Otj'fdZddlmZddlmZmZmZmZmZm Z m Z m Z ddl m Z mZddlmZddlmZdZedd Zd Zdd Zd ZdZdZdZeddZeddZdZdZeddZeddZ dZ!eddZ"dZ#eddZ$dZ%dZ&ddZ'dS) z:Efficient functions for generating orthogonal polynomials.)Dummy)dup_muldup_mul_ground dup_lshiftdup_subdup_add dup_sub_termdup_sub_grounddup_sqr)ZZQQ) named_poly)publicc ~|dkr|jgS|jg||z|dz |jz||z |dz g}}td|dzD]g}||||z|zz||z|d|zz|dz z}||z|d|zz|jz ||z||zz z|d|zz }||z|d|zz|jz ||z|d|zz|dz z||z|d|zzz|d|zz } ||z|jz ||z|jz z||z|d|zzz|z } t|||} tt|d|| |} t|| |} |t t | | || |}}i|S)z/Low-level implementation of Jacobi polynomials.)onerangerrrr)nabKm2m1idenf0f1f2p0p1p2s T/var/www/html/CA-Chatbot/venv/lib/python3.11/site-packages/sympy/polys/orthopolys.py dup_jacobir$ s1uuweW!QQqTTzAE)AaC1:6B 1ac]]88addAEAIA!Q1 56!eaadd1fnqu$1qs 3qqttCx @!eaadd1fnqu$Q1a!!A$$)> ?1q511Q44PQ6> RVWVWXYVZVZ[^V^ _!eaema!eaem ,a!eaadd1fn = C BA & & Jr1a00"a 8 8 BA & &WWRQ//Q77B INFc :t|tdd|||f|S)aGenerates the Jacobi polynomial `P_n^{(a,b)}(x)`. Parameters ========== n : int Degree of the polynomial. a Lower limit of minimal domain for the list of coefficients. b Upper limit of minimal domain for the list of coefficients. x : optional polys : bool, optional If True, return a Poly, otherwise (default) return an expression. NzJacobi polynomial)rr$)rrrxpolyss r# jacobi_polyr)s#" aT+>Aq 5 Q QQr%c|dkr|jgS|jg|d|z|jg}}td|dzD]}tt |d||d||jz z||z |dz|}t||d||jz z||z |jz|}|t |||}}|S)z3Low-level implementation of Gegenbauer polynomials.rrrzerorrrr)rrrrrrr!r"s r#dup_gegenbauerr-,s1uuweWqqttAvqv&B 1ac]](( Jr1a00!!A$$!%.12E!2La P P B!agqqtt 3ae ;Q ? ?WRQ''B Ir%c8t|tdd||f|S)a?Generates the Gegenbauer polynomial `C_n^{(a)}(x)`. Parameters ========== n : int Degree of the polynomial. x : optional a Decides minimal domain for the list of coefficients. polys : bool, optional If True, return a Poly, otherwise (default) return an expression. NzGegenbauer polynomial)rr-)rrr'r(s r#gegenbauer_polyr/7s" a/FAPU V VVr%cj|dkr|jgS|dkrt||St||S)zDLow-level implementation of Chebyshev polynomials of the first kind.r@)r_dup_chebyshevt_rec_dup_chebyshevt_prod)rrs r#dup_chebyshevtr4Gs=1uuw2vv"1a((( 1 % %%r%c |jg|j|jg}}t|dz D]<}|tt t |d||d|||}}=|S)a Chebyshev polynomials of the first kind using recurrence. Explanation =========== Chebyshev polynomials of the first kind are defined by the recurrence relation: .. math:: T_0(x) &= 1\\ T_1(x) &= x\\ T_n(x) &= 2xT_{n-1}(x) - T_{n-2}(x) This function calculates the Chebyshev polynomial of the first kind using the above recurrence relation. Parameters ========== n : int n is a nonnegative integer. K : domain rrrr,rrrr)rrrr_s r#r2r2Psq2eWquafoB 1q5\\SSW^Jr1a,@,@!!A$$JJBPQRRB Ir%c  |j|jg|d|j|j g}}t|ddD]}tt t ||||d||jd|}|dkr?|t t t|||d||j|}}t t t|||d||j||}}|S)a Chebyshev polynomials of the first kind using recursive products. Explanation =========== Computes Chebyshev polynomials of the first kind using .. math:: T_{2n}(x) &= 2T_n^2(x) - 1\\ T_{2n+1}(x) &= 2T_{n+1}(x)T_n(x) - x This is faster than ``_dup_chebyshevt_rec`` for large ``n``. Parameters ========== n : int n is a nonnegative integer. K : domain rNr1)rr,binr rrr r )rrrrrcs r#r3r3ns,eQV_qqttQVaeV4B VVABBZZZ B(:(:AAaDD!DDaeQPQ R R #II~gb!nnaaddA'N'NPQPUWXYYBB#N72q>>11Q44$K$KQUTUVVXYBB Ir%c |dkr|jgS|jg|d|jg}}td|dzD]<}|tt t |d||d|||}}=|S)zELow-level implementation of Chebyshev polynomials of the second kind.rrr6rrrrrs r#dup_chebyshevur?s1uuweWqqttQVnB 1ac]]SSW^Jr1a,@,@!!A$$JJBPQRRB Ir%c@t|ttd|f|S)aGenerates the Chebyshev polynomial of the first kind `T_n(x)`. Parameters ========== n : int Degree of the polynomial. x : optional polys : bool, optional If True, return a Poly, otherwise (default) return an expression. z&Chebyshev polynomial of the first kind)rr4r rr'r(s r#chebyshevt_polyrBs( a 4qdE C CCr%c@t|ttd|f|S)aGenerates the Chebyshev polynomial of the second kind `U_n(x)`. Parameters ========== n : int Degree of the polynomial. x : optional polys : bool, optional If True, return a Poly, otherwise (default) return an expression. z'Chebyshev polynomial of the second kind)rr?r rAs r#chebyshevu_polyrDs( a 5tU D DDr%c 4|dkr|jgS|jg|d|jg}}td|dzD][}t|d|}t |||dz |}|t t ||||d|}}\|S)z0Low-level implementation of Hermite polynomials.rrrr,rrrrrrrrrrrs r# dup_hermiterHs1uuweWqqttQVnB 1ac]]?? r1a  2qq1vvq ) )^GAq!$4$4aaddA>>B Ir%c|dkr|jgS|jg|j|jg}}td|dzD]C}t|d|}t |||dz |}|t |||}}D|S)z>Low-level implementation of probabilist's Hermite polynomials.rrrFrGs r#dup_hermite_probrJs1uuweWquafoB 1ac]]&& r1a  2qq1vvq ) )WQ1%%B Ir%c@t|ttd|f|S)zGenerates the Hermite polynomial `H_n(x)`. Parameters ========== n : int Degree of the polynomial. x : optional polys : bool, optional If True, return a Poly, otherwise (default) return an expression. zHermite polynomial)rrHr rAs r# hermite_polyrLs ab*>e L LLr%c@t|ttd|f|S)aGenerates the probabilist's Hermite polynomial `He_n(x)`. Parameters ========== n : int Degree of the polynomial. x : optional polys : bool, optional If True, return a Poly, otherwise (default) return an expression. z probabilist's Hermite polynomial)rrJr rAs r#hermite_prob_polyrNs& a)2 .e = ==r%c<|dkr|jgS|jg|j|jg}}td|dzD]c}tt |d||d|zdz ||}t|||dz ||}|t |||}}d|S)z1Low-level implementation of Legendre polynomials.rrr+rGs r# dup_legendrerPs1uuweWquafoB 1ac]]&& :b!Q//1Q3q5!a @ @ 2qq1ayy! , ,WQ1%%B Ir%c@t|ttd|f|S)zGenerates the Legendre polynomial `P_n(x)`. Parameters ========== n : int Degree of the polynomial. x : optional polys : bool, optional If True, return a Poly, otherwise (default) return an expression. zLegendre polynomial)rrPr rAs r# legendre_polyrRs ar+@1$ N NNr%c \|jg|jg}}td|dzD]}t||j ||z ||jz ||z |dzg|}t |||jz ||z |jz|}|t |||}}|S)z1Low-level implementation of Laguerre polynomials.rr)r,rrrrr)ralpharrrrrrs r# dup_laguerrerUsfXwB 1ac]]&& B!%!uQU{AAaDD&811Q44&?@! D D 2ae QQqTT1AE91 = =WQ1%%B Ir%c8t|tdd||f|S)aQGenerates the Laguerre polynomial `L_n^{(\alpha)}(x)`. Parameters ========== n : int Degree of the polynomial. x : optional alpha : optional Decides minimal domain for the list of coefficients. polys : bool, optional If True, return a Poly, otherwise (default) return an expression. NzLaguerre polynomial)rrU)rr'rTr(s r# laguerre_polyrWs" at-BQJPU V VVr%c $|dkr|j|jgS|jg|j|jg}}td|dzD]B}|tt t |d||d|zdz |||}}Ct |d|S)z%Low-level implementation of fn(n, x).rrr6r>s r#dup_spherical_bessel_fnrYs1uuqveWquafoB 1ac]]WWW^Jr1a,@,@!!AaCE((ANNPRTUVVB b!Q  r%c |j|jg|jg}}td|dzD]B}|tt t |d||dd|zz |||}}C|S)z&Low-level implementation of fn(-n, x).rrr9r6r>s r#dup_spherical_bessel_fn_minusr[(szeQV_qvhB 1ac]]WWW^Jr1a,@,@!!AacE((ANNPRTUVVB Ir%c |td}|dkrtnt}tt ||t dt d|z f|S)a Coefficients for the spherical Bessel functions. These are only needed in the jn() function. The coefficients are calculated from: fn(0, z) = 1/z fn(1, z) = 1/z**2 fn(n-1, z) + fn(n+1, z) == (2*n+1)/z * fn(n, z) Parameters ========== n : int Degree of the polynomial. x : optional polys : bool, optional If True, return a Poly, otherwise (default) return an expression. Examples ======== >>> from sympy.polys.orthopolys import spherical_bessel_fn as fn >>> from sympy import Symbol >>> z = Symbol("z") >>> fn(1, z) z**(-2) >>> fn(2, z) -1/z + 3/z**3 >>> fn(3, z) -6/z**2 + 15/z**4 >>> fn(4, z) 1/z - 45/z**3 + 105/z**5 Nr'rr)rr[rYrabsr r )rr'r(fs r#spherical_bessel_fnr`/sSJ y #JJ)*Q%%4KA c!ffaR"Q%%'U ; ;;r%)NF)NrF)(__doc__sympy.core.symbolrsympy.polys.densearithrrrrrr r r sympy.polys.domainsr r sympy.polys.polytoolsrsympy.utilitiesrr$r)r-r/r4r2r3r?rBrDrHrJrLrNrPrRrUrWrYr[r`r%r#rhs@@######IIIIIIIIIIIIIIIIIIII&&&&&&&&,,,,,,"""""" RRRR$   WWWW &&&<> C C C C D D D D       M M M M = = = =    O O O OWWWW    (<(<(<(<(<(