B ['@s@dZddlmZmZddlmZddlmZddlm Z ddl m Z m Z ddl mZddlmZmZmZmZmZdd lmZmZdd lmZd d Zed/ddZddZd0ddZddZed1ddZddZ ed2ddZ!ddZ"ed3dd Z#d!d"Z$ed4d#d$Z%d%d&Z&ed5d'd(Z'd)d*Z(d+d,Z)d6d-d.Z*d S)7z;Efficient functions for generating orthogonal polynomials. )print_functiondivision)Dummy)public)construct_domain)PolyPurePoly)DMP)dup_muldup_mul_ground dup_lshiftdup_subdup_add)ZZQQ)rangec Cs|jg|||d|d|||dgg}xZtd|dD]F}||||||||d||d}|||d||j|||||d|}|||d||j|||d||d|||d||d|}|||j|||j|||d||} t|d||} tt|dd|||} t|d| |} |tt| | || |qBW||S)z0Low-level implementation of Jacobi polynomials. )onerr r appendr r) nabKseqiZdenZf0f1f2Zp0p1p2r"5lib/python3.7/site-packages/sympy/polys/orthopolys.py dup_jacobis006V4r$NFcCs~|dkrtd|t||gdd\}}ttt||d|d||}|dk r^t||}nt|td}|rv|S| S)aGenerates Jacobi polynomial of degree `n` in `x`. Parameters ---------- n : int `n` decides the degree of polynomial a Lower limit of minimal domain for the list of coefficients. b Upper limit of minimal domain for the list of coefficients. x : optional polys : bool, optional ``polys=True`` returns an expression, otherwise (default) returns an expression. rz-can't generate Jacobi polynomial of degree %sT)fieldrNx) ValueErrorrr r$intrnewrras_expr)rrrr&polysrvpolyr"r"r# jacobi_poly's  r.c Cs|jg|d||jgg}xtd|dD]t}|d|||j|}||d||d|}tt|dd|||}t|d||}|t|||q*W||S)z4Low-level implementation of Gegenbauer polynomials. rrrr)rzerorr r rr ) rrrrrrrr r!r"r"r#dup_gegenbauerHsr0cCsp|dkrtd|t|dd\}}ttt||||}|dk rPt||}nt|td}|rh|S| S)ajGenerates 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 %sT)r%Nr&) r'rr r0r(rr)rrr*)rrr&r+rr-r"r"r#gegenbauer_polyVs r1cCsf|jg|j|jgg}xHtd|dD]6}tt|dd||d|}|t||d|q$W||S)zCLow-level implementation of Chebyshev polynomials of the 1st kind. rrrr)rr/rr r rr )rrrrrr"r"r#dup_chebyshevtts r2cCs^|dkrtd|ttt|tt}|dk r>t||}nt|td}|rV|S| S)a0Generates Chebyshev polynomial of the first kind of degree `n` in `x`. Parameters ---------- n : int `n` decides the degree of polynomial x : optional polys : bool, optional ``polys=True`` returns an expression, otherwise (default) returns an expression. rz9can't generate 1st kind Chebyshev polynomial of degree %sNr&) r'r r2r(rrr)rrr*)rr&r+r-r"r"r#chebyshevt_polys  r3cCsh|jg|d|jgg}xHtd|dD]6}tt|dd||d|}|t||d|q&W||S)zCLow-level implementation of Chebyshev polynomials of the 2nd kind. rrrr)rr/rr r rr )rrrrrr"r"r#dup_chebyshevus r4cCs^|dkrtd|ttt|tt}|dk r>t||}nt|td}|rV|S| S)a1Generates Chebyshev polynomial of the second kind of degree `n` in `x`. Parameters ---------- n : int `n` decides the degree of polynomial x : optional polys : bool, optional ``polys=True`` returns an expression, otherwise (default) returns an expression. rz9can't generate 2nd kind Chebyshev polynomial of degree %sNr&) r'r r4r(rrr)rrr*)rr&r+r-r"r"r#chebyshevu_polys  r5cCs|jg|d|jgg}x`td|dD]N}t|dd|}t|d||d|}tt||||d|}||q&W||S)z1Low-level implementation of Hermite polynomials. rrrr)rr/rr r r r)rrrrrrcr"r"r# dup_hermitesr7cCs^|dkrtd|ttt|tt}|dk r>t||}nt|td}|rV|S| S)aGenerates Hermite polynomial of degree `n` in `x`. Parameters ---------- n : int `n` decides the degree of polynomial x : optional polys : bool, optional ``polys=True`` returns an expression, otherwise (default) returns an expression. rz.can't generate Hermite polynomial of degree %sNr&) r'r r7r(rrr)rrr*)rr&r+r-r"r"r# hermite_polys  r8cCs|jg|j|jgg}xhtd|dD]V}tt|dd||d|d||}t|d||d||}|t|||q$W||S)z2Low-level implementation of Legendre polynomials. rrrr)rr/rr r rr )rrrrrrr"r"r# dup_legendres &r9cCs^|dkrtd|ttt|tt}|dk r>t||}nt|td}|rV|S| 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. rz/can't generate Legendre polynomial of degree %sNr&) r'r r9r(rrr)rrr*)rr&r+r-r"r"r# legendre_polys  r:cCs|jg|jgg}xtd|dD]n}t|d|j ||||d|d|g|}t|d||||d||}|t|||q W|dS)z2Low-level implementation of Laguerre polynomials. rrrr)r/rrr r rr )ralpharrrrrr"r"r# dup_laguerres 4$r<cCs|dkrtd||dk r.t|dd\}}nttd}}ttt||||}|dk rht||}nt|t d}|r|S| S)alGenerates Laguerre polynomial of degree `n` in `x`. Parameters ---------- n : int `n` decides the degree of polynomial x : optional alpha Decides minimal domain for the list of coefficients. polys : bool, optional ``polys=True`` returns an expression, otherwise (default) returns an expression. rz/can't generate Laguerre polynomial of degree %sNT)r%r&) r'rrr r<r(rr)rrr*)rr&r;r+rr-r"r"r# laguerre_polys r=cCsv|jg|j|jgg}xPtd|dD]>}tt|dd||d|d|}|t||d|q$Wt||d|S)z& Low-level implementation of fn(n, x) rrrr)rr/rr r rr )rrrrrr"r"r#dup_spherical_bessel_fn@s $r>cCsn|j|jg|jgg}xPtd|dD]>}tt|dd||dd||}|t||d|q$W||S)z' Low-level implementation of fn(-n, x) rrrr)rr/rr r rr )rrrrrr"r"r#dup_spherical_bessel_fn_minusKs $r@cCsp|dkrtt| t}ntt|t}t|t}|dk rLt|d|}nt|dtd}|rh|S| S)a  Coefficients for the spherical Bessel functions. Those are only needed in the jn() function. The coefficients are calculated from: fn(0, z) = 1/z fn(1, z) = 1/z**2 fn(n-1, z) + fn(n+1, z) == (2*n+1)/z * fn(n, z) Parameters ---------- n : int `n` decides the degree of polynomial x : optional polys : bool, optional ``polys=True`` returns an expression, otherwise (default) returns an expression. Examples ======== >>> from sympy.polys.orthopolys import spherical_bessel_fn as fn >>> from sympy import Symbol >>> z = Symbol("z") >>> fn(1, z) z**(-2) >>> fn(2, z) -1/z + 3/z**3 >>> fn(3, z) -6/z**2 + 15/z**4 >>> fn(4, z) 1/z - 45/z**3 + 105/z**5 rNrr&) r@r(rr>r rr)rrr*)rr&r+dupr-r"r"r#spherical_bessel_fnVs& rB)NF)NF)NF)NF)NF)NF)NNF)NF)+__doc__Z __future__rrZsympyrZsympy.utilitiesrZsympy.polys.constructorrZsympy.polys.polytoolsrrZsympy.polys.polyclassesr Zsympy.polys.densearithr r r r rZsympy.polys.domainsrrZsympy.core.compatibilityrr$r.r0r1r2r3r4r5r7r8r9r:r<r=r>r@rBr"r"r"r#s@                "