Ed'LdZddlmZddlmZddlmZmZmZm Z m Z ddl m Z m Z ddlmZddlmZmZddlmZd Zedd Zd ZddZdZeddZdZeddZdZeddZdZeddZ dZ!eddZ"dZ#dZ$ddZ%d S)z;Efficient functions for generating orthogonal polynomials. )Dummy)construct_domain)dup_muldup_mul_ground dup_lshiftdup_subdup_add)ZZQQ)DMP)PolyPurePoly)publicc |jg||z|dz|dz ||z |dz gg}td|dzD]}||||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|d||} tt|dd|||} t|d| |} |t t | | || |||S)z0Low-level implementation of Jacobi polynomials. )onerangerrappendrr ) nabKseqidenf0f1f2p0p1p2s 6lib/python3.11/site-packages/sympy/polys/orthopolys.py dup_jacobir&s E7a!eaaddlAAaDD(1q5!!A$$,7 8C 1a!e__77addAEAIA!Q1 56!eaadd1fnqu$1qs 3qqttCx @!eaadd1fnqu$Q1a!!A$$)> ?1q511Q44PQ6> RVWVWXYVZVZ[^V^ _!eaema!eaem ,a!eaadd1fn = C CGR + + Js2w155r1 = = CGR + + 772r1--r1556666 q6MNFct|dkrtd|zt||gd\}}ttt ||d|d||}|t j||}n"tj|td}|r|n| S)aGenerates Jacobi polynomial of degree `n` in `x`. Parameters ========== n : int `n` decides the degree of 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 ``polys=True`` returns an expression, otherwise (default) returns an expression. rz.Cannot generate Jacobi polynomial of degree %sTfieldrNx) ValueErrorrr r&intr newrras_expr)rrrr+polysrvpolys r% jacobi_polyr3 s( 1uOIAMNNN QF$ / / /DAq z#a&&!A$!a00! 4 4D.xa  |D%**-- ,44dllnn,r'c|jg|d|z|jgg}td|dzD]}|d||z|jz z|z }||d|zz|dz |z }tt |dd|||}t|d||}|t |||||S)z4Low-level implementation of Gegenbauer polynomials. rrrrrzerorrrrr) rrrrrr r!r#r$s r%dup_gegenbauerr7Bs E7QQqTT!VQV$ %C 1a!e__'' QqTTQUQU] #a '!!A$$q&j11Q441 $ Js2w155r1 = = CGR + + 72r1%%&&&& q6Mr'cV|dkrtd|zt|d\}}ttt ||||}|t j||}n"tj|td}|r|n| S)akGenerates Gegenbauer polynomial of degree `n` in `x`. Parameters ========== n : int `n` decides the degree of polynomial x : optional a Decides minimal domain for the list of coefficients. polys : bool, optional ``polys=True`` returns an expression, otherwise (default) returns an expression. rz2Cannot generate Gegenbauer polynomial of degree %sTr)Nr+) r,rr r7r-r r.rrr/)rrr+r0rr2s r%gegenbauer_polyr9Ps  1uF @1 DFF F AT * * *DAq ~c!ffa++Q / /D.xa  |D%**-- ,44dllnn,r'c|jg|j|jgg}td|dzD][}tt |dd||d|}|t ||d|\||S)zCLow-level implementation of Chebyshev polynomials of the 1st kind. rrrrr5rrrrrs r%dup_chebyshevtr<os E7QUAFO $C 1a!e__++ :c"gq!44aaddA > > 71c"gq))**** q6Mr'c@|dkrtd|zttt|tt}|t j||}n"tj|td}|r|n| S)a1Generates Chebyshev polynomial of the first kind of degree `n` in `x`. Parameters ========== n : int `n` decides the degree of polynomial x : optional polys : bool, optional ``polys=True`` returns an expression, otherwise (default) returns an expression. rz:Cannot generate 1st kind Chebyshev polynomial of degree %sNr+) r,r r<r-r r r.rrr/rr+r0r2s r%chebyshevt_polyr?z 1uN H1 LNN N ~c!ffb))2 . .D.xa  |D%**-- ,44dllnn,r'c$|jg|d|jgg}td|dzD][}tt |dd||d|}|t ||d|\||S)zCLow-level implementation of Chebyshev polynomials of the 2nd kind. rrrrr5r;s r%dup_chebyshevurBs E7QQqTT16N #C 1a!e__++ :c"gq!44aaddA > > 71c"gq))**** q6Mr'c@|dkrtd|zttt|tt}|t j||}n"tj|td}|r|n| S)a2Generates Chebyshev polynomial of the second kind of degree `n` in `x`. Parameters ========== n : int `n` decides the degree of polynomial x : optional polys : bool, optional ``polys=True`` returns an expression, otherwise (default) returns an expression. rz:Cannot generate 2nd kind Chebyshev polynomial of degree %sNr+) r,r rBr-r r r.rrr/r>s r%chebyshevu_polyrDr@r'cb|jg|d|jgg}td|dzD]z}t|dd|}t |d||dz |}t t ||||d|}||{||S)z1Low-level implementation of Hermite polynomials. rrrr)rr6rrrrr)rrrrrrcs r% dup_hermiterGs E7QQqTT16N #C 1a!e__ s2w1 % % 3r7AAa!eHHa 0 0 71a++QQqTT1 5 5 1 q6Mr'c@|dkrtd|zttt|tt}|t j||}n"tj|td}|r|n| S)aGenerates Hermite polynomial of degree `n` in `x`. Parameters ========== n : int `n` decides the degree of polynomial x : optional polys : bool, optional ``polys=True`` returns an expression, otherwise (default) returns an expression. rz/Cannot generate Hermite polynomial of degree %sNr+) r,r rGr-r r r.rrr/r>s r% hermite_polyrIs 1uPJQNOOO {3q662&& + +D.xa  |D%**-- ,44dllnn,r'cf|jg|j|jgg}td|dzD]}tt |dd||d|zdz ||}t|d||dz ||}|t |||||S)z2Low-level implementation of Legendre polynomials. rrrrr5)rrrrrrs r% dup_legendrerKs E7QUAFO $C 1a!e__%% :c"gq!44aa!ammQ G G 3r7AAa!eQKK 3 3 71a##$$$$ q6Mr'c@|dkrtd|zttt|tt}|t j||}n"tj|td}|r|n| S)aGenerates Legendre polynomial of degree `n` in `x`. Parameters ========== n : int `n` decides the degree of polynomial x : optional polys : bool, optional ``polys=True`` returns an expression, otherwise (default) returns an expression. rz0Cannot generate Legendre polynomial of degree %sNr+) r,r rKr-r r r.rrr/r>s r% legendre_polyrMs 1uQKaOPPP |CFFB'' , ,D.xa  |D%**-- ,44dllnn,r'c r|jg|jgg}td|dzD]}t|d|j |z ||z |d|zdz |z zg|}t |d||z ||dz |z z|}|t ||||dS)z2Low-level implementation of Laguerre polynomials. rrrr)r6rrrrrr)ralpharrrrrs r% dup_laguerrerPs F8aeW C 1a!e__%% CGqufQha!!AaC!G**Q,(>? C C 3r7E!GaaAhhqj$8! < < 71a##$$$$ r7Nr'c|dkrtd|z|t|d\}}nttd}}tt t ||||}|t j||}n"tj|td}|r|n| S)amGenerates Laguerre polynomial of degree `n` in `x`. Parameters ========== n : int `n` decides the degree of polynomial x : optional alpha Decides minimal domain for the list of coefficients. polys : bool, optional ``polys=True`` returns an expression, otherwise (default) returns an expression. rz0Cannot generate Laguerre polynomial of degree %sNTr)r+) r,rr r rPr-r r.rrr/)rr+rOr0rr2s r% laguerre_polyrRs" 1uQKaOPPP # 55r!uu5 |CFFE1--q 1 1D.xa  |D%**-- ,44dllnn,r'cF|jg|j|jgg}td|dzD]a}tt |dd||d|zdz |}|t ||d|bt ||d|S)z& Low-level implementation of fn(n, x) rrrrr5r;s r%dup_spherical_bessel_fnrT@s E7QUAFO $C 1a!e__++ :c"gq!44aa!ajj! D D 71c"gq))**** c!fa # ##r'c (|j|jg|jgg}td|dzD]a}tt |dd||dd|zz |}|t ||d|b||S)z' Low-level implementation of fn(-n, x) rrrrr5r;s r%dup_spherical_bessel_fn_minusrWKs E16?QVH %C 1a!e__++ :c"gq!44aaAaCjj! D D 71c"gq))**** q6Mr'ct|dkr$tt| t}n"tt|t}t |t}|t j|d|z }n%tj|dtdz }|r|n| S)a  Coefficients for the spherical Bessel functions. Those are only needed in the jn() function. The coefficients are calculated from: fn(0, z) = 1/z fn(1, z) = 1/z**2 fn(n-1, z) + fn(n+1, z) == (2*n+1)/z * fn(n, z) Parameters ========== n : int `n` decides the degree of polynomial x : optional polys : bool, optional ``polys=True`` returns an expression, otherwise (default) returns an expression. Examples ======== >>> from sympy.polys.orthopolys import spherical_bessel_fn as fn >>> from sympy import Symbol >>> z = Symbol("z") >>> fn(1, z) z**(-2) >>> fn(2, z) -1/z + 3/z**3 >>> fn(3, z) -6/z**2 + 15/z**4 >>> fn(4, z) 1/z - 45/z**3 + 105/z**5 rNrr+) rWr-r rTr r r.rrr/)rr+r0dupr2s r%spherical_bessel_fnrZVsN 1u2+SVVGR88%c!ffb11 sB<rdsHAA$#####444444'&&&&&&&''''''00000000"""""""----B   ---->----6----6   ----4   ----4    - - - -F$$$3-3-3-3-3-3-r'