EdkdZddlmZddlmZddlmZddlmZddl m Z m Z m Z m Z ddlmZmZddlmZdd lmZmZmZmZmZmZmZd Zd Zdd ZdZddZddZ dZ!dZ"dZ#dZ$dZ%dZ&ddZ'dZ(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)imre)sqrt)gcdex_diophantinefrac_in derivation splitfactorNonElementaryIntegralExceptionDecrementLevelrecognize_log_derivativecn|jrtS|t||kr3||ddSg}|}||}d}|jr=|||f||z}|dz}||}|j=d}td|}t|dkr[|} || dz} || }|jr || dz }| }t|dk[|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_powernproductfinalproductfs 3lib/python3.11/site-packages/sympy/integrals/rde.pyorder_atr+)sG y DAJJ'yy||  #A&& J B b AL )2l+,,, U EE"II ) A1ajjG j//Q   58# EE(OO 9  qMAG j//Q  Hct|jrtS||||z 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)rdr!s r* order_at_oor0Ts2 y 88A;;! $$r,Ncv|ptd}t|\}}t||j}||}|t||t |jjj\}} t|jtzz j j} t| |} | j |stdj|ffSd| D} ttfd| Dtdj} t| } | z|| zz }| |z}||d\}}| ||ffS)a Weak normalization. Explanation =========== Given a derivation D on k[t] and f == a/d in k(t), return q in k[t] such that f - Dq/q is weakly normalized with respect to t. f in k(t) is said to be "weakly normalized" with respect to t if residue_p(f) is not a positive integer for any normal irreducible p in k[t] such that f is in R_p (Definition 6.1.1). If f has an elementary integral, this is equivalent to no logarithm of integral(f) whose argument depends on t has a positive integer coefficient, where the arguments of the logarithms not in k(t) are in k[t]. Returns (q, f - Dq/q) zrc0g|]}|tv |dk|S)r)r ).0is r* z#weak_normalizer..s*888qa2g8!a%8888r,c ~g|]9}tt|jtzz :S)rrr!r)r4r&DErd1s r*r6z#weak_normalizer..sANNNqST!RT]]:b"+=+===rBBNNNr,Tinclude)rrrdiffr!quorrrr resultantexprhas real_rootsrrr )rr/r9r2dndsg d_sqf_parta1br$Nqdqsnsdr:s` ` @r*weak_normalizerrN`s( U3ZZA B  FB B AJ J** + +B aeeBii//55rzz"$7G7G "$  EB T!RT]]:b"-- - -66rt<<FF 24  A Q A 6::a=='Q 1v&&88ALLNN888AsNNNNNNANNN Q   A Ar  B 1qtB 1B YYr4Y ( (FB Bx=r,cdt||\}}t||\}}||} |||j| | |j} || z} | | z} | |drt | |z} | |d\} }| |z|t| |z|zz }||d\}}| ||f| |f| fS)a Normal part of the denominator. Explanation =========== Given a derivation D on k[t] and f, g in k(t) with f weakly normalized with respect to t, either raise NonElementaryIntegralException, in which case the equation Dy + f*y == g has no solution in k(t), or the quadruplet (a, b, c, h) such that a, h in k[t], b, c in k, and for any solution y in k(t) of Dy + f*y == g, q = y*h in k satisfies a*Dq + b*q == c. This constitutes step 1 in the outline given in the rde.py docstring. rTr;) rrr=r!r>divrr r)fafdgagdr9rCrDenesr hrccacdbabds r* normal_denomr]s  R FB R FB r A rwwrt}}!!!%%rt "5"566A 1A !AuuRyy|-,, 2B YYr4Y ( (FB 2:a$$$R' 'B YYr4Y ( (FB Bx"b1 %%r,autoc  |dkr|j}|dkrt|j|j}n|dkr!t|jdzdz|j}n}|dvrg||}||} ||| td|jfSt d|zt |||jt |||jz } t |||jt |||jz } td| td| z } | sdd lm } |dkr|j t|j|j}t|5t| d | dz | dz |j\}}t||j\}}| |||||}||\}}}|dkrt| |} d d d n #1swxYwYn|dkr|j t|jdzdz|j}t|5tt| td  | td z | td z |j\}}tt!| td  | td z | td z |j\}}t||j\}}t#td|j|z||r\| |ttd |jz|z||zz||z|||}||\}}}|dkrt| |} d d d n #1swxYwYt%d| | | z }||z}|| z}||z}|||zt| |j|zt'|||z|zz}||z|z|} |}||| |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. r^exptanrr) primitivebasez@case must be one of {'exp', 'tan', 'primitive', 'base'}, not %s.rparametric_log_derivN)caserr!to_fieldr> ValueErrorr+minprderer/rrevalr r r rmaxr)rr[r\rYrZr9rgr BCnbncr&redcoeffalphaaalphadetaaetadAQmr2betaabetadrIpNpnrWs r* special_denomr~sJ. v~w u} ' rt   ' q1bd # # & &' KKMM  b ! ! KKMM  b ! !1aa''!%&'' ' "a  "a!6!6 6B "a  "a!6!6 6B ArC2JJA )...... 5= )TXXd24..//F## & &!("''!**RWWQZZ)?q )I24!P!P$VRT22 d((tRHH&GAq!Av&1II & & & & & & & & & & & & & & &U] )TXXd24719bd3344F## ) )!(RWWT"XX->->,>rwwtBxx?P?P,PQRQWQWX\]_X`X`QaQa,a)b)bdfdh!i!i&r277488+<+<*.(s%---Q!((24..---r,rcrrrb)limited_integratezLength of m should be 1!is_log_deriv_k_t_radical_in_fieldNr`rd)raother_nonlinearzScase must be one of {'exp', 'tan', 'primitive', 'other_nonlinear', 'base'}, not %s.)rgr.r!rmr rLCas_expr is_Integerrr/Tlevelrrkrrrirrrrer>r)rrHcQr9rg parametricdadbdcalphar&rurvt1rsrtrzazdryrrwaar2betarzr{redeltalams ` r* bound_degreer sH( v~w "$B "$B ----"--- . . YYrt__ AIIbdOO&&((00222 "$$$&&' ( (E v~O: 2BQ' ( ( a< 'E, 'Aub2g&&A  J: 7 $ArBwAAArBw{##ART241 #566 d T B  % 7% 7$UBD11NFFR!V|# 7333333%"3"3FFdD\N##KHRa 1vv{D()BCCCAqt AA 6Dr 7 DCCCCC55ffbII7EBQw 7!":a#4#4#<#!!! D!A% 7% 7% 7% 7% 7% 7% 7% 7% 7% 7% 7% 7% 7% 7% 7N :...... 2B # $ $ 8 & $rtRT*:*:!;!;RT"(Q,=OPPJD$## & &!(!5!5((tRHH& GAq!Av&1II & & & & & & & & & & & & & & & + + : BD!!dggiiuSy!! 2BJNB/// 0 0 ea  'E$4 'Aub2g&&A2489:: : Hso(N7I:N INICNM?4A N? N  N N  NN"NARRRcXtd|j}td|j}td|j} |jr||d||fS|dkdurt||}||jst||||||}}}||jdkrU||}||}|||||fSt|||\} } |t||z }| t| |z }|||jz}||| zz }||z}k)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) rr!rrrrr>r.rhrr) rrHrXr&r9zerorrrEr$r2s r*spders" 24==D BDMME 24==D 9 /$4. . Ed? 10 0 EE!HHuuQxx 10 0%%((AEE!HHaeeAhha1 88BD>>Q  *   ##A   ##Aq!UD) ) Aq))1 Z2   1b!! ! QXXbd^^    /r,ctd|j}|js||j||jz }d|cxkr|ks ntt||j||jz |j|zz|jd}||z}|dz }|t||z ||zz }|j|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)rr!rr.rrrrrHrXr&r9rJryr s r*no_cancel_b_largers Q Ai ( HHRTNNQXXbd^^ +A 1 1 1 1 1 1 1 10 0 24##%%aiioo&8&8&:&::247BBD    E E 1b!! !AaC 'i ( Hr,ctd|j}|js*|dkrd}n=||j|j|jz dz}d|cxkr|ks nt |dkrt||j||j|jzz |j|zz|jd}n||j||jkrt ||jdkrQ|||j|j dz ||j|j dz fSt||j||jz |jd}||z}|dz }|t||z ||zz }|j*|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) rr!rr.r/rrrrrrrs r*no_cancel_b_smallrs$ Q Ai( 6 7AART!2!22Q6AA 1 1 1 1 1 1 1 10 0 q5 QYYrt__''))1RT\\"$-?-?-B-B-D-D+DEbdAgMU$$$AAxx~~"$/ 544xx~~" 3199RT"(Q,%788IIbd28a<01133QYYrt__''))!))BD//*<*<*>*>>A E E 1b!! !AaC '/i(2 Hr,c2td|j}t||j |j|jz }|jr |jr|}nd}|jst|| |j|j |jz dz}d|cxkr|ks ntt||j|jz||jz}|jr|||fS|dkrPt||j|z |j|zz|jd} n| |j|j |jdz krt||j||jz } || z}|dz }|t| |z || zz }|j|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. rrfrFr) rr!r rrr/r is_positiverrmr.rr) rHrXr&r9rJlcMryur s r*no_cancel_equalrsL Q A 24##%%%bdll24&8&8&;&;&=&== > >B }  i( 188BD>>BDKK$5$559 : :A 1 1 1 1 1 1 1 10 0 1RT\\"$''**,,,qyy/A/A/C/CC D D 9 q!9  q5 >QYYrt__''))!+BD!G3RT%HHHAAxx~~RT!2!2Q!66 >44IIbdOO&&((24););)=)== E E 1b!! !AaC ''i(* Hr,c|ddlm}t|5t||j\}}||||}||\}}|dkrt ddddn #1swxYwY|jr|S|||jkrttd|j} |js||j} || krtt|5t| |j\} } t||| | |\} }dddn #1swxYwYt| | z |j| zz|jd}| |z } | dz }|||zt||zz}|j| 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. rrNz7is_deriv_in_field() is required to solve this problem.rFr)rkrrrr!NotImplementedErrorrr.rrrrischDErr)rHrXr&r9rr[r\rwr2rJrya2aa2dsarMstms r*cancel_primitiver.sU877777   ,,BD!!B - -b"b 9 9  ,DAqAv ,)++,,, ,,,,,,,,,,,,,,, y188BD>>-,, Q Ai ) HHRTNN q5 10 0 B   3 3qttvvrt,,HCRS#r22FB 3 3 3 3 3 3 3 3 3 3 3 3 3 3 32::<< ,RT1W4bd5III S E QsUZR(( ((i ) Hs%AA$$A(+A(0AD==EEcddlm}|jt |j|j}t|5t||j\}}t||j\}} ||| |||} | | \} } } | dkrtddddn #1swxYwY|j r|S|| |jkrtt d|j}|j sT| |j} || krt|}t|5t||j\}}||z||zt | |jzz}||z}t| |j\}}t|||||\}}dddn #1swxYwYt ||z |j| zz|jd}||z }| dz }|||zt||zz}|j T|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. rrdNz6is_deriv_in_field() is required to solve this problem.rFr)rkrer/r>rr!rrrrrr.rrrr)rHrXr&r9reetarurvr[r\rwrryr2rJrGa1aa1drrrrMrs r* cancel_expr`s+***** $((4bd## $ $ , , . .C   ++S"$'' dBD!!B RtR 8 8  +GAq!Av +)+*++++++++++++++++++ y188BD>>-,, Q Ai) HHRTNN q5 10 0 YY[[ B   5 5r24((HCd(T#Xd1bdmm33Cd(Cqttvvrt,,HCS#sC44FB 5 5 5 5 5 5 5 5 5 5 5 5 5 5 52::<< ,RT1W4bd5III S E QsUZR(( ((%i)& Hs&ACC C#A?G..G25G2c|js|jdksL||jt d|j|jdz kr(|rddlm}|||||St||||S|js?||j|j|jdz kr|jdks#|j|jdkr|rddlm }|||||St||||}t|tr|S|\}} } t|5| |j| |j} } | td| td t!| | |||j} dddn #1swxYwY|| zS|j|jdkr8||j|j|jdz kr|||j |j|jz kr||jjst'd |rt)d t+||||}t|tr|S|\}} } t!|| | |} || zS|jrt)d |jd kr#|rt)dt-||||S|jdkr#|rt)dt/||||St)d|jz)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. rcrr)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).r`zIParametric RDE cancellation hyperexponential case is not yet implemented.rbzBParametric RDE cancellation primitive case is not yet implemented.zBOther Poly (P)RDE cancellation cases are not yet implemented (%s).)rrgr.r!rmr/rkrrrr isinstancerrrrisolve_poly_rder is_number TypeErrorrrrr)rHrr&r9rrrRrWb0c0yryros r*rrs 9P"'V+P HHRTNNSBDKK$5$5$9:: :P  8 4 4 4 4 4 4))!RB77 7 B2... )Gqxx~~ BD(9(9A(==G W G"$$++bd"3"3q"8G  8 4 4 4 4 4 4))!RB77 7 aQ + + a   HIAr2## @ @BD))2::bd+;+;BF$%DEEEF$%DEEE"2r1b1199"$??  @ @ @ @ @ @ @ @ @ @ @ @ @ @ @q5L RT  a 0AHHRTNNbdkk"$6G6G!6K$K0 24##%%%bdll24&8&8&;&;&=&== =0 yy!!##- 9788 8  %'    Ar1b ) ) a   HGAq!q!Q++Aq5L 9 E%'BCC Cw% EI-/HIII!!RB///K' EB-/ABBB'2q"555*+:<>G+DEEEsBGG #G cxt|||\}\}}t|||||\}\}}\} } } t|||| | |\} } }} t| | ||}n#t$r t }YnwxYwt | | |||\} }}}}|jr|}nt| |||}||z|z| |zfS)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. ) rNr]r~rrrrrr)rQrRrSrTr9_rr[r\rYrZhnrwrnrohsr&ryrrrs r*rrs""b"--KAxB ,RRR @ @AxB"b22r2r266KAq!R  Aq" % %    1aB//Aq!UDy(  1aB ' ' !GdNBrE ""sA##A76A7)N)r^)r^F)F))__doc__operatorr functoolsr sympy.corersympy.core.symbolr sympy.polysrrr r $sympy.functions.elementary.complexesr r (sympy.functions.elementary.miscellaneousr sympy.integrals.rischrrrrrrrr+r0rNr]r~rrrrrrrrrr8r,r*rs.######------------99999999999999[[[[[[[[[[[[[[[[[[ ( ( ( V % % %0000f"&"&"&JQQQQht t t t n---^   <+ + + ^, , , ^/ / / d9 9 9 xZZZZz'#'#'#'#'#r,