ó ¡¼™\c@ s´dZddlmZmZddlmZddlmZddlm Z ddl m Z m Z m Z mZmZddlmZmZddlmZdd lmZmZdd lmZd „Zeded „ƒZd „Zded„Zd„Z eded„ƒZ!d„Z"eded„ƒZ#d„Z$eded„ƒZ%d„Z&eded„ƒZ'd„Z(edded„ƒZ)d„Z*d„Z+ded„Z,dS(s;Efficient functions for generating orthogonal polynomials. iÿÿÿÿ(tprint_functiontdivision(tDummy(trange(tconstruct_domain(tdup_multdup_mul_groundt dup_lshifttdup_subtdup_add(tZZtQQ(tDMP(tPolytPurePoly(tpublicc C sé|jg|||dƒ|dƒ|||dƒgg}x td|dƒD]‹}||ƒ||||||dƒ||dƒ}|||dƒ||j|||||dƒ|}|||dƒ||j|||dƒ||dƒ|||dƒ||dƒ|}|||j|||j|||dƒ||} t|d||ƒ} tt|dd|ƒ||ƒ} t|d| |ƒ} |jtt| | |ƒ| |ƒƒqRW||S(s0Low-level implementation of Jacobi polynomials. iiiÿÿÿÿiþÿÿÿ(toneRRRtappendRR ( tntatbtKtseqtitdentf0tf1tf2tp0tp1tp2((s5lib/python2.7/site-packages/sympy/polys/orthopolys.pyt dup_jacobis>8=c:")cC s¶|dkrtd|ƒ‚nt||gdtƒ\}}ttt|ƒ|d|d|ƒ|ƒ}|dk rŠtj||ƒ}nt j|t dƒƒ}|r¬|S|j ƒS(sÀGenerates Jacobi polynomial of degree `n` in `x`. Parameters ========== n : int `n` decides the degree of polynomial a Lower limit of minimal domain for the list of coefficients. b Upper limit of minimal domain for the list of coefficients. x : optional polys : bool, optional ``polys=True`` returns an expression, otherwise (default) returns an expression. is-can't generate Jacobi polynomial of degree %stfielditxN( t ValueErrorRtTrueR RtinttNoneR tnewRRtas_expr(RRRR!tpolysRtvtpoly((s5lib/python2.7/site-packages/sympy/polys/orthopolys.pyt jacobi_poly"s , c C sÝ|jg|dƒ||jgg}x­td|dƒD]˜}|dƒ|||j|}||dƒ||dƒ|}tt|dd|ƒ||ƒ}t|d||ƒ}|jt|||ƒƒq9W||S(s4Low-level implementation of Gegenbauer polynomials. iiiÿÿÿÿiþÿÿÿ(RtzeroRRRRR( RRRRRRRRR((s5lib/python2.7/site-packages/sympy/polys/orthopolys.pytdup_gegenbauerDs%""cC s¥|dkrtd|ƒ‚nt|dtƒ\}}ttt|ƒ||ƒ|ƒ}|dk rytj||ƒ}nt j|t dƒƒ}|r›|S|j ƒS(skGenerates Gegenbauer polynomial of degree `n` in `x`. Parameters ========== n : int `n` decides the degree of polynomial x : optional a Decides minimal domain for the list of coefficients. polys : bool, optional ``polys=True`` returns an expression, otherwise (default) returns an expression. is1can't generate Gegenbauer polynomial of degree %sR R!N( R"RR#R R-R$R%R R&RRR'(RRR!R(RR*((s5lib/python2.7/site-packages/sympy/polys/orthopolys.pytgegenbauer_polyRs ! cC s‰|jg|j|jgg}x`td|dƒD]K}tt|dd|ƒ|dƒ|ƒ}|jt||d|ƒƒq2W||S(sCLow-level implementation of Chebyshev polynomials of the 1st kind. iiiÿÿÿÿiþÿÿÿ(RR,RRRRR(RRRRR((s5lib/python2.7/site-packages/sympy/polys/orthopolys.pytdup_chebyshevtqs (!cC sŠ|dkrtd|ƒ‚nttt|ƒtƒtƒ}|dk r^tj||ƒ}ntj|t dƒƒ}|r€|S|j ƒS(s1Generates 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. is9can't generate 1st kind Chebyshev polynomial of degree %sR!N( R"R R/R$R R%R R&RRR'(RR!R(R*((s5lib/python2.7/site-packages/sympy/polys/orthopolys.pytchebyshevt_poly|s  cC sŒ|jg|dƒ|jgg}x`td|dƒD]K}tt|dd|ƒ|dƒ|ƒ}|jt||d|ƒƒq5W||S(sCLow-level implementation of Chebyshev polynomials of the 2nd kind. iiiÿÿÿÿiþÿÿÿ(RR,RRRRR(RRRRR((s5lib/python2.7/site-packages/sympy/polys/orthopolys.pytdup_chebyshevu˜s !(!cC sŠ|dkrtd|ƒ‚nttt|ƒtƒtƒ}|dk r^tj||ƒ}ntj|t dƒƒ}|r€|S|j ƒS(s2Generates Chebyshev polynomial of the second kind of degree `n` in `x`. Parameters ========== n : int `n` decides the degree of polynomial x : optional polys : bool, optional ``polys=True`` returns an expression, otherwise (default) returns an expression. is9can't generate 2nd kind Chebyshev polynomial of degree %sR!N( R"R R1R$R R%R R&RRR'(RR!R(R*((s5lib/python2.7/site-packages/sympy/polys/orthopolys.pytchebyshevu_poly£s  cC s®|jg|dƒ|jgg}x‚td|dƒD]m}t|dd|ƒ}t|d||dƒ|ƒ}tt|||ƒ|dƒ|ƒ}|j|ƒq5W||S(s1Low-level implementation of Hermite polynomials. iiiÿÿÿÿiþÿÿÿ(RR,RRRRR(RRRRRRtc((s5lib/python2.7/site-packages/sympy/polys/orthopolys.pyt dup_hermite¿s! $cC sŠ|dkrtd|ƒ‚nttt|ƒtƒtƒ}|dk r^tj||ƒ}ntj|t dƒƒ}|r€|S|j ƒS(sGenerates Hermite polynomial of degree `n` in `x`. Parameters ========== n : int `n` decides the degree of polynomial x : optional polys : bool, optional ``polys=True`` returns an expression, otherwise (default) returns an expression. is.can't generate Hermite polynomial of degree %sR!N( R"R R4R$R R%R R&RRR'(RR!R(R*((s5lib/python2.7/site-packages/sympy/polys/orthopolys.pyt hermite_polyÎs  cC s³|jg|j|jgg}xŠtd|dƒD]u}tt|dd|ƒ|d|d|ƒ|ƒ}t|d||d|ƒ|ƒ}|jt|||ƒƒq2W||S(s2Low-level implementation of Legendre polynomials. iiiÿÿÿÿiþÿÿÿ(RR,RRRRR(RRRRRR((s5lib/python2.7/site-packages/sympy/polys/orthopolys.pyt dup_legendreés 3#cC sŠ|dkrtd|ƒ‚nttt|ƒtƒtƒ}|dk r^tj||ƒ}ntj|t dƒƒ}|r€|S|j ƒS(sGenerates 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. is/can't generate Legendre polynomial of degree %sR!N( R"R R6R$R R%R R&RRR'(RR!R(R*((s5lib/python2.7/site-packages/sympy/polys/orthopolys.pyt legendre_polyös  cC sÁ|jg|jgg}xžtd|dƒD]‰}t|d|j ||||d|dƒ|g|ƒ}t|d||||dƒ||ƒ}|jt|||ƒƒq,W|dS(s2Low-level implementation of Laguerre polynomials. iiÿÿÿÿiiþÿÿÿ(R,RRRRRR(RtalphaRRRRR((s5lib/python2.7/site-packages/sympy/polys/orthopolys.pyt dup_laguerres >,cC sÇ|dkrtd|ƒ‚n|dk rFt|dtƒ\}}nttdƒ}}ttt|ƒ||ƒ|ƒ}|dk r›tj ||ƒ}nt j |t dƒƒ}|r½|S|j ƒS(smGenerates Laguerre polynomial of degree `n` in `x`. Parameters ========== n : int `n` decides the degree of polynomial x : optional alpha Decides minimal domain for the list of coefficients. polys : bool, optional ``polys=True`` returns an expression, otherwise (default) returns an expression. is/can't generate Laguerre polynomial of degree %sR R!N( R"R%RR#R R R9R$R R&RRR'(RR!R8R(RR*((s5lib/python2.7/site-packages/sympy/polys/orthopolys.pyt laguerre_polys  ! cC s|jg|j|jgg}xhtd|dƒD]S}tt|dd|ƒ|d|dƒ|ƒ}|jt||d|ƒƒq2Wt||d|ƒS(s& Low-level implementation of fn(n, x) iiiÿÿÿÿiþÿÿÿ(RR,RRRRR(RRRRR((s5lib/python2.7/site-packages/sympy/polys/orthopolys.pytdup_spherical_bessel_fnBs 0!cC s‘|j|jg|jgg}xhtd|dƒD]S}tt|dd|ƒ|dd|ƒ|ƒ}|jt||d|ƒƒq2W||S(s' Low-level implementation of fn(-n, x) iiiÿÿÿÿiiþÿÿÿ(RR,RRRRR(RRRRR((s5lib/python2.7/site-packages/sympy/polys/orthopolys.pytdup_spherical_bessel_fn_minusMs 0!cC sž|dkr%tt|ƒ tƒ}ntt|ƒtƒ}t|tƒ}|dk rntj|d|ƒ}ntj|dt dƒƒ}|r”|S|j ƒS(s  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 iiR!N( R<R$R R;R R%R R&RRR'(RR!R(tdupR*((s5lib/python2.7/site-packages/sympy/polys/orthopolys.pytspherical_bessel_fnXs'  N(-t__doc__t __future__RRtsympyRtsympy.core.compatibilityRtsympy.polys.constructorRtsympy.polys.densearithRRRRR tsympy.polys.domainsR R tsympy.polys.polyclassesR tsympy.polys.polytoolsR Rtsympy.utilitiesRRR%tFalseR+R-R.R/R0R1R2R4R5R6R7R9R:R;R<R>(((s5lib/python2.7/site-packages/sympy/polys/orthopolys.pyts@( !      #