ó ¡¼™\c@ s=dZddlmZmZddlmZddlmZddlm Z ddl m Z ddl m Z mZmZmZddlmZmZmZmZmZmZd „Zd „Zdd „Zd „Zd d„Zd ed„Zd„Z d„Z!d„Z"d„Z#d„Z$d„Z%ed„Z&d„Z'dS(sô Algorithms for solving the Risch differential equation. Given a differential field K of characteristic 0 that is a simple monomial extension of a base field k and f, g in K, the Risch Differential Equation problem is to decide if there exist y in K such that Dy + f*y == g and to find one if there are some. If t is a monomial over k and the coefficients of f and g are in k(t), then y is in k(t), and the outline of the algorithm here is given as: 1. Compute the normal part n of the denominator of y. The problem is then reduced to finding y' in k, where y == y'/n. 2. Compute the special part s of the denominator of y. The problem is then reduced to finding y'' in k[t], where y == y''/(n*s) 3. Bound the degree of y''. 4. Reduce the equation Dy + f*y == g to a similar equation with f, g in k[t]. 5. Find the solutions in k[t] of bounded degree of the reduced equation. See Chapter 6 of "Symbolic Integration I: Transcendental Functions" by Manuel Bronstein. See also the docstring of risch.py. iÿÿÿÿ(tprint_functiontdivision(tmul(too(treduce(tDummy(tPolytgcdtZZtcancel(tgcdex_diophantinetfrac_int derivationt splitfactortNonElementaryIntegralExceptiontDecrementLevelc C s|jr tS|t||ƒkr=|j|ƒjƒddSg}|}|j|ƒ}d}xC|jr£|j||fƒ||}|d9}|j|ƒ}qaWd}td|ƒ}x_t|ƒdkr|jƒ} || d} |j| ƒ}|jr¼|| d7}| }q¼q¼W|S(s8 Computes the order of a at p, with respect to t. For a, p in k[t], the order of a at p is defined as nu_p(a) = max({n in Z+ such that p**n|a}), where a != 0. If a == 0, nu_p(a) = +oo. To compute the order at a rational function, a/b, use the fact that nu_p(a/b) == nu_p(a) - nu_p(b). iii( tis_zeroRRtas_polytETtremtappendtlentpop( tatpttt power_listtp1trt tracks_powertntproducttfinaltproductf((s2lib/python2.7/site-packages/sympy/integrals/rde.pytorder_at's.       cC s'|jr tS|j|ƒ|j|ƒS(s¤ Computes the order of a/d at oo (infinity), with respect to t. For f in k(t), the order or f at oo is defined as deg(d) - deg(a), where f == a/d. (RRtdegree(RtdR((s2lib/python2.7/site-packages/sympy/integrals/rde.pyt order_at_ooOs c C s|ptdƒ}t||ƒ\}}t||j|jƒƒ}|j|ƒ}|jt||ƒƒ}t|j|ƒj|jƒ|j|jƒ|j|jƒƒ\} } |t||jƒt ||ƒj|jƒj |j|jƒƒ} t| |ƒ} | j |ƒs,td|jƒ||ffSg| j ƒD]$} | t kr9| dkr9| ^q9} ttg| D]2}t|t||jƒt ||ƒ|ƒ^qptd|jƒƒ}t ||ƒ}||||}||}|j|dtƒ\}}|||ffS(sg Weak normalization. Given a derivation D on k[t] and f == a/d in k(t), return q in k[t] such that f - Dq/q is weakly normalized with respect to t. f in k(t) is said to be "weakly normalized" with respect to t if residue_p(f) is not a positive integer for any normal irreducible p in k[t] such that f is in R_p (Definition 6.1.1). If f has an elementary integral, this is equivalent to no logarithm of integral(f) whose argument depends on t has a positive integer coefficient, where the arguments of the logarithms not in k(t) are in k[t]. Returns (q, f - Dq/q) tziitinclude(RR RtdiffRtquoR RRR t resultantthast real_rootsRRRR tTrue(RR$tDER&tdntdstgt d_sqf_parttd1ta1tbRtitNRtqtdqtsntsd((s2lib/python2.7/site-packages/sympy/integrals/rde.pytweak_normalizer[s(*/7B cC st||ƒ\}}t||ƒ\}}|j|ƒ} |j|j|jƒƒj| j| j|jƒƒƒ} || } | | } | j|ƒdr¢t‚n| |} | j|dtƒ\} }| ||t | |ƒ|}|j|dtƒ\}}| ||f| |f| fS(s÷ Normal part of the denominator. Given a derivation D on k[t] and f, g in k(t) with f weakly normalized with respect to t, either raise NonElementaryIntegralException, in which case the equation Dy + f*y == g has no solution in k(t), or the quadruplet (a, b, c, h) such that a, h in k[t], b, c in k, and for any solution y in k(t) of Dy + f*y == g, q = y*h in k satisfies a*Dq + b*q == c. This constitutes step 1 in the outline given in the rde.py docstring. iR'( R RR(RR)tdivRR R-R (tfatfdtgatgdR.R/R0tentesRthRtctcatcdtbatbd((s2lib/python2.7/site-packages/sympy/integrals/rde.pyt normal_denom‹s 9    tautoc C spddlm}|dkr(|j}n|dkrLt|j|jƒ}nŽ|dkrxt|jdd|jƒ}nb|d krÊ|jƒj|ƒ} |jƒj|ƒ} || | td|jƒfStd |ƒ‚t|||jƒt|||jƒ} t|||jƒt|||jƒ} t d | t d | ƒƒ} | sÌ|dkr7|j jt|j|jƒƒ}t |ƒ­t |j d ƒ |j d ƒ|j d ƒ|jƒ\}}t ||jƒ\}}||||||ƒ}|d k r.|\}}}|dkr.t | |ƒ} q.nWd QXqÌ|dkrÌ|j jt|jdd|jƒƒ}t |ƒLt t|j tdƒƒ |j tdƒƒ|j tdƒƒƒ|jƒ\}}t t|j tdƒƒ |j tdƒƒ|j tdƒƒƒ|jƒ\}}t ||jƒ\}}td|||ƒrÀ||tdƒ||||||||ƒ}|d k rÀ|\}}}|dkr½t | |ƒ} q½qÀnWd QXqÌntd | | | ƒ}||}|| }||}||j|ƒt| |jƒ|t||ƒj|ƒ|} |||j|ƒ} |}|| | |fS(së Special part of the denominator. case is one of {'exp', 'tan', 'primitive'} for the hyperexponential, hypertangent, and primitive cases, respectively. For the hyperexponential (resp. hypertangent) case, given a derivation D on k[t] and a in k[t], b, c, in k with Dt/t in k (resp. Dt/(t**2 + 1) in k, sqrt(-1) not in k), a != 0, and gcd(a, t) == 1 (resp. gcd(a, t**2 + 1) == 1), return the quadruplet (A, B, C, 1/h) such that A, B, C, h in k[t] and for any solution q in k of a*Dq + b*q == c, r = qh in k[t] satisfies A*Dr + B*r == C. For case == 'primitive', k == k[t], so it returns (a, b, c, 1) in this case. This constitutes step 2 of the outline given in the rde.py docstring. iÿÿÿÿ(tparametric_log_derivRKtexpttaniit primitivetbases@case must be one of {'exp', 'tan', 'primitive', 'base'}, not %s.iN(RORP(tsympy.integrals.prdeRLtcaseRRtto_fieldR)t ValueErrorR"tminR$RR tevaltNonetimtsqrttretrecognize_log_derivativetmaxR (RRHRIRFRGR.RRRLRtBtCtnbtncRtdcoefftalphaatalphadtetaatetadtAtQtmR&tbetaatbetadR7tpNtpnRD((s2lib/python2.7/site-packages/sympy/integrals/rde.pyt special_denom­s^       (( ! <   ) TT2  $   Ac C s4ddlm}m}m}|dkr4|j}n|j|jƒ} |j|jƒ} |rŒtg|D]} | j|jƒ^qhƒ} n|j|jƒ} t|j |jƒj ƒj ƒ |j |jƒj ƒj ƒƒ} |dkr@td| t| | dƒƒ}| | dkr0| j r0td| | | ƒ}q0nð|dkrŒ| | krntd| | ƒ}ntd| | dƒ}t |j|j|jdƒ\}}|j}t|ƒÇt | |jƒ\}}| | dkrby.|||||fg|ƒ\\}}}Wntk r*qƒXt|ƒdkrLtdƒ‚nt||dƒ}n!| | krƒ||||ƒ}|d k rƒ|\}}|dkr€|t||ƒj |ƒ||j |ƒj ƒ |j ƒ|j ƒ}t ||jƒ\}}y.|||||fg|ƒ\\}}}Wntk rEq}Xt|ƒdkrgtdƒ‚nt||dƒ}q€qƒnWd QXn¤|d krƒtd| t| | ƒƒ}| | kr0t |jjt|j|jƒƒ|j|jdƒ\}}t|ƒqt | |jƒ\}}||||||ƒ}|d k rw|\}}}|dkrwt||ƒ}qwnWd QXq0n­|dkr |jj|jƒ}|jj ƒ}t| |ƒ} td| t| |d| ƒƒ}| | |dkr0| j r0td| | | ƒ}q0ntd |ƒ‚|S(s$ Bound on polynomial solutions. Given a derivation D on k[t] and a, b, c in k[t] with a != 0, return n in ZZ such that deg(q) <= n for any solution q in k[t] of a*Dq + b*q == c, when parametric=False, or deg(q) <= n for any solution c1, ..., cm in Const(k) and q in k[t] of a*Dq + b*q == Sum(ci*gi, (i, 1, m)) when parametric=True. For parametric=False, cQ is c, a Poly; for parametric=True, cQ is Q == [q1, ..., qm], a list of Polys. This constitutes step 3 of the outline given in the rde.py docstring. iÿÿÿÿ(RLtlimited_integratet!is_log_deriv_k_t_radical_in_fieldRKRPiiROsLength of m should be 1NRMRNtother_nonlinearsScase must be one of {'exp', 'tan', 'primitive', 'other_nonlinear', 'base'}, not %s.(RNRp(RQRLRnRoRRR#RR\R RtLCtas_exprt is_IntegerR R$tTtlevelRRRRTRWR R)R(RR5tcQR.RRt parametricRLRnRotdatdbR6tdctalphaRRdRett1RbRctzatzdRhRftaaR&tbetaRiRjtdeltatlam((s2lib/python2.7/site-packages/sympy/integrals/rde.pyt bound_degreeÿsˆ  ."    &       / % %  >   ! $ c C s˜td|jƒ}td|jƒ}td|jƒ}x[tr“|jr[||d||fS|dktkrvt‚n|j|ƒ}|j|ƒjs t‚n|j|ƒ|j|ƒ|j|ƒ}}}|j|jƒdkr$|j ƒj|ƒ}|j ƒj|ƒ}|||||fSt |||ƒ\} } |t ||ƒ7}| t | |ƒ}||j|jƒ8}||| 7}||9}q9WdS(sw Rothstein's Special Polynomial Differential Equation algorithm. Given a derivation D on k[t], an integer n and a, b, c in k[t] with a != 0, either raise NonElementaryIntegralException, in which case the equation a*Dq + b*q == c has no solution of degree at most n in k[t], or return the tuple (B, C, m, alpha, beta) such that B, C, alpha, beta in k[t], m in ZZ, and any solution q in k[t] of degree at most n of a*Dq + b*q == c must be of the form q == alpha*h + beta, where h in k[t], deg(h) <= m, and Dh + B*h == C. This constitutes step 4 of the outline given in the rde.py docstring. iiN( RRR-RRRRR)R#RSR R ( RR5RERR.tzeroR{R€R1RR&((s2lib/python2.7/site-packages/sympy/integrals/rde.pytspdeps,    /cC sçtd|jƒ}xÎ|jsâ|j|jƒ|j|jƒ}d|koW|knset‚nt|j|jƒjƒ|j|jƒjƒ|j||jdtƒ}||}|d}|t||ƒ||}qW|S(s´ Poly Risch Differential Equation - No cancellation: deg(b) large enough. Given a derivation D on k[t], n either an integer or +oo, and b, c in k[t] with b != 0 and either D == d/dt or deg(b) > max(0, deg(D) - 1), either raise NonElementaryIntegralException, in which case the equation Dq + b*q == c has no solution of degree at most n in k[t], or a solution q in k[t] of this equation with deg(q) < n. itexpandi( RRRR#RRRqtFalseR (R5RERR.R8RhR((s2lib/python2.7/site-packages/sympy/integrals/rde.pytno_cancel_b_largeœs  " B   cC sÙtd|jƒ}xÀ|jsÔ|dkr3d}n)|j|jƒ|jj|jƒd}d|kos|knst‚n|dkrât|j|jƒjƒ||jj|jƒjƒ|j||jdtƒ}nÀ|j|jƒ|j|jƒkrt‚n|j|jƒdkrb||j|j |j dƒ|j|j |j dƒfSt|j|jƒjƒ|j|jƒjƒ|jdtƒ}||}|d}|t ||ƒ||}qW|S(sl Poly Risch Differential Equation - No cancellation: deg(b) small enough. Given a derivation D on k[t], n either an integer or +oo, and b, c in k[t] with deg(b) < deg(D) - 1 and either D == d/dt or deg(D) >= 2, either raise NonElementaryIntegralException, in which case the equation Dq + b*q == c has no solution of degree at most n in k[t], or a solution q in k[t] of this equation with deg(q) <= n, or the tuple (h, b0, c0) such that h in k[t], b0, c0, in k, and for any solution q in k[t] of degree at most n of Dq + bq == c, y == q - h is a solution in k of Dy + b0*y == c0. iiR†( RRRR#R$RRRqR‡RtRuR (R5RERR.R8RhR((s2lib/python2.7/site-packages/sympy/integrals/rde.pytno_cancel_b_small·s*    )  @$ 7   c C std|jƒ}t|j|jƒjƒ |jj|jƒjƒƒ}|jre|jre|}nd}xŽ|jsût ||j |jƒ|jj |jƒdƒ}d|koÀ|knsÎt ‚nt||jj|jƒjƒ|j|jƒjƒƒ}|jr|||fS|dkrgt|j|jƒjƒ||j||jdt ƒ} nb|j |jƒ|jj |jƒdkr›t ‚n.|j|jƒjƒ|j|jƒjƒ} || }|d}|t | |ƒ|| }qnW|S(sx Poly Risch Differential Equation - No cancellation: deg(b) == deg(D) - 1 Given a derivation D on k[t] with deg(D) >= 2, n either an integer or +oo, and b, c in k[t] with deg(b) == deg(D) - 1, either raise NonElementaryIntegralException, in which case the equation Dq + b*q == c has no solution of degree at most n in k[t], or a solution q in k[t] of this equation with deg(q) <= n, or the tuple (h, m, C) such that h in k[t], m in ZZ, and C in k[t], and for any solution q in k[t] of degree at most n of Dq + b*q == c, y == q - h is a solution in k[t] of degree at most m of Dy + b*y == C. iiÿÿÿÿiR†(RRR RRqR$Rst is_positiveRR\R#RR‡R ( R5RERR.R8tlctMRhtuR((s2lib/python2.7/site-packages/sympy/integrals/rde.pytno_cancel_equalãs* 8  2 ;   <+ .  c C s¯ddlm}t|ƒet||jƒ\}}||||ƒ}|dk r}|\}}|dkr}tdƒ‚q}nWdQX|jr|S||j|jƒkr±t ‚nt d|jƒ} xå|jsª|j|jƒ} || kröt ‚nt|ƒAt|j ƒ|jƒ\} } t ||| | |ƒ\} }WdQXt | j ƒ|j ƒ|j| |jdtƒ}| |7} | d}|||t||ƒ8}qÆW| S(s† Poly Risch Differential Equation - Cancellation: Primitive case. Given a derivation D on k[t], n either an integer or +oo, b in k, and c in k[t] with Dt in k and b != 0, either raise NonElementaryIntegralException, in which case the equation Dq + b*q == c has no solution of degree at most n in k[t], or a solution q in k[t] of this equation with deg(q) <= n. iÿÿÿÿ(Rois7is_deriv_in_field() is required to solve this problem.NiR†(RQRoRR RRWtNotImplementedErrorRR#RRRqtrischDERrR‡R (R5RERR.RoRHRIRfR&R8Rhta2ata2dtsaR;tstm((s2lib/python2.7/site-packages/sympy/integrals/rde.pytcancel_primitives2           $3  c C sGddlm}|jjt|j|jƒƒjƒ}t|ƒ†t||jƒ\}}t||jƒ\}} ||| |||ƒ} | dk rÅ| \} } } | dkrÅt dƒ‚qÅnWdQX|j rØ|S||j |jƒkrùt ‚ntd|jƒ}x5|j sB|j |jƒ} || kr>t ‚n|jƒ}t|ƒ…t||jƒ\}}||||t| |jƒ}||}t|jƒ|jƒ\}}t|||||ƒ\}}WdQXt|jƒ|jƒ|j| |jdtƒ}||7}| d}|||t||ƒ8}qW|S(s Poly Risch Differential Equation - Cancellation: Hyperexponential case. Given a derivation D on k[t], n either an integer or +oo, b in k, and c in k[t] with Dt/t in k and b != 0, either raise NonElementaryIntegralException, in which case the equation Dq + b*q == c has no solution of degree at most n in k[t], or a solution q in k[t] of this equation with deg(q) <= n. iÿÿÿÿ(RLis6is_deriv_in_field() is required to solve this problem.NiR†(RQRLR$R)RRRrRR RWRRR#RRqRR‡R (R5RERR.RLtetaRdReRHRIRfRRhR&R8R4ta1ata1dR‘R’R“R;R”((s2lib/python2.7/site-packages/sympy/integrals/rde.pyt cancel_exp>s> '          " $3  c C s¦ddlm}m}|j r|jdksc|j|jƒtd|jj|jƒdƒkr|r||||||ƒSt ||||ƒS|jsÃ|j|jƒ|jj|jƒdkr×|jdksí|jj|jƒdkr×|r|||||ƒSt ||||ƒ}t |t ƒr.|S|\}} } t |ƒ| j|jƒ| j|jƒ} } | d krŠtdƒ‚n| d kr¥tdƒ‚nt| | ||ƒj|jƒ} Wd QX|| SnË|jj|jƒdkrë|j|jƒ|jj|jƒdkrë||j|jƒjƒ |jj|jƒjƒkrë|j|jƒjƒjstd ƒ‚n|r”td ƒ‚nt||||ƒ}t |t ƒr¼|S|\}} } t|| | |ƒ} || Sn·|jrtd ƒ‚n~|jd kr:|r'tdƒ‚nt||||ƒS|jdkrq|r^tdƒ‚nt||||ƒStdtƒ‚|r–tdƒ‚ntdƒ‚d S(s Solve a Polynomial Risch Differential Equation with degree bound n. This constitutes step 4 of the outline given in the rde.py docstring. For parametric=False, cQ is c, a Poly; for parametric=True, cQ is Q == [q1, ..., qm], a list of Polys. iÿÿÿÿ(tprde_no_cancel_b_largetprde_no_cancel_b_smallRPiiisb0 should be a non-Null valuesc0 should be a non-Null valueNsResult should be a numbers0prde_no_cancel_b_equal() is not yet implemented.sWRemaining cases for Poly (P)RDE are not yet implemented (is_deriv_in_field() required).RMsIParametric RDE cancellation hyperexponential case is not yet implemented.ROsBParametric RDE cancellation primitive case is not yet implemented.sBOther Poly (P)RDE cancellation cases are not yet implemented (%s).s2Remaining cases for Poly PRDE not yet implemented.s1Remaining cases for Poly RDE not yet implemented.(RQRšR›RRRR#RR\R$RˆR‰t isinstanceRRRRWRTtsolve_poly_rdeRqt is_numbert TypeErrorRRŽR™R•(R5RvRR.RwRšR›tRRDtb0tc0tyRhR^((s2lib/python2.7/site-packages/sympy/integrals/rde.pyRxsb 44* %  ' F8   cC s t|||ƒ\}\}}t|||||ƒ\}\}}\} } } t|||| | |ƒ\} } }}yt| | ||ƒ}Wntk r§t}nXt| | |||ƒ\} }}}}|jrá|}nt| |||ƒ}|||| |fS(sê Solve a Risch Differential Equation: Dy + f*y == g. See the outline in the docstring of rde.py for more information about the procedure used. Either raise NonElementaryIntegralException, in which case there is no solution y in the given differential field, or return y in k(t) satisfying Dy + f*y == g, or raise NotImplementedError, in which case, the algorithms necessary to solve the given Risch Differential Equation have not yet been implemented. ( R<RJRmRƒRRR…RR(R>R?R@RAR.t_RRHRIRFRGthnRfR]R^thsRRhR{R€R£((s2lib/python2.7/site-packages/sympy/integrals/rde.pyRÕs 0'  '  N((t__doc__t __future__RRtoperatorRt sympy.coreRtsympy.core.compatibilityRtsympy.core.symbolRt sympy.polysRRRR tsympy.integrals.rischR R R R RRR"R%RWR<RJRmR‡RƒR…RˆR‰RŽR•R™RR(((s2lib/python2.7/site-packages/sympy/integrals/rde.pyts*". ( 0 " Rq ,  , , / : ]