o o hj@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|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_polyETremappendlenpop) apt power_listp1r tracks_powernproductfinalproductfr(g/var/www/html/construction_image-detection-poc/venv/lib/python3.10/site-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_ooTsr-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)r0r$DErd1r(r)r2s,Tinclude)rrrdiffrquor rrr resultantexprhas real_rootsrrr )rr,r4r.dndsg d_sqf_parta1br"Nqdqsnsdr(r3r)weak_normalizer`s.   "      rIcCst||\}}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. rTr6) rrr8rr9divrr r)fafdgagdr4r>r?enesrhrccacdbabdr(r(r) normal_denoms &rWautocCsl|dkr|j}|dkrt|j|j}n2|dkr#t|jdd|j}n"|dvr?||}||} ||| td|jfStd|t|||jt|||j} t|||jt|||j} td| td| } | svdd lm } |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. rXexptanrr) primitivebasez@case must be one of {'exp', 'tan', 'primitive', 'base'}, not %s.rparametric_log_derivN)caserrto_fieldr9 ValueErrorr*minprder^r,rrevalr r r rmaxr)rrUrVrSrTr4r`rBCnbncr$r^dcoeffalphaaalphadetaaetadAQmr.betaabetadrDpNpnrQr(r(r) special_denomsj  ,    <<0      2 rwFc s|dkrj}|j}|j}|r!tfdd|D}n|j}t|j |j} |dkratd|t||d} ||dkr_| jr_td| ||} | S|dkr||krrtd||} n td||d} t j j j d\} } j} t t | j\}}||dkrddlm}z|||| | fg\\}}}Wn tyYnwt|dkrtd t| |d} n||krQdd lm}|||}|d uri|\}}|dkrq|t|| |||  ||}t |j\}}ddlm}z|||| | fg\\}}}Wn ty4YnEwt|dkr@td t| |d} Wd | SWd | SWd | SWd | SWd | SWd | SWd | S1swY| S|d krdd lm}td|t||} ||krt j tjjj j d\} } t =t | j\}}|||| | }|d ur|\}}}|dkrt| |} Wd | SWd | SWd | S1swY| S|dvrDj j}j }t| |} td|t||d|} |||dkrB| jrBtd| ||} | 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. rXc3s|] }|jVqdSN)r+rr/r4r(r) (szbound_degree..r\rrr[)limited_integratezLength of m should be 1!is_log_deriv_k_t_radical_in_fieldNrYr])rZother_nonlinearzScase must be one of {'exp', 'tan', 'primitive', 'other_nonlinear', 'base'}, not %s.)r`r+rrfr rLCas_expr is_Integerrr,Tlevelrrdr{rrrbr}rr^r9r)rrCcQr4r` parametricdadbdcalphar$rnrot1rlrmr{zazdrrr}rpaar.betarsrtr^deltalamr(ryr) bound_degree s   N               C C C C C C CC  ,           rc Cstd|j}td|j}td|j} |jr||d||fS|dkdur%t||}||js2t||||||}}}||jdkr`||}||}|||||fSt |||\} } |t ||7}| t | |}|||j8}||| 7}||9}q)a Rothstein's Special Polynomial Differential Equation algorithm. Explanation =========== Given a derivation D on k[t], an integer n and ``a``,``b``,``c`` in k[t] with ``a != 0``, either raise NonElementaryIntegralException, in which case the equation a*Dq + b*q == c has no solution of degree at most ``n`` in k[t], or return the tuple (B, C, m, alpha, beta) such that B, C, alpha, beta in k[t], m in ZZ, and any solution q in k[t] of degree at most n of a*Dq + b*q == c must be of the form q == alpha*h + beta, where h in k[t], deg(h) <= m, and Dh + B*h == C. This constitutes step 4 of the outline given in the rde.py docstring. rrT) rrrrrrr9r+rar r) rrCrRr$r4zerorrr@r"r.r(r(r)spdes.      " rcCstd|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+rrrrrCrRr$r4rErrrr(r(r)no_cancel_b_larges . rcCsVtd|j}|js|dkrd}n||j|j|jd}d|kr*|ks-tt|dkrPt||j||j|j|j||jdd}nC||j||jkr^t||jdkr}|||j|j d||j|j dfSt||j||j|jdd}||}|d}|t ||||}|jr |S)a Poly Risch Differential Equation - No cancellation: deg(b) small enough. Explanation =========== Given a derivation D on k[t], ``n`` either an integer or +oo, and ``b``,``c`` in k[t] with deg(b) < deg(D) - 1 and either D == d/dt or deg(D) >= 2, either raise NonElementaryIntegralException, in which case the equation Dq + b*q == c has no solution of degree at most n in k[t], or a solution q in k[t] of this equation with deg(q) <= n, or the tuple (h, b0, c0) such that h in k[t], b0, c0, in k, and for any solution q in k[t] of degree at most n of Dq + bq == c, y == q - h is a solution in k of Dy + b0*y == c0. rrFr) rrrr+r,rrrrrrrr(r(r)no_cancel_b_smalls6 0$rc 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. rr_rFr) rrr rrr,r is_positiverrfr+rr) rCrRr$r4rElcMrrurr(r(r)no_cancel_equals0 ( $* , rcCs^ddlm}t|&t||j\}}||||}|dur)|\}}|dkr)tdWdn1s3wY|jr=|S|||jkrGtt d|j} |js||j} || kr\tt|t| |j\} } t ||| | |\} }Wdn1swYt | | |j| |jdd}| |7} | d}|||t ||8}|jrP| S)a Poly Risch Differential Equation - Cancellation: Primitive case. Explanation =========== Given a derivation D on k[t], n either an integer or +oo, ``b`` in k, and ``c`` in k[t] with Dt in k and ``b != 0``, either raise NonElementaryIntegralException, in which case the equation Dq + b*q == c has no solution of degree at most n in k[t], or a solution q in k[t] of this equation with deg(q) <= n. rr|Nz7is_deriv_in_field() is required to solve this problem.rFr)rdr}rrrNotImplementedErrorrr+rrrrischDErr)rCrRr$r4r}rUrVrpr.rErra2aa2dsarHstmr(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. rr]Nz6is_deriv_in_field() is required to solve this problem.rFr)rdr^r,r9rrrrrrrr+rrrr)rCrRr$r4r^etarnrorUrVrprrrr.rErBa1aa1drrrrHrr(r(r) cancel_exp`sF      & rcCs|js0|jdks||jtd|j|jdkr0|r)ddlm}|||||St||||S|jsB||j|j|jdkr|jdksP|j|jdkr|r_ddlm }|||||St ||||}t |t rm|S|\}} } t |3| |j| |j} } | durtd| durtd t| | |||j} Wd|| S1swY|| S|j|jdkr||j|j|jdkr|||j |j|jkr||jjstd |rtd t||||}t |t r|S|\}} } t|| | |} || S|jrtd |jd kr2|r+tdt||||S|jdkrF|r?tdt||||Std|j)a  Solve a Polynomial Risch Differential Equation with degree bound ``n``. This constitutes step 4 of the outline given in the rde.py docstring. For parametric=False, cQ is c, a Poly; for parametric=True, cQ is Q == [q1, ..., qm], a list of Polys. r\rr)prde_no_cancel_b_larger)prde_no_cancel_b_smallNzb0 should be a non-Null valuezc0 should be a non-Null valuezResult should be a numberz0prde_no_cancel_b_equal() is not yet implemented.zWRemaining cases for Poly (P)RDE are not yet implemented (is_deriv_in_field() required).rYzIParametric RDE cancellation hyperexponential case is not yet implemented.r[zBParametric RDE cancellation primitive case is not yet implemented.zBOther Poly (P)RDE cancellation cases are not yet implemented (%s).)rr`r+rrfr,rdrrrr isinstancerrrrbsolve_poly_rder is_number TypeErrorrrrr)rCrr$r4rrrRrQb0c0yrrrhr(r(r)rsf $ $     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. ) rIrWrwrrrrrr)rKrLrMrNr4_rrUrVrSrThnrprgrhhsr$rrrrrr(r(r)rs  rrx)rX)rXF)F))__doc__operatorr functoolsr sympy.corersympy.core.symbolr sympy.polysrrrr $sympy.functions.elementary.complexesr r (sympy.functions.elementary.miscellaneousr sympy.integrals.rischr rrrrrrr*r-rIrWrwrrrrrrrrrr(r(r(r)s.     $+ 3 % Tw///2 < ]