B ˜‘[Žã@s‚dZddlmZmZddlmZmZmZddlm Z m Z ddl m Z m Z mZddlmZddlmZdd d „Zd d „Zd d„ZdS)z1Gosper's algorithm for hypergeometric summation. é)Úprint_functionÚdivision)ÚSÚDummyÚsymbols)Ú is_sequenceÚrange)ÚPolyÚparallel_poly_from_exprÚfactor)Úsolve)Ú hypersimpTcCsTt||f|ddd\\}}}| ¡| ¡}}| ¡| ¡} } |j|| } } tdƒ} t|| || |jd}| |  |¡¡}t |  ¡  ¡ƒ}x(t |ƒD]}|j rª|dkr˜|  |¡q˜Wxht|ƒD]\}| |  | ¡¡}| |¡}|  | | ¡¡} x(td|dƒD]}| | | ¡9} qWqÂW| | ¡}|sJ| ¡}|  ¡} |  ¡} || | fS)a? 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) T)ZfieldÚ extensionÚh)Údomainré)r ÚLCZmonicZonerr rZ resultantZcomposeÚsetZ ground_rootsÚkeysÚ is_IntegerÚremoveÚsortedZgcdÚshiftZquorZ mul_groundÚas_expr)ÚfÚgÚnZpolysÚpÚqZoptÚaÚAÚbÚBÚCÚZrÚDÚRÚrootsÚrÚiÚdÚj©r,ú4lib/python3.7/site-packages/sympy/concrete/gosper.pyÚ gosper_normal s0#  r.cCsÔt||ƒ}|dkrdS| ¡\}}t|||ƒ\}}}| d¡}t| ¡ƒ}t| ¡ƒ} t| ¡ƒ} || ksz| ¡| ¡krŒ| t|| ƒh} nJ|s¦| |dtdƒh} n0| |d| |d¡| |d¡| ¡h} x(t | ƒD]} | j rò| dkrà|   | ¡qàW| s dSt| ƒ} t d| dt d} | ¡j| Ž}t| ||d}|| d¡|||}t| ¡| ƒ}|dkrxdS| ¡ |¡}x$| D]}||krŒ| |d¡}qŒW|tjkr¼dS| ¡|| ¡SdS)a 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) Néÿÿÿÿrrzc:%s)Úcls)r)r Zas_numer_denomr.rrZdegreerÚmaxZnthrrrrrZ get_domainZinjectr r ÚcoeffsrÚsubsZZero)rrr(rrr r"r#ÚNÚMÚKr%r*r2rÚxÚHZsolutionZcoeffr,r,r-Ú gosper_termSsD      0    r9cCs¸d}t|ƒr|\}}}nd}t||ƒ}|dkr2dS|r@||}np||d ||¡|| ||¡}|tjkr°y(||d ||¡|| ||¡}Wntk r®d}YnXt|ƒS)a. 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 FTNr)rr9r3rZNaNÚlimitÚNotImplementedErrorr )rÚkZ indefiniterr!rÚresultr,r,r-Ú gosper_sumŸs %   $ ( r>N)T)Ú__doc__Z __future__rrZ sympy.corerrrZsympy.core.compatibilityrrZ sympy.polysr r r Z sympy.solversr Zsympy.simplifyr r.r9r>r,r,r,r-Ús   HL