ó ¡¼™\c@ sÆdZddlmZmZddlmZmZmZddlm Z ddl m Z m Z ddl mZd„Zdefd „ƒYZd „Zd efd „ƒYZd efd„ƒYZdS(sRecurrence Operatorsiÿÿÿÿ(tprint_functiontdivision(tsymbolstSymboltS(tsstr(tranget string_types(tsympifycC st||ƒ}||jfS(s9 Returns an Algebra of Recurrence Operators and the operator for shifting i.e. the `Sn` operator. The first argument needs to be the base polynomial ring for the algebra and the second argument must be a generator which can be either a noncommutative Symbol or a string. Examples ======== >>> from sympy.polys.domains import ZZ >>> from sympy import symbols >>> from sympy.holonomic.recurrence import RecurrenceOperators >>> n = symbols('n', integer=True) >>> R, Sn = RecurrenceOperators(ZZ.old_poly_ring(n), 'Sn') (tRecurrenceOperatorAlgebratshift_operator(tbaset generatortring((s9lib/python2.7/site-packages/sympy/holonomic/recurrence.pytRecurrenceOperators sR cB s/eZdZd„Zd„ZeZd„ZRS(sì A Recurrence Operator Algebra is a set of noncommutative polynomials in intermediate `Sn` and coefficients in a base ring A. It follows the commutation rule: Sn * a(n) = a(n + 1) * Sn This class represents a Recurrence Operator Algebra and serves as the parent ring for Recurrence Operators. Examples ======== >>> from sympy.polys.domains import ZZ >>> from sympy import symbols >>> from sympy.holonomic.recurrence import RecurrenceOperators >>> n = symbols('n', integer=True) >>> R, Sn = RecurrenceOperators(ZZ.old_poly_ring(n), 'Sn') >>> R Univariate Recurrence Operator Algebra in intermediate Sn over the base ring ZZ[n] See Also ======== RecurrenceOperator cC s‘||_t|j|jg|ƒ|_|dkrKtddtƒ|_nBt |t ƒrrt|dtƒ|_nt |t ƒr||_ndS(NtSnt commutative( R tRecurrenceOperatortzerotoneR tNoneRtFalset gen_symbolt isinstanceRR(tselfR R ((s9lib/python2.7/site-packages/sympy/holonomic/recurrence.pyt__init__=s  cC s(dt|jƒd|jjƒ}|S(Ns7Univariate Recurrence Operator Algebra in intermediate s over the base ring (RRR t__str__(Rtstring((s9lib/python2.7/site-packages/sympy/holonomic/recurrence.pyRLscC s0|j|jkr(|j|jkr(tStSdS(N(R RtTrueR(Rtother((s9lib/python2.7/site-packages/sympy/holonomic/recurrence.pyt__eq__Us$(t__name__t __module__t__doc__RRt__repr__R(((s9lib/python2.7/site-packages/sympy/holonomic/recurrence.pyR !s   cC s“t|ƒt|ƒkrUgt||ƒD]\}}||^q(|t|ƒ}n:gt||ƒD]\}}||^qe|t|ƒ}|S(N(tlentzip(tlist1tlist2tatbtsol((s9lib/python2.7/site-packages/sympy/holonomic/recurrence.pyt _add_lists\s=:RcB sqeZdZdZd„Zd„Zd„Zd„ZeZd„Z d„Z d„Z d „Z e Z d „ZRS( st The Recurrence Operators are defined by a list of polynomials in the base ring and the parent ring of the Operator. Takes a list of polynomials for each power of Sn and the parent ring which must be an instance of RecurrenceOperatorAlgebra. A Recurrence Operator can be created easily using the operator `Sn`. See examples below. Examples ======== >>> from sympy.holonomic.recurrence import RecurrenceOperator, RecurrenceOperators >>> from sympy.polys.domains import ZZ, QQ >>> from sympy import symbols >>> n = symbols('n', integer=True) >>> R, Sn = RecurrenceOperators(ZZ.old_poly_ring(n),'Sn') >>> RecurrenceOperator([0, 1, n**2], R) (1)Sn + (n**2)Sn**2 >>> Sn*n (n + 1)Sn >>> n*Sn*n + 1 - Sn**2*n (1) + (n**2 + n)Sn + (-n - 2)Sn**2 See Also ======== DifferentialOperatorAlgebra icC sÀ||_t|tƒr¦xt|ƒD]q\}}t|tƒrb|jjjt|ƒƒ|||j|jkr7|j|jkr7tStSnL|jd|kr†x.|jdD]}||jjjk r_tSq_WtStSdS(Nii(RRR1R+RRR R(RRR4((s9lib/python2.7/site-packages/sympy/holonomic/recurrence.pyR+s$(RR R!t _op_priorityRR>R?RBt__radd__RCRDRGRR"R(((s9lib/python2.7/site-packages/sympy/holonomic/recurrence.pyRds!  6      tHolonomicSequencecB s2eZdZgd„Zd„ZeZd„ZRS(sõ A Holonomic Sequence is a type of sequence satisfying a linear homogeneous recurrence relation with Polynomial coefficients. Alternatively, A sequence is Holonomic if and only if its generating function is a Holonomic Function. cC st||_t|tƒs'|g|_n ||_t|jƒdkrQt|_n t|_|jj j d|_ dS(Ni( t recurrenceRR,tu0R#Rt_have_init_condRR+R R;RE(RRORP((s9lib/python2.7/site-packages/sympy/holonomic/recurrence.pyRBs    cC s‹d|jjƒt|jƒf}|js/|Sd}d}x;|jD]0}|dt|ƒt|ƒf7}|d7}qEW||}|SdS(NsHolonomicSequence(%s, %s)RHis , u(%s) = %si(ROR"RRERQRP(Rtstr_soltcond_strtseq_strR4R)((s9lib/python2.7/site-packages/sympy/holonomic/recurrence.pyR"Os"   cC si|j|jkra|j|jkrZ|jrS|jrS|j|jkrLtStSq^tSqetSntSdS(N(RORERQRPRR(RR((s9lib/python2.7/site-packages/sympy/holonomic/recurrence.pyR_s(RR R!RR"RR(((s9lib/python2.7/site-packages/sympy/holonomic/recurrence.pyRN;s  N(R!t __future__RRtsympyRRRtsympy.printingRtsympy.core.compatibilityRRtsympy.core.sympifyRRtobjectR R*RRN(((s9lib/python2.7/site-packages/sympy/holonomic/recurrence.pyts ; ×