B [h@sdZddlmZmZddlmZddlmZddlm Z ddl m Z ddl m Z mZmZmZddlmZmZmZmZmZmZd d Zd d Zd(ddZddZd)ddZd*ddZddZddZddZ 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. )print_functiondivision)mul)oo)reduce)Dummy)PolygcdZZcancel)gcdex_diophantinefrac_in derivation splitfactorNonElementaryIntegralExceptionDecrementLevelc Cs|jr tS|t||kr.||ddSg}|}||}d}x2|jrv|||f||}|d9}||}qFWd}td|}xDt|dkr|} || d} || }|jr|| d7}| }qW|S)a8 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). r) is_zerorras_polyZETremappendlenpop) aptZ power_listZp1rZ tracks_powernproductfinalZproductfr!2lib/python3.7/site-packages/sympy/integrals/rde.pyorder_at's.      r#cCs|jr tS||||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_ooOsr&NcsD|p td}t|\}}t||j}||}|t||t|jjj\}} t|jt j j} t| |} | |stdj|ffSdd| D} t tfdd| Dtdj} t | } | || }| |}|j|dd\}}| ||ffS)ag 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) zrcSs g|]}|tkr|dkr|qS)r)r ).0ir!r!r" ~sz#weak_normalizer..cs,g|]$}tt|jtqSr!)r rrr)r(r)DErd1r!r"r*sT)include)rrr diffrquor rrrZ resultantZhasZ real_rootsrrr )rr%r+r'dndsgZ d_sqf_parta1brNqZdqZsnsdr!)r+rr,r"weak_normalizer[s(  "   r8cCst||\}}t||\}}||} |||j| | |j} || } | | } | |drnt| |} | j|dd\} }| ||t| ||}|j|dd\}}| ||f| |f| fS)a 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. rT)r-) rr r.rr/Zdivrr r)fafdgagdr+r0r1ZenZesrhrccacdbabdr!r!r" normal_denoms  &rCautoc Cs:ddlm}|dkr|j}|dkr2t|j|j}nd|dkrRt|jdd|j}nD|dkr||} ||} || | td|jfStd |t|||jt|||j} t|||jt|||j} t d| t d| } | s|dkr|j t|j|j}t |zt | d | d| d|j\}}t ||j\}}||||||}|d k r|\}}}|dkrt | |} Wd QRXn&|dkr|j t|jdd|j}t |t t| td  | td | td |j\}}t t| td  | td | td |j\}}t ||j\}}td|||r||td ||||||||}|d k r|\}}}|dkrt | |} Wd QRXtd| | | }||}|| }||}|||t| |j|t||||} ||||} |}|| | |fS) a 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. r)parametric_log_derivrDexptanrr) primitivebasez@case must be one of {'exp', 'tan', 'primitive', 'base'}, not %s.N)sympy.integrals.prderEcaserrto_fieldr/ ValueErrorr#minr%rr evalZimZsqrtreZrecognize_log_derivativemaxr)rrArBr?r@r+rLrErBCZnbZncrZdcoeffalphaaalphadetaaetadAQmr'betaabetadr5ZpNZpnr=r!r!r" special_denoms^   ,     <<(    2r^Fc sddlm}m}m}|dkr"j}|j} |j} |rVtfdd|D} n |j} t| j | j } |dkrtd| t| | d} | | dkr| j rtd| | | } n|dkr| | krtd| | } ntd| | d} t jjjd\}}j}tXt | j\}}| | dkry |||||fg\\}}}Wntk rYn&Xt|dkrtd t| |d} n| | kr|||}|d k r|\}}|dkr|t| ||| | | | }t |j\}}y |||||fg\\}}}Wntk rdYn&Xt|dkr|td t| |d} Wd QRXn4|d krJtd| t| | } | | krt jtjjjjd\}}tNt | j\}}|||||}|d k r>|\}}}|dkr>t| |} Wd QRXn|d krjj}j }t| |} td| t| |d| } | | |dkr| j rtd| | | } n td || S)a$ 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. r)rElimited_integrate!is_log_deriv_k_t_radical_in_fieldrDcsg|]}|jqSr!)r$r)r(r))r+r!r"r*sz bound_degree..rIrrHzLength of m should be 1NrF)rGZother_nonlinearzScase must be one of {'exp', 'tan', 'primitive', 'other_nonlinear', 'base'}, not %s.)rKrEr_r`rLr$rrRr rLCas_expr is_Integerr r%TlevelrrrrNrr/r)rr4cQr+rL parametricrEr_r`ZdaZdbZdcalpharrWrXZt1rUrVZzaZzdr[rYZaar'betar\r]ZdeltaZlamr!)r+r" bound_degrees         &  ,       rjc Cstd|j}td|j}td|j}x|jr:||d||fS|dkdkrJt||}||jsdt||||||}}}||jdkr||}||}|||||fSt |||\} } |t ||7}| t | |}|||j8}||| 7}||9}q&WdS)aw 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. rrTN) rrrrr rr/r$rMr r) rr4r>rr+Zzerorhrir2rr'r!r!r"spdeps,      " rkcCstd|j}x|js||j||j}d|kr@|ksFntt||j||j|j||jdd}||}|d}|t||||}qW|S)a Poly Risch Differential Equation - No cancellation: deg(b) large enough. Given a derivation D on k[t], n either an integer or +oo, and b, c in k[t] with b != 0 and either D == d/dt or deg(b) > max(0, deg(D) - 1), either raise NonElementaryIntegralException, in which case the equation Dq + b*q == c has no solution of degree at most n in k[t], or a solution q in k[t] of this equation with deg(q) < n. rF)expandr)rrrr$rrrar)r4r>rr+r6r[rr!r!r"no_cancel_b_larges .rmcCsXtd|j}xD|jsR|dkr&d}n||j|j|jd}d|krX|ks^nt|dkrt||j||j|j|j||jdd}n||j||jkrt||jdkr|||j|j d||j|j dfSt||j||j|jdd}||}|d}|t ||||}qW|S)al 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. rrF)rl) rrrr$r%rrrardrer)r4r>rr+r6r[rr!r!r"no_cancel_b_smalls*  0$rnc Csxtd|j}t||j |j|j}|jrF|jrF|}nd}x&|jsrt || |j|j |jd}d|kr|ksnt t||j|j||j}|jr|||fS|dkrt||j||j||jdd} nF| |j|j |jdkr*t n ||j||j} || }|d}|t | ||| }qNW|S)ax 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. rrJrF)rl) rrr rrar%rcZ is_positiverrRr$rr) r4r>rr+r6ZlcMr[urr!r!r"no_cancel_equals* (  $*  ,  rqc Cs8ddlm}t|Bt||j\}}||||}|dk rR|\}}|dkrRtdWdQRX|jrf|S|||jkrztt d|j} x|js2||j} || krtt|.t| |j\} } t ||| | |\} }WdQRXt | | |j| |jdd}| |7} | d}|||t ||8}qW| S)a Poly Risch Differential Equation - Cancellation: Primitive case. Given a derivation D on k[t], n either an integer or +oo, b in k, and c in k[t] with Dt in k and b != 0, either raise NonElementaryIntegralException, in which case the equation Dq + b*q == c has no solution of degree at most n in k[t], or a solution q in k[t] of this equation with deg(q) <= n. r)r`Nrz7is_deriv_in_field() is required to solve this problem.F)rl)rKr`rr rNotImplementedErrorrr$rrrarischDErbr)r4r>rr+r`rArBrYr'r6r[a2aa2dsar7stmr!r!r"cancel_primitives2       &rxc Csddlm}|jt|j|j}t|Xt||j\}}t||j\}} ||| |||} | dk r| \} } } | dkrt dWdQRX|j r|S|| |jkrt td|j}x|j s| |j} || krt |}t|bt||j\}}||||t| |j}||}t| |j\}}t|||||\}}WdQRXt|||j| |jdd}||7}| d}|||t||8}qW|S)a 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. r)rENrz6is_deriv_in_field() is required to solve this problem.F)rl)rKrEr%r/rrrbrr rrrr$rrarsr)r4r>rr+rEZetarWrXrArBrYrr[r'r6r3Za1aZa1drtrurvr7rwr!r!r" cancel_exp>s>       &ryc Csddlm}m}|jsd|jdksD||jtd|j|jdkrd|rV|||||St ||||S|js||j|j|jdkrT|jdks|j|jdkrT|r|||||St ||||}t |t r|S|\}} } t |Z| |j| |j} } | dkrtd| dkr(tdt| | |||j} WdQRX|| SnL|j|jdkr||j|j|jdkr|||j |j|jkr||jjstd |rtd t||||}t |t r|S|\}} } t|| | |} || Sn|jr.td n\|jd krV|rHtd t||||S|jdkr~|rptdt||||Stdt|rtdtddS)a Solve a Polynomial Risch Differential Equation with degree bound n. This constitutes step 4 of the outline given in the rde.py docstring. For parametric=False, cQ is c, a Poly; for parametric=True, cQ is Q == [q1, ..., qm], a list of Polys. r)prde_no_cancel_b_largeprde_no_cancel_b_smallrIrrNzb0 should be a non-Null valuezc0 should be a non-Null valuezResult should be a numberz0prde_no_cancel_b_equal() is not yet implemented.zWRemaining cases for Poly (P)RDE are not yet implemented (is_deriv_in_field() required).rFzIParametric RDE cancellation hyperexponential case is not yet implemented.rHzBParametric RDE cancellation primitive case is not yet implemented.zBOther Poly (P)RDE cancellation cases are not yet implemented (%s).z2Remaining cases for Poly PRDE not yet implemented.z1Remaining cases for Poly RDE not yet implemented.)rKrzr{rrLr$rrRr%rmrn isinstancerrrrNsolve_poly_rderaZ is_number TypeErrorrrrqryrx)r4rfrr+rgrzr{Rr=Zb0Zc0yr[rTr!r!r"r}xsb $&       4*      r}cCst|||\}\}}t|||||\}\}}\} } } t|||| | |\} } }}yt| | ||}Wntk rxt}YnXt| | |||\} }}}}|jr|}nt| |||}|||| |fS)a 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. ) r8rCr^rjrrrrkrr})r9r:r;r<r+_rrArBr?r@ZhnrYrSrTZhsrr[rhrirr!r!r"rss   rs)N)rD)rDF)F)&__doc__Z __future__rroperatorrZ sympy.corerZsympy.core.compatibilityrZsympy.core.symbolrZ sympy.polysrr r r Zsympy.integrals.rischr r rrrrr#r&r8rCr^rjrkrmrnrqrxryr}rsr!r!r!r"s*     ( 0" R q,,,/: ]