ó ¡¼™\c@ s¬dZddlmZmZddlmZmZmZddlm Z m Z ddl m Z m Z mZddlmZddlmZed„Zd „Zd „Zd S( s1Gosper's algorithm for hypergeometric summation. iÿÿÿÿ(tprint_functiontdivision(tStDummytsymbols(t is_sequencetrange(tPolytparallel_poly_from_exprtfactor(tsolve(t hypersimpcC sÞt||f|dtdtƒ\\}}}|jƒ|jƒ}}|jƒ|jƒ} } |j|| } } tdƒ} t|| || d|jƒ}|j| j |ƒƒ}t |j ƒj ƒƒ}x:t |ƒD],}|j s÷|dkrÛ|j|ƒqÛqÛWx‡t|ƒD]y}|j| j| ƒƒ}|j|ƒ}| j|j| ƒƒ} x/td|dƒD]}| |j| ƒ9} qsWqW|j| ƒ}|sÑ|jƒ}| jƒ} | jƒ} n|| | fS(s? Compute the Gosper's normal form of ``f`` and ``g``. Given relatively prime univariate polynomials ``f`` and ``g``, rewrite their quotient to a normal form defined as follows: .. math:: \frac{f(n)}{g(n)} = Z \cdot \frac{A(n) C(n+1)}{B(n) C(n)} where ``Z`` is an arbitrary constant and ``A``, ``B``, ``C`` are monic polynomials in ``n`` with the following properties: 1. `\gcd(A(n), B(n+h)) = 1 \forall h \in \mathbb{N}` 2. `\gcd(B(n), C(n+1)) = 1` 3. `\gcd(A(n), C(n)) = 1` This normal form, or rational factorization in other words, is a crucial step in Gosper's algorithm and in solving of difference equations. It can be also used to decide if two hypergeometric terms are similar or not. This procedure will return a tuple containing elements of this factorization in the form ``(Z*A, B, C)``. Examples ======== >>> from sympy.concrete.gosper import gosper_normal >>> from sympy.abc import n >>> gosper_normal(4*n+5, 2*(4*n+1)*(2*n+3), n, polys=False) (1/4, n + 3/2, n + 1/4) tfieldt extensionthtdomainii(RtTruetLCtmonictoneRRRt resultanttcomposetsett ground_rootstkeyst is_IntegertremovetsortedtgcdtshifttquoRt mul_groundtas_expr(tftgtntpolystptqtopttatAtbtBtCtZRtDtRtrootstrtitdtj((s4lib/python2.7/site-packages/sympy/concrete/gosper.pyt gosper_normal s0#*   cC s~t||ƒ}|dkrdS|jƒ\}}t|||ƒ\}}}|jdƒ}t|jƒƒ}t|jƒƒ} t|jƒƒ} || ksµ|jƒ|jƒkrÎ| t|| ƒh} n_|sñ| |dtdƒh} n<| |d|j |dƒ|j |dƒ|jƒh} x:t | ƒD],} | j sV| dkr:| j | ƒq:q:W| stdSt| ƒ} t d| ddtƒ} |jƒj| Œ}t| |d|ƒ}||jdƒ|||}t|jƒ| ƒ}|dkrdS|jƒj|ƒ}x/| D]'}||kr$|j|dƒ}q$q$W|tjkrbdS|jƒ||jƒSdS(s Compute Gosper's hypergeometric term for ``f``. Suppose ``f`` is a hypergeometric term such that: .. math:: s_n = \sum_{k=0}^{n-1} f_k and `f_k` doesn't depend on `n`. Returns a hypergeometric term `g_n` such that `g_{n+1} - g_n = f_n`. Examples ======== >>> from sympy.concrete.gosper import gosper_term >>> from sympy.functions import factorial >>> from sympy.abc import n >>> gosper_term((4*n + 1)*factorial(n)/factorial(2*n + 1), n) (-n - 1/2)/(n + 1/4) iÿÿÿÿiisc:%stclsRN(R tNonetas_numer_denomR5RRtdegreeRtmaxtnthRRRRRt get_domaintinjectRR tcoeffsR tsubstZero(R!R#R1R%R&R)R+R,tNtMtKR.R3R>RtxtHtsolutiontcoeff((s4lib/python2.7/site-packages/sympy/concrete/gosper.pyt gosper_termSsD $<    cC sòt}t|ƒr$|\}}}nt}t||ƒ}|dkrIdS|r\||}nŒ||dj||ƒ||j||ƒ}|tjkrèy2||dj||ƒ||j||ƒ}Wqèt k räd}qèXnt |ƒS(s. Gosper's hypergeometric summation algorithm. Given a hypergeometric term ``f`` such that: .. math :: s_n = \sum_{k=0}^{n-1} f_k and `f(n)` doesn't depend on `n`, returns `g_{n} - g(0)` where `g_{n+1} - g_n = f_n`, or ``None`` if `s_n` can not be expressed in closed form as a sum of hypergeometric terms. Examples ======== >>> from sympy.concrete.gosper import gosper_sum >>> from sympy.functions import factorial >>> from sympy.abc import i, n, k >>> f = (4*k + 1)*factorial(k)/factorial(2*k + 1) >>> gosper_sum(f, (k, 0, n)) (-factorial(n) + 2*factorial(2*n + 1))/factorial(2*n + 1) >>> _.subs(n, 2) == sum(f.subs(k, i) for i in [0, 1, 2]) True >>> gosper_sum(f, (k, 3, n)) (-60*factorial(n) + factorial(2*n + 1))/(60*factorial(2*n + 1)) >>> _.subs(n, 5) == sum(f.subs(k, i) for i in [3, 4, 5]) True References ========== .. [1] Marko Petkovsek, Herbert S. Wilf, Doron Zeilberger, A = B, AK Peters, Ltd., Wellesley, MA, USA, 1997, pp. 73--100 iN( tFalseRRRHR7R?RtNaNtlimittNotImplementedErrorR (R!tkt indefiniteR(R*R"tresult((s4lib/python2.7/site-packages/sympy/concrete/gosper.pyt gosper_sumŸs %   .2  N(t__doc__t __future__RRt sympy.coreRRRtsympy.core.compatibilityRRt sympy.polysRRR t sympy.solversR tsympy.simplifyR RR5RHRP(((s4lib/python2.7/site-packages/sympy/concrete/gosper.pyts H L