o à8Va·ã@sndZddlmZmZmZddlmZddlmZm Z m Z ddl m Z ddl mZddd „Zd d „Zd d „ZdS)z1Gosper's algorithm for hypergeometric summation. é)ÚSÚDummyÚsymbols)Ú is_sequence)ÚPolyÚparallel_poly_from_exprÚfactor)Úsolve)Ú hypersimpTcCsDt||f|ddd\\}}}| ¡| ¡}}| ¡| ¡} } |j|| } } tdƒ} t|| || |jd}| |  |¡¡}t |  ¡  ¡ƒ}t |ƒD]}|j rT|dkrY|  |¡qKt|ƒD]+}| |  | ¡¡}| |¡}|  | | ¡¡} td|dƒD] }| | | ¡9} q~q^| | ¡}|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)ZfieldÚ extensionÚh©Údomainré)rÚLCZmonicZonerrrZ resultantZcomposeÚsetZ ground_rootsÚkeysÚ is_IntegerÚremoveÚsortedZgcdÚshiftZquoÚrangeZ mul_groundÚas_expr)ÚfÚgÚnZpolysÚpÚqZoptÚaÚAÚbÚBÚCÚZr ÚDÚRÚrootsÚrÚiÚdÚj©r+ú7/usr/lib/python3/dist-packages/sympy/concrete/gosper.pyÚ gosper_normal s6& ÿ  €  ÿ  r-cCs¸t||ƒ}|dur dS| ¡\}}t|||ƒ\}}}| d¡}t| ¡ƒ}t| ¡ƒ} t| ¡ƒ} || ks=| ¡| ¡krF| t|| ƒh} n$|sR| |dtjh} n| |d|  |d¡|  |d¡| ¡h} t | ƒD]} | j rw| dkr||   | ¡qn| sdSt| ƒ} t d| dtd} | ¡j| Ž}t| ||d}|| d¡|||}t| ¡| ƒ}|dur·dS| ¡ |¡}| D] }||vrÌ| |d¡}qÀ|jrÒdS| ¡|| ¡S)a/ Compute Gosper's hypergeometric term for ``f``. Explanation =========== Suppose ``f`` is a hypergeometric term such that: .. math:: s_n = \sum_{k=0}^{n-1} f_k and `f_k` doesn't depend on `n`. Returns a hypergeometric term `g_n` such that `g_{n+1} - g_n = f_n`. Examples ======== >>> from sympy.concrete.gosper import gosper_term >>> from sympy.functions import factorial >>> from sympy.abc import n >>> gosper_term((4*n + 1)*factorial(n)/factorial(2*n + 1), n) (-n - 1/2)/(n + 1/4) Néÿÿÿÿrrzc:%s)Úclsr )r Zas_numer_denomr-rrZdegreerÚmaxZZeroZnthrrrrrZ get_domainZinjectrr ÚcoeffsrÚsubsZis_zero)rrr'rrrr!r"ÚNÚMÚKr$r)r1rÚxÚHZsolutionZcoeffr+r+r,Ú gosper_termUsH      0  € €r8cCsÈd}t|ƒr |\}}}nd}t||ƒ}|durdS|r#||}t|ƒS||d ||¡|| ||¡}|tjur`z||d ||¡|| ||¡}Wt|ƒSty_d}Yt|ƒSwt|ƒS)aL 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)` 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 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)rr8r2rZNaNÚlimitÚNotImplementedErrorr)rÚkZ indefiniterr rÚresultr+r+r,Ú gosper_sum¤s((   $ø & ýýr=N)T)Ú__doc__Z sympy.corerrrZsympy.core.compatibilityrZ sympy.polysrrrZ sympy.solversr Zsympy.simplifyr r-r8r=r+r+r+r,Ús    K O