True, return a Poly, otherwise (default) return an expression. zHermite polynomial)r r;r r4s r hermite_polyr?s ab*>e L LLr"c@t|ttd|f|S)aGenerates the probabilist's Hermite polynomial `He_n(x)`. Parameters ========== n : int Degree of the polynomial. x : optional polys : bool, optional If True, return a Poly, otherwise (default) return an expression. z probabilist's Hermite polynomial)r r=r r4s r hermite_prob_polyrAs& a)2 .e = ==r"c<|dkr|jgS|jg|j|jg}}td|dzD]c}tt |d||d|zdz ||}t|||dz ||}|t |||}}d|S)z1Low-level implementation of Legendre polynomials.rrr(r:s r dup_legendrerCs1uuweWquafoB 1ac]]&& :b!Q//1Q3q5!a @ @ 2qq1ayy! , ,WQ1%%B Ir"c@t|ttd|f|S)zGenerates the Legendre polynomial `P_n(x)`. Parameters ========== n : int Degree of the polynomial. x : optional polys : bool, optional If True, return a Poly, otherwise (default) return an expression. zLegendre polynomial)r rCr r4s r legendre_polyrEs ar+@1$ N NNr"c \|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_laguerrerHsfXwB 1ac]]&& B!%!uQU{AAaDD&811Q44&?@! D D 2ae QQqTT1AE91 = =WQ1%%B Ir"c8t|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 V VVr"c $|dkr|j|jgS|jg|j|jg}}td|dzD]B}|tt t |d||d|zdz |||}}Ct |d|S)z%Low-level implementation of fn(n, x).rrr.r/s r dup_spherical_bessel_fnrLs1uuqveWquafoB 1ac]]WWW^Jr1a,@,@!!AaCE((ANNPRTUVVB b!Q  r"c |j|jg|jg}}td|dzD]B}|tt t |d||dd|zz |||}}C|S)z&Low-level implementation of fn(-n, x).rrr.r/s r dup_spherical_bessel_fn_minusrOszeQV_qvhB 1ac]]WWW^Jr1a,@,@!!AacE((ANNPRTUVVB Ir"c |td}|dkrtnt}tt ||t dt d|z f|S)a Coefficients for the spherical Bessel functions. These are only needed in the jn() function. The coefficients are calculated from: fn(0, z) = 1/z fn(1, z) = 1/z**2 fn(n-1, z) + fn(n+1, z) == (2*n+1)/z * fn(n, z) Parameters ========== n : int Degree of the polynomial. x : optional polys : bool, optional If True, return a Poly, otherwise (default) return an expression. Examples ======== >>> from sympy.polys.orthopolys import spherical_bessel_fn as fn >>> from sympy import Symbol >>> z = Symbol("z") >>> fn(1, z) z**(-2) >>> fn(2, z) -1/z + 3/z**3 >>> fn(3, z) -6/z**2 + 15/z**4 >>> fn(4, z) 1/z - 45/z**3 + 105/z**5 Nr$rr)rrOrLr absr r )rr$r%fs r spherical_bessel_fnrTsSJ y #JJ)*Q%%4KA c!ffaR"Q%%'U ; ;;r")NF)NrF)#__doc__sympy.core.symbolrsympy.polys.densearithrrrrrsympy.polys.domainsr r sympy.polys.polytoolsr sympy.utilitiesr r!r&r*r,r0r2r5r7r;r=r?rArCrErHrJrLrOrTr"r r\sc@@######""""""""""""""&&&&&&&&,,,,,,"""""" RRRR$   WWWW  C C C C D D D D       M M M M = = = =    O O O OWWWW    (<(<(<(<(<(