U —9%eµã@sVdZddlmZmZmZddlmZmZmZddl m Z d dd„Z dd „Z d d „Z d S)z1Gosper's algorithm for hypergeometric summation. é)ÚSÚDummyÚsymbols)ÚPolyÚparallel_poly_from_exprÚfactor)Ú is_sequenceTcCsHt||f|ddd\\}}}| ¡| ¡}}| ¡| ¡} } |j|| } } tdƒ} t|| || |jd}| |  |¡¡}t |  ¡  ¡ƒ}t |ƒD]}|j r¨|dkr–|  |¡q–t|ƒD]X}| |  | ¡¡}| |¡}|  | | ¡¡} td|dƒD]}| | | ¡9} qüq¼| | ¡}|s>| ¡}|  ¡} |  ¡} || | fS)a` Compute the Gosper's normal form of ``f`` and ``g``. Explanation =========== 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)ÚfieldÚ extensionÚh©Údomainré)rÚLCZmonicÚonerrr Z resultantZcomposeÚsetZ ground_rootsÚkeysÚ is_IntegerÚremoveÚsortedÚgcdÚshiftZquoÚrangeZ mul_groundÚas_expr)ÚfÚgÚnZpolysÚpÚqÚoptÚaÚAÚbÚBÚCÚZr ÚDÚRÚrootsÚrÚiÚdÚj©r-úT/var/www/html/Darija-Ai-API/env/lib/python3.8/site-packages/sympy/concrete/gosper.pyÚ gosper_normals8&ÿ     r/cCsÞddlm}|||ƒ}|dkr"dS| ¡\}}t|||ƒ\}}}| d¡}t| ¡ƒ} t| ¡ƒ} t| ¡ƒ} | | ks†| ¡| ¡kr˜| t| | ƒh} nH| s°| | dtj h} n0| | d|  | d¡|  | d¡| ¡h} t | ƒD]} | j rú| dkrè|   | ¡qè| sdSt| ƒ} td| dtd}| ¡j|Ž}t|||d}|| d¡|||}dd lm}|| ¡|ƒ}|dkrŠdS| ¡ |¡}|D]}||krœ| |d¡}qœ|jrÆdS| ¡|| ¡SdS) a& Compute Gosper's hypergeometric term for ``f``. Explanation =========== Suppose ``f`` is a hypergeometric term such that: .. math:: s_n = \sum_{k=0}^{n-1} f_k and `f_k` does not 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 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) r)Ú hypersimpNéÿÿÿÿrzc:%s)Úclsr )Úsolve)Zsympy.simplifyr0Zas_numer_denomr/rrZdegreerÚmaxZZeroZnthrrrrrZ get_domainZinjectrZsympy.solvers.solversr3ÚcoeffsrÚsubsÚis_zero)rrr0r)rrr!r#r$ÚNÚMÚKr&r+r5r ÚxÚHr3ZsolutionZcoeffr-r-r.Ú gosper_termSsH       0     r=cCs¸d}t|ƒr|\}}}nd}t||ƒ}|dkr2dS|r@||}np||d ||¡|| ||¡}|tjkr°z(||d ||¡|| ||¡}Wntk r®d}YnXt|ƒS)aB Gosper's hypergeometric summation algorithm. Explanation =========== Given a hypergeometric term ``f`` such that: .. math :: s_n = \sum_{k=0}^{n-1} f_k and `f(n)` does not depend on `n`, returns `g_{n} - g(0)` where `g_{n+1} - g_n = f_n`, or ``None`` if `s_n` cannot be expressed in closed form as a sum of hypergeometric terms. Examples ======== >>> from sympy.concrete.gosper import gosper_sum >>> from sympy import factorial >>> from sympy.abc import 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)rr=r6rÚNaNÚlimitÚNotImplementedErrorr)rÚkZ indefiniter r"rÚresultr-r-r.Ú gosper_sum¤s (   $ ( rCN)T)Ú__doc__Z sympy.corerrrZ sympy.polysrrrZsympy.utilities.iterablesrr/r=rCr-r-r-r.Ús   KQ