o 8Va.j@sdZddlmZddlmZddlmZddlmZddl m Z m Z m Z m Z ddlmZmZmZddlmZmZmZmZmZmZmZd d Zd d Zd(ddZddZd)ddZd*ddZddZ ddZ!ddZ"ddZ#d d!Z$d"d#Z%d+d$d%Z&d&d'Z'd S),a 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. )mul)reduce)oo)Dummy)PolygcdZZcancel)sqrtreim)gcdex_diophantinefrac_in derivation splitfactorNonElementaryIntegralExceptionDecrementLevelrecognize_log_derivativec Cs|jrtS|t||kr||ddSg}|}||}d}|jr<|||f||}|d9}||}|js%d}td|}t|dkri|} || d} || }|jrc|| d7}| }t|dksI|S)aY Computes the order of a at p, with respect to t. Explanation =========== 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). r) is_zerorras_polyZETremappendlenpop) aptZ power_listZp1rZ tracks_powernproductfinalZproductfr#5/usr/lib/python3/dist-packages/sympy/integrals/rde.pyorder_at(s2         r%cCs|jrtS||||S)z 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. )rrdegree)rdrr#r#r$ order_at_ooSsr(NcsF|ptd}t|\}}t||j}||}|t||t|jjj\}} t|jt j j} t| |} | j |sgtdj|ffSdd| D} ttfdd| Dtdj} t | } | || }| |}|j|dd\}}| ||ffS)a Weak normalization. Explanation =========== 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) zrcSs g|] }|tvr|dkr|qS)r)r.0ir#r#r$ s z#weak_normalizer..cs,g|]}tt|jtqSr#)rrrr)r+r DErZd1r#r$r-s,TZinclude)rrrdiffrquor rrrZ resultantexprZhasZ real_rootsrrr )rr'r/r)dndsgZ d_sqf_parta1brNqZdqZsnsdr#r.r$weak_normalizer_s.   "      r<cCst||\}}t||\}}||} |||j| | |j} || } | | } | |dr7t| |} | j|dd\} }| ||t| ||}|j|dd\}}| ||f| |f| fS)a Normal part of the denominator. Explanation =========== 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. rTr0) rrr1rr2Zdivrr r)fafdgagdr/r4r5enesrhrccacdbabdr#r#r$ normal_denoms &rIautocCslddlm}|dkr |j}|dkrt|j|j}n2|dkr)t|jdd|j}n"|dvrE||} ||} || | td|jfStd |t|||jt|||j} t|||jt|||j} t d| t d| } | sv|dkr|j t|j|j}t |@t | d | d| d|j\}}t ||j\}}||||||}|d ur|\}}}|dkrt | |} Wd n1swYn|dkrv|j t|jdd|j}t |t t| td  | td | td |j\}}t t| td  | td | td |j\}}t ||j\}}ttd|j|||rf||ttd |j||||||||}|d urf|\}}}|dkrft | |} Wd n 1sqwYtd| | | }||}|| }||}|||t| |j|t||||} ||||} |}|| | |fS) a Special part of the denominator. Explanation =========== 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. rparametric_log_derivrJexptanrr) primitivebasez@case must be one of {'exp', 'tan', 'primitive', 'base'}, not %s.N)sympy.integrals.prderLcaserrto_fieldr2 ValueErrorr%minr'rrevalr r r rmaxr)rrGrHrErFr/rSrLrBCnbZncr ZdcoeffalphaaalphadetaaetadAQmr)betaabetadr9ZpNZpnrCr#r#r$ special_denomsj  ,    <<0      2 reFc sxddlm}m}m}|dkrj}|j} |j} |r+tfdd|D} n|j} t| j | j } |dkrktd| t| | d} | | dkri| j ritd| | | } | S|dkr| | kr|td| | } n td| | d} t jjjd\}}j}tt | j\}}| | dkrz|||||fg\\}}}Wn tyYnwt|dkrtd t| |d} n|| | krI|||}|d ura|\}}|dkri|t| ||| | | | }t |j\}}z|||||fg\\}}}Wn ty,YnEwt|dkr8td t| |d } Wd | SWd | SWd | SWd | SWd | SWd | SWd | S1s}wY| S|d krtd| t| | } | | krt jtjjjjd\}}t=t | j\}}|||||}|d ur|\}}}|dkrt| |} Wd | SWd | SWd | S1swY| S|d vr6jj}j }t| |} td| t| |d| } | | |dkr4| j r4td| | | } | Std |)am Bound on polynomial solutions. Explanation =========== 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. r)rLlimited_integrate!is_log_deriv_k_t_radical_in_fieldrJcsg|]}|jqSr#)r&rr*r/r#r$r-*sz bound_degree..rPrrOzLength of m should be 1NrM)rNZother_nonlinearzScase must be one of {'exp', 'tan', 'primitive', 'other_nonlinear', 'base'}, not %s.)rRrLrfrgrSr&rrXr rLCas_expr is_Integerrr'TlevelrrrrUrr2r)rr8cQr/rS parametricrLrfrgdaZdbZdcalphar r^r_t1r\r]ZzaZzdrbr`Zaar)betarcrdZdeltaZlamr#rhr$ bound_degree s   I            > > > > > > >>  ,           rtc Cstd|j}td|j}td|j} |jr||d||fS|dkdur%t||}||js2t||||||}}}||jdkr`||}||}|||||fSt |||\} } |t ||7}| t | |}|||j8}||| 7}||9}q)a Rothstein's Special Polynomial Differential Equation algorithm. Explanation =========== 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. rrT) rrrrrrr2r&rTr r) rr8rDr r/Zzerorqrsr6rr)r#r#r$spdes.      " rucCstd|j}|jsT||j||j}d|kr |ks#ttt||j||j|j||jdd}||}|d}|t||||}|jr |S)a Poly Risch Differential Equation - No cancellation: deg(b) large enough. Explanation =========== Given a derivation D on k[t], ``n`` either an integer or +oo, and ``b``,``c`` in k[t] with ``b != 0`` and either D == d/dt or deg(b) > max(0, deg(D) - 1), either raise NonElementaryIntegralException, in which case the equation ``Dq + b*q == c`` has no solution of degree at most n in k[t], or a solution q in k[t] of this equation with ``deg(q) < n``. rFexpandr)rrrr&rrrirr8rDr r/r:rbrr#r#r$no_cancel_b_larges . rycCsVtd|j}|js|dkrd}n||j|j|jd}d|kr*|ks-tt|dkrPt||j||j|j|j||jdd}nC||j||jkr^t||jdkr}|||j|j d||j|j dfSt||j||j|jdd}||}|d}|t ||||}|jr |S)a Poly Risch Differential Equation - No cancellation: deg(b) small enough. Explanation =========== 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. rrFrv) rrrr&r'rrrirlrmrrxr#r#r$no_cancel_b_smalls6 0$rzc Csptd|j}t||j |j|j}|jr#|jr#|}nd}|jst || |j|j |jd}d|krE|ksHt t t||j|j||j}|jre|||fS|dkrt||j||j||jdd} n!| |j|j |jdkrt ||j||j} || }|d}|t | ||| }|jr(|S)a Poly Risch Differential Equation - No cancellation: deg(b) == deg(D) - 1 Explanation =========== 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. rrQrFrv) rrr rrir'rkZ is_positiverrXr&rr) r8rDr r/r:ZlcMrburr#r#r$no_cancel_equals0 ( $* , r}cCs^ddlm}t|&t||j\}}||||}|dur)|\}}|dkr)tdWdn1s3wY|jr=|S|||jkrGtt d|j} |js||j} || kr\tt|t| |j\} } t ||| | |\} }Wdn1swYt | | |j| |jdd}| |7} | d}|||t ||8}|jrP| S)a Poly Risch Differential Equation - Cancellation: Primitive case. Explanation =========== Given a derivation D on k[t], n either an integer or +oo, ``b`` in k, and ``c`` in k[t] with Dt in k and ``b != 0``, either raise NonElementaryIntegralException, in which case the equation Dq + b*q == c has no solution of degree at most n in k[t], or a solution q in k[t] of this equation with deg(q) <= n. r)rgNrz7is_deriv_in_field() is required to solve this problem.Frv)rRrgrrrNotImplementedErrorrr&rrririschDErjr)r8rDr r/rgrGrHr`r)r:rba2aa2dsar;stmr#r#r$cancel_primitive+s:      & rcCsddlm}|jt|j|j}t|1t||j\}}t||j\}} ||| |||} | durA| \} } } | dkrAt dWdn1sKwY|j rU|S|| |jkr_t td|j}|j s| |j} || krtt |}t|6t||j\}}||||t| |j}||}t| |j\}}t|||||\}}Wdn1swYt|||j| |jdd}||7}| d}|||t||8}|j rh|S)a Poly Risch Differential Equation - Cancellation: Hyperexponential case. Explanation =========== 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. rrKNrz6is_deriv_in_field() is required to solve this problem.Frv)rRrLr'r2rrrjrrr~rr&rrirr)r8rDr r/rLZetar^r_rGrHr`rrbr)r:r7Za1aZa1drrrr;rr#r#r$ cancel_exp]sF      & rcCsddlm}m}|js2|jdks"||jtd|j|jdkr2|r+|||||St ||||S|jsD||j|j|jdkr|jdksR|j|jdkr|r[|||||St ||||}t |t ri|S|\}} } t |3| |j| |j} } | durtd| durtdt| | |||j} Wd|| S1swY|| S|j|jdkr||j|j|jdkr|||j |j|jkr||jjstd |rtd t||||}t |t r|S|\}} } t|| | |} || S|jrtd |jd kr.|r'td t||||S|jdkrB|r;tdt||||Std|j)a  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. r)prde_no_cancel_b_largeprde_no_cancel_b_smallrPrrNzb0 should be a non-Null valuezc0 should be a non-Null valuezResult should be a numberz0prde_no_cancel_b_equal() is not yet implemented.zWRemaining cases for Poly (P)RDE are not yet implemented (is_deriv_in_field() required).rMzIParametric RDE cancellation hyperexponential case is not yet implemented.rOzBParametric RDE cancellation primitive case is not yet implemented.zBOther Poly (P)RDE cancellation cases are not yet implemented (%s).)rRrrrrSr&rrXr'ryrz isinstancerrrrUsolve_poly_rderiZ is_number TypeErrorr~r}rr)r8rnr r/rorrRrCZb0Zc0yrbrZr#r#r$rsn "$    4&    rcCst|||\}\}}t|||||\}\}}\} } } t|||| | |\} } }}z t| | ||}Wn ty;t}Ynwt| | |||\} }}}}|jrO|}nt| |||}|||| |fS)a  Solve a Risch Differential Equation: Dy + f*y == g. Explanation =========== 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<rIrertr~rrurr)r=r>r?r@r/_rrGrHrErFZhnr`rYrZZhsr rbrqrsrr#r#r$rs  r)N)rJ)rJF)F)(__doc__operatorr functoolsrZ sympy.corerZsympy.core.symbolrZ sympy.polysrrrr Zsympyr r r Zsympy.integrals.rischr rrrrrrr%r(r<rIrertruryrzr}rrrrr#r#r#r$s,    $+ 3 % Ut///2 = ]