o *ήc'@s$dZddlmZddlmZddlmZmZmZm Z m Z ddl m Z m Z ddlmZddlmZmZddlmZd d Zed-d dZddZd-ddZddZed-ddZddZed-ddZddZed-ddZdd Zed-d!d"Z d#d$Z!ed.d%d&Z"d'd(Z#d)d*Z$d-d+d,Z%d S)/z;Efficient functions for generating orthogonal polynomials. )Dummy)construct_domain)dup_muldup_mul_ground dup_lshiftdup_subdup_add)ZZQQ)DMP)PolyPurePoly)publicc Cs|jg|||d|d|||dgg}td|dD]}||||||||d||d}|||d||j|||||d|}|||d||j|||d||d|||d||d|}|||j|||j|||d||} t|d||} tt|dd|||} t|d| |} |tt| | || |q||S)z0Low-level implementation of Jacobi polynomials. )onerangerrappendrr) nabKseqidenf0f1f2p0p1p2r#=/tmp/pip-target-vg8gfxp4/lib/python/sympy/polys/orthopolys.py dup_jacobis006V4r%NFcCs~|dkr td|t||gdd\}}ttt||d|d||}|dur/t||}nt|td}|r;|S| S)aGenerates Jacobi polynomial of degree `n` in `x`. Parameters ========== n : int `n` decides the degree of polynomial a Lower limit of minimal domain for the list of coefficients. b Upper limit of minimal domain for the list of coefficients. x : optional polys : bool, optional ``polys=True`` returns an expression, otherwise (default) returns an expression. rz.Cannot generate Jacobi polynomial of degree %sTfieldrNx) ValueErrorrr r%intr newr ras_expr)rrrr(polysrvpolyr#r#r$ jacobi_poly s  r0c Cs|jg|d||jgg}td|dD]:}|d|||j|}||d||d|}tt|dd|||}t|d||}|t|||q||S)z4Low-level implementation of Gegenbauer polynomials. rrrrrzerorrrrr) rrrrrrrr!r"r#r#r$dup_gegenbauerBsr3cCsp|dkr td|t|dd\}}ttt||||}|dur(t||}nt|td}|r4|S| S)akGenerates Gegenbauer polynomial of degree `n` in `x`. Parameters ========== n : int `n` decides the degree of polynomial x : optional a Decides minimal domain for the list of coefficients. polys : bool, optional ``polys=True`` returns an expression, otherwise (default) returns an expression. rz2Cannot generate Gegenbauer polynomial of degree %sTr&Nr() r)rr r3r*r r+r rr,)rrr(r-rr/r#r#r$gegenbauer_polyPsr4cCsb|jg|j|jgg}td|dD]}tt|dd||d|}|t||d|q||S)zCLow-level implementation of Chebyshev polynomials of the 1st kind. rrrrr1rrrrrr#r#r$dup_chebyshevtos r6cC^|dkr td|ttt|tt}|durt||}nt|td}|r+|S| S)a1Generates Chebyshev polynomial of the first kind of degree `n` in `x`. Parameters ========== n : int `n` decides the degree of polynomial x : optional polys : bool, optional ``polys=True`` returns an expression, otherwise (default) returns an expression. rz:Cannot generate 1st kind Chebyshev polynomial of degree %sNr() r)r r6r*r r r+r rr,rr(r-r/r#r#r$chebyshevt_polyzr9cCsd|jg|d|jgg}td|dD]}tt|dd||d|}|t||d|q||S)zCLow-level implementation of Chebyshev polynomials of the 2nd kind. rrrrr1r5r#r#r$dup_chebyshevus r;cCr7)a2Generates Chebyshev polynomial of the second kind of degree `n` in `x`. Parameters ========== n : int `n` decides the degree of polynomial x : optional polys : bool, optional ``polys=True`` returns an expression, otherwise (default) returns an expression. rz:Cannot generate 2nd kind Chebyshev polynomial of degree %sNr() r)r r;r*r r r+r rr,r8r#r#r$chebyshevu_polyr:r<cCs||jg|d|jgg}td|dD]'}t|dd|}t|d||d|}tt||||d|}||q||S)z1Low-level implementation of Hermite polynomials. rrrr)rr2rrrrr)rrrrrrcr#r#r$ dup_hermites r>cCr7)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 r>r*r r r+r rr,r8r#r#r$ hermite_poly r?cCs|jg|j|jgg}td|dD]+}tt|dd||d|d||}t|d||d||}|t|||q||S)z2Low-level implementation of Legendre polynomials. rrrrr1)rrrrrrr#r#r$ dup_legendres &rAcCr7)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 rAr*r r r+r rr,r8r#r#r$ legendre_polyr@rBcCs|jg|jgg}td|dD]7}t|d|j ||||d|d|g|}t|d||||d||}|t|||q|dS)z2Low-level implementation of Laguerre polynomials. rrrr)r2rrrrrr)ralpharrrrrr#r#r$ dup_laguerres 4$rDcCs|dkr td||durt|dd\}}nttd}}ttt||||}|dur4t||}nt|t d}|r@|S| S)amGenerates Laguerre polynomial of degree `n` in `x`. Parameters ========== n : int `n` decides the degree of polynomial x : optional alpha Decides minimal domain for the list of coefficients. polys : bool, optional ``polys=True`` returns an expression, otherwise (default) returns an expression. rz0Cannot generate Laguerre polynomial of degree %sNTr&r() r)rr r rDr*r r+r rr,)rr(rCr-rr/r#r#r$ laguerre_polys  rEcCsr|jg|j|jgg}td|dD]}tt|dd||d|d|}|t||d|qt||d|S)z& Low-level implementation of fn(n, x) rrrrr1r5r#r#r$dup_spherical_bessel_fn@s $rFcCsj|j|jg|jgg}td|dD]}tt|dd||dd||}|t||d|q||S)z' Low-level implementation of fn(-n, x) rrrrr1r5r#r#r$dup_spherical_bessel_fn_minusKs $rHcCsp|dkr tt| t}ntt|t}t|t}|dur&t|d|}n t|dtd}|r4|S| S)a  Coefficients for the spherical Bessel functions. Those are only needed in the jn() function. The coefficients are calculated from: fn(0, z) = 1/z fn(1, z) = 1/z**2 fn(n-1, z) + fn(n+1, z) == (2*n+1)/z * fn(n, z) Parameters ========== n : int `n` decides the degree of polynomial x : optional polys : bool, optional ``polys=True`` returns an expression, otherwise (default) returns an expression. Examples ======== >>> from sympy.polys.orthopolys import spherical_bessel_fn as fn >>> from sympy import Symbol >>> z = Symbol("z") >>> fn(1, z) z**(-2) >>> fn(2, z) -1/z + 3/z**3 >>> fn(3, z) -6/z**2 + 15/z**4 >>> fn(4, z) 1/z - 45/z**3 + 105/z**5 rNrr() rHr*r rFr r r+r rr,)rr(r-dupr/r#r#r$spherical_bessel_fnVs' rJ)NF)NNF)&__doc__sympy.core.symbolrsympy.polys.constructorrsympy.polys.densearithrrrrrsympy.polys.domainsr r sympy.polys.polyclassesr sympy.polys.polytoolsr r sympy.utilitiesrr%r0r3r4r6r9r;r<r>r?rArBrDrErFrHrJr#r#r#r$s>     !      #