a khk@sdZddlmZddlmZddlmZddlmZddl m Z m Z m Z m Z ddlmZmZddlmZdd lmZmZmZmZmZmZmZd d Zd d Zd)ddZddZd*ddZd+ddZ ddZ!ddZ"ddZ#dd Z$d!d"Z%d#d$Z&d,d%d&Z'd'd(Z(dS)-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)imre)sqrt)gcdex_diophantinefrac_in derivation splitfactorNonElementaryIntegralExceptionDecrementLevelrecognize_log_derivativec Cs|jr tS|t||kr.||ddSg}|}||}d}|jrt|||f||}|d9}||}qDd}td|}t|dkr|} || d} || }|jr|| d7}| }q|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_polyETremappendlenpop) aptZ power_listp1rZ tracks_powernproductfinalZproductfr%A/var/www/auris/lib/python3.9/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_ooTsr*NcsF|p td}t|\}}t||j}||}|t||t|jjj\}} t|jt j j} t| |} | j |stdj|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& z#weak_normalizer..cs,g|]$}tt|jtqSr%)rrrr)r-r"DErd1r%r&r/r0Tinclude)rrrdiffrquor rrrZ resultantexprZhasZ real_rootsrrr )rr)r2r+dndsgZ d_sqf_parta1br!NqZdqZsnsdr%r1r&weak_normalizer`s.   "     rAcCst||\}}t||\}}||} |||j| | |j} || } | | } | |drnt| |} | 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. rTr4) rrr6rr7divrr r)fafdgagdr2r9r:enesrhrccacdbabdr%r%r& normal_denoms &rOautocCsx|dkr|j}|dkr&t|j|j}nd|dkrFt|jdd|j}nD|dvr~||}||} ||| td|jfStd|t|||jt|||j} t|||jt|||j} td| td| } | sdd lm } |dkr|j t|j|j}t |t | d | d| d|j\}}t ||j\}}| |||||}|d ur|\}}}|dkrt| |} Wd n1s0YnN|dkr|j t|jdd|j}t | t t| td  | td | td |j\}}t t| td  | td | td |j\}}t ||j\}}ttd|j|||r| |ttd |j||||||||}|d ur|\}}}|dkrt| |} Wd n1s0Ytd| | | }||}|| }||}|||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. rPexptanrr) primitivebasez@case must be one of {'exp', 'tan', 'primitive', 'base'}, not %s.rparametric_log_derivN)caserrto_fieldr7 ValueErrorr'minprderVr)rrevalr r r rmaxr)rrMrNrKrLr2rXrBCnbZncr"rVZdcoeffalphaaalphadetaaetadAQmr+betaabetadr>ZpNZpnrIr%r%r& special_denoms`   ,   .  <<0   * 2rkFc s|dkrj}|j}|j}|rBtfdd|D}n |j}t|j |j} |dkrtd|t||d} ||dkr| jrtd| ||} nR|dkr||krtd||} ntd||d} t j j j d\} } j} t t | j\}}||dkrddlm}z |||| | fg\\}}}WntyYn&0t|dkrtd t| |d} n||krdd lm}|||}|d ur|\}}|dkr|t|| |||  ||}t |j\}}ddlm}z |||| | fg\\}}}WntypYn*0t|dkrtd t| |d} Wd n1s0YnV|d krdd lm}td|t||} ||krt j tjjj j d\} } t Xt | j\}}|||| | }|d urp|\}}}|dkrpt| |} Wd n1s0Yn|dvrj j}j }t| |} td|t||d|} |||dkr| jrtd| ||} n td|| S)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. rPc3s|]}|jVqdS)N)r(rr,r2r%r& (r0zbound_degree..rTrrrS)limited_integratezLength of m should be 1!is_log_deriv_k_t_radical_in_fieldNrQrU)rRZother_nonlinearzScase must be one of {'exp', 'tan', 'primitive', 'other_nonlinear', 'base'}, not %s.)rXr(rr^r rLCas_expr is_Integerrr)Tlevelrr\rnrrrZrprrVr7r)rr=cQr2rX parametricdadbdcalphar"rdret1rbrcrnZzaZzdrhrprfZaar+betarirjrVdeltaZlamr%rlr& bound_degree s              6   ,    ,   rc Cstd|j}td|j}td|j}|jr8||d||fS|dkdurHt||}||jsbt||||||}}}||jdkr||}||}|||||fSt |||\} } |t ||7}| t | |}|||j8}||| 7}||9}q$dS)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. rrTN) rrrrrrr7r(rYr r) rr=rJr"r2Zzeror{r}r;r!r+r%r%r&spdes*      " rcCstd|j}|js||j||j}d|kr>|ksDntt||j||j|j||jdd}||}|d}|t||||}q |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(rrrqrr=rJr"r2r?rhrr%r%r&no_cancel_b_larges .rcCsRtd|j}|jsN|dkr"d}n||j|j|jd}d|krT|ksZnt|dkrt||j||j|j|j||jdd}n||j||jkrt||jdkr|||j|j d||j|j dfSt||j||j|jdd}||}|d}|t ||||}q |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. rrFr) rrrr(r)rrrqrtrurrr%r%r&no_cancel_b_smalls0 0$rc Csptd|j}t||j |j|j}|jrF|jrF|}nd}|jslt || |j|j |jd}d|kr|ksnt t||j|j||j}|jr|||fS|dkrt||j||j||jdd} nF| |j|j |jdkr$t n ||j||j} || }|d}|t | ||| }qJ|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. rrWrFr) rrr rrqr)rsZ is_positiverr^r(rr) r=rJr"r2r?lcMrhurr%r%r&no_cancel_equals* ( $* ,  rcCs^ddlm}t|Lt||j\}}||||}|durR|\}}|dkrRtdWdn1sf0Y|jrz|S|||jkrtt d|j} |jsZ||j} || krtt|8t| |j\} } t ||| | |\} }Wdn1s0Yt | | |j| |jdd}| |7} | d}|||t ||8}q| 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. rroNz7is_deriv_in_field() is required to solve this problem.rFr)r\rprrrNotImplementedErrorrr(rrrqrischDErrr)r=rJr"r2rprMrNrfr+r?rha2aa2dsar@stmr%r%r&cancel_primitive.s2   &   4&rcCsddlm}|jt|j|j}t|bt||j\}}t||j\}} ||| |||} | dur| \} } } | dkrt dWdn1s0Y|j r|S|| |jkrt td|j}|j s| |j} || krt |}t|lt||j\}}||||t| |j}||}t| |j\}}t|||||\}}Wdn1sn0Yt|||j| |jdd}||7}| d}|||t||8}q|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. rrUNz6is_deriv_in_field() is required to solve this problem.rFr)r\rVr)r7rrrrrrrrr(rrqrr)r=rJr"r2rVetardrerMrNrfrrhr+r?r<Za1aZa1drrrr@rr%r%r& cancel_exp`s>   &   4&rcCs|js`|jdks4||jtd|j|jdkr`|rRddlm}|||||St||||S|js||j|j|jdkrr|jdks|j|jdkrr|rddlm }|||||St ||||}t |t r|S|\}} } t |d| |j| |j} } | durtd| dur0td t| | |||j} Wdn1s\0Y|| SnN|j|jdkr:||j|j|jdkr:|||j |j|jkr:||jjstd |rtd t||||}t |t r|S|\}} } t|| | |} || Sn|jrLtd n^|jd krt|rftdt||||S|jdkr|rtdt||||Std|j|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. rTrr)prde_no_cancel_b_larger)prde_no_cancel_b_smallNzb0 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).rQzIParametric RDE cancellation hyperexponential case is not yet implemented.rSzBParametric 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.)rrXr(rr^r)r\rrrr isinstancerrrrZsolve_poly_rderqZ is_number TypeErrorrrrr)r=rvr"r2rwrrRrIZb0Zc0yrhr`r%r%r&rsp " &      6 4&      rcCst|||\}\}}t|||||\}\}}\} } } t|||| | |\} } }}zt| | ||}Wntyvt}Yn0t| | |||\} }}}}|jr|}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. ) rArOrkrrrrrr)rCrDrErFr2_rrMrNrKrLZhnrfr_r`Zhsr"rhr{r}rr%r%r&rs   r)N)rP)rPF)F))__doc__operatorr functoolsrZ sympy.corerZsympy.core.symbolrZ sympy.polysrrrr Z$sympy.functions.elementary.complexesr r Z(sympy.functions.elementary.miscellaneousr Zsympy.integrals.rischr rrrrrrr'r*rArOrkrrrrrrrrrr%r%r%r&s,     $+ 3% T w///2< ]