o 8Vap'@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$d-d+d,Z%d S)/z;Efficient functions for generating orthogonal polynomials. )Dummy)construct_domain)dup_muldup_mul_ground dup_lshiftdup_subdup_add)ZZQQ)DMP)PolyPurePoly)publicc Cs|jg|||d|d|||dgg}td|dD]}||||||||d||d}|||d||j|||||d|}|||d||j|||d||d|||d||d|}|||j|||j|||d||} t|d||} tt|dd|||} t|d| |} |tt| | || |q||S)z0Low-level implementation of Jacobi polynomials. )onerangerrappendrr) nabKseqiZdenZf0f1f2Zp0p1p2r 8/usr/lib/python3/dist-packages/sympy/polys/orthopolys.py dup_jacobis006V4r"NFcCs~|dkr td|t||gdd\}}ttt||d|d||}|dur/t||}nt|td}|r;|S| S)aGenerates Jacobi polynomial of degree `n` in `x`. Parameters ========== n : int `n` decides the degree of polynomial a Lower limit of minimal domain for the list of coefficients. b Upper limit of minimal domain for the list of coefficients. x : optional polys : bool, optional ``polys=True`` returns an expression, otherwise (default) returns an expression. rz-can't generate Jacobi polynomial of degree %sTZfieldrNx) ValueErrorrr r"intr newr ras_expr)rrrr$polysrvpolyr r r! jacobi_poly s  r,c Cs|jg|d||jgg}td|dD]:}|d|||j|}||d||d|}tt|dd|||}t|d||}|t|||q||S)z4Low-level implementation of Gegenbauer polynomials. rrrrrzerorrrrr) rrrrrrrrrr r r!dup_gegenbauerBsr/cCsp|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. rz1can't generate Gegenbauer polynomial of degree %sTr#Nr$) r%rr r/r&r r'r rr()rrr$r)rr+r r r!gegenbauer_polyPsr0cCsb|jg|j|jgg}td|dD]}tt|dd||d|}|t||d|q||S)zCLow-level implementation of Chebyshev polynomials of the 1st kind. rrrrr-rrrrrr r r!dup_chebyshevtos r2cC^|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. rz9can't generate 1st kind Chebyshev polynomial of degree %sNr$) r%r r2r&r r r'r rr(rr$r)r+r r r!chebyshevt_polyzr5cCsd|jg|d|jgg}td|dD]}tt|dd||d|}|t||d|q||S)zCLow-level implementation of Chebyshev polynomials of the 2nd kind. rrrrr-r1r r r!dup_chebyshevus r7cCr3)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. rz9can't generate 2nd kind Chebyshev polynomial of degree %sNr$) r%r r7r&r r r'r rr(r4r r r!chebyshevu_polyr6r8cCs||jg|d|jgg}td|dD]'}t|dd|}t|d||d|}tt||||d|}||q||S)z1Low-level implementation of Hermite polynomials. rrrr)rr.rrrrr)rrrrrrcr r r! dup_hermites r:cCr3)aGenerates Hermite polynomial of degree `n` in `x`. Parameters ========== n : int `n` decides the degree of polynomial x : optional polys : bool, optional ``polys=True`` returns an expression, otherwise (default) returns an expression. rz.can't generate Hermite polynomial of degree %sNr$) r%r r:r&r r r'r rr(r4r 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. rrrrr-)rrrrrrr r r! dup_legendres &r=cCr3)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. rz/can't generate Legendre polynomial of degree %sNr$) r%r r=r&r r r'r rr(r4r r r! legendre_polyr<r>cCs|jg|jgg}td|dD]7}t|d|j ||||d|d|g|}t|d||||d||}|t|||q|dS)z2Low-level implementation of Laguerre polynomials. rrrr)r.rrrrrr)ralpharrrrrr r r! dup_laguerres 4$r@cCs|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. rz/can't generate Laguerre polynomial of degree %sNTr#r$) r%rr r r@r&r r'r rr()rr$r?r)rr+r r r! laguerre_polys  rAcCsr|jg|j|jgg}td|dD]}tt|dd||d|d|}|t||d|qt||d|S)z& Low-level implementation of fn(n, x) rrrrr-r1r r r!dup_spherical_bessel_fn@s $rBcCsj|j|jg|jgg}td|dD]}tt|dd||dd||}|t||d|q||S)z' Low-level implementation of fn(-n, x) rrrrr-r1r r r!dup_spherical_bessel_fn_minusKs $rDcCsp|dkr tt| t}ntt|t}t|t}|dur&t|d|}n t|dtd}|r4|S| S)a  Coefficients for the spherical Bessel functions. Those are only needed in the jn() function. The coefficients are calculated from: fn(0, z) = 1/z fn(1, z) = 1/z**2 fn(n-1, z) + fn(n+1, z) == (2*n+1)/z * fn(n, z) Parameters ========== n : int `n` decides the degree of polynomial x : optional polys : bool, optional ``polys=True`` returns an expression, otherwise (default) returns an expression. Examples ======== >>> from sympy.polys.orthopolys import spherical_bessel_fn as fn >>> from sympy import Symbol >>> z = Symbol("z") >>> fn(1, z) z**(-2) >>> fn(2, z) -1/z + 3/z**3 >>> fn(3, z) -6/z**2 + 15/z**4 >>> fn(4, z) 1/z - 45/z**3 + 105/z**5 rNrr$) rDr&r rBr r r'r rr()rr$r)dupr+r r r!spherical_bessel_fnVs' rF)NF)NNF)&__doc__ZsympyrZsympy.polys.constructorrZsympy.polys.densearithrrrrrZsympy.polys.domainsr r Zsympy.polys.polyclassesr Zsympy.polys.polytoolsr r Zsympy.utilitiesrr"r,r/r0r2r5r7r8r:r;r=r>r@rArBrDrFr r r r!s>     !      #