U Zʹjj@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%R/var/www/html/bwave/python/venv/lib/python3.8/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|]}|tkr|dkr|qS)r)r.0ir%r%r& sz#weak_normalizer..cs,g|]$}tt|jtqSr%)rrrr)r-r"DErZd1r%r&r/sTinclude)rrrdiffrquor rrrZ resultantexprhasZ real_rootsrrr )rr)r1r+dndsgZ d_sqf_parta1br!NqZdqZsnsdr%r0r&weak_normalizer`s.   "     r@cCst||\}}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. rTr2) rrr4rr5divrr r)fafdgagdr1r8r9enesrhrccacdbabdr%r%r& normal_denoms &rNautoc CsL|dkr|j}|dkr&t|j|j}nd|dkrFt|jdd|j}nD|dkr~||}||} ||| td|jfStd|t|||jt|||j} t|||jt|||j} td| td| } | sdd lm } |dkr|j t|j|j}t |zt | d | d| d|j\}}t ||j\}}| |||||}|d k r|\}}}|dkrt| |} W5QRXn8|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 k r|\}}}|dkrt| |} W5QRXtd| | | }||}|| }||}|||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. rOexptanrr) primitivebasez@case must be one of {'exp', 'tan', 'primitive', 'base'}, not %s.rparametric_log_derivN)caserrto_fieldr5 ValueErrorr'minprderUr)rrevalr r r rmaxr)rrLrMrJrKr1rWrBCnbncr"rUZdcoeffalphaaalphadetaaetadAQmr+betaabetadr=ZpNZpnrHr%r%r& special_denoms`   ,     <<0    2rkFc s|dkrj}|j}|j}|rBtfdd|D}n |j}t|j |j} |dkrtd|t||d} ||dkr| jrtd| ||} n*|dkr||krtd||} ntd||d} t j j j d\} } j} t t | j\}}||dkrddlm}z |||| | fg\\}}}Wntk rYn&Xt|dkrtd t| |d} n||krdd lm}|||}|d k r|\}}|dkr|t|| |||  ||}t |j\}}ddlm}z |||| | fg\\}}}Wntk rtYn*Xt|dkrtd t| |d} W5QRXn@|d krjdd lm}td|t||} ||krt j tjjj j d\} } t Nt | j\}}|||| | }|d k r^|\}}}|dkr^t| |} W5QRXn|dkrj 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. rOc3s|]}|jVqdS)N)r(rr,r1r%r& (szbound_degree..rSrrrR)limited_integratezLength of m should be 1!is_log_deriv_k_t_radical_in_fieldNrPrT)rQZother_nonlinearzScase must be one of {'exp', 'tan', 'primitive', 'other_nonlinear', 'base'}, not %s.)rWr(rr]r rLCas_expr is_Integerrr)Tlevelrr[rnrrrYrprrUr5r)rr<cQr1rW parametricdadbZdcalphar"rdret1rbrcrnZzazdrhrprfZaar+betarirjrUdeltaZlamr%rlr& bound_degree s                 ,       rc Cstd|j}td|j}td|j}|jr8||d||fS|dkdkrHt||}||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) rrrrrrr5r(rXr r) rr<rIr"r1zerorzr}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<rIr"r1r>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_smalls2 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. rrVrFr) rrr rrqr)rsZ is_positiverr]r(rr) r<rIr"r1r>lcMrhurr%r%r&no_cancel_equals* ( $* ,  rc Cs4ddlm}t|Bt||j\}}||||}|dk rR|\}}|dkrRtdW5QRX|jrf|S|||jkrztt d|j} |js0||j} || krtt|.t| |j\} } t ||| | |\} }W5QRXt | | |j| |jdd}| |7} | d}|||t ||8}q| S)a Poly Risch Differential Equation - Cancellation: Primitive case. Explanation =========== Given a derivation D on k[t], n either an integer or +oo, ``b`` in k, and ``c`` in k[t] with Dt in k and ``b != 0``, either raise NonElementaryIntegralException, in which case the equation Dq + b*q == c has no solution of degree at most n in k[t], or a solution q in k[t] of this equation with deg(q) <= n. rroNz7is_deriv_in_field() is required to solve this problem.rFr)r[rprrrNotImplementedErrorrr(rrrqrischDErrr)r<rIr"r1rprLrMrfr+r>rha2aa2dsar?stmr%r%r&cancel_primitive.s2      &rc Csddlm}|jt|j|j}t|Xt||j\}}t||j\}} ||| |||} | dk r| \} } } | dkrt dW5QRX|j r|S|| |jkrt td|j}|j s| |j} || krt |}t|bt||j\}}||||t| |j}||}t| |j\}}t|||||\}}W5QRXt|||j| |jdd}||7}| d}|||t||8}q|S)a Poly Risch Differential Equation - Cancellation: Hyperexponential case. Explanation =========== Given a derivation D on k[t], n either an integer or +oo, ``b`` in k, and ``c`` in k[t] with Dt/t in k and ``b != 0``, either raise NonElementaryIntegralException, in which case the equation Dq + b*q == c has no solution of degree at most n in k[t], or a solution q in k[t] of this equation with deg(q) <= n. rrTNz6is_deriv_in_field() is required to solve this problem.rFr)r[rUr)r5rrrrrrrrr(rrqrr)r<rIr"r1rUetardrerLrMrfrrhr+r>r;Za1aZa1drrrr?rr%r%r& cancel_exp`s>      &rc Cs|js`|jdks4||jtd|j|jdkr`|rRddlm}|||||St||||S|js||j|j|jdkr\|jdks|j|jdkr\|rddlm }|||||St ||||}t |t r|S|\}} } t |Z| |j| |j} } | dkrtd| dkr0td t| | |||j} W5QRX|| SnN|j|jdkr$||j|j|jdkr$|||j |j|jkr$||jjstd |rtd t||||}t |t r|S|\}} } t|| | |} || Sn|jr6td n^|jd kr^|rPtdt||||S|jdkr|rxtdt||||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. rSrr)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).rPzIParametric RDE cancellation hyperexponential case is not yet implemented.rRzBParametric 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.)rrWr(rr]r)r[rrrr isinstancerrrrYsolve_poly_rderqZ is_number TypeErrorrrrr)r<rvr"r1rwrrRrHZb0Zc0yrhr_r%r%r&rsp " &        4&      rcCst|||\}\}}t|||||\}\}}\} } } t|||| | |\} } }}zt| | ||}Wntk rxt}YnXt| | |||\} }}}}|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. ) r@rNrkrrrrrr)rBrCrDrEr1_rrLrMrJrKZhnrfr^r_hsr"rhrzr}rr%r%r&rs  r)N)rO)rOF)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*r@rNrkrrrrrrrrrr%r%r%r&s,     $+ 3% T w///2< ]