\hrSSKJr SSKJr SSKJr SSKJrJrJ r SSK J r SSK J r SSKJrJr SSKJrJr SS KJr SS KJr SS KJr SS KJr SS KJr SSKJr SSK J!r! SSK"J#r#J$r$J%r%J&r&J'r'J(r(J)r)J*r*J+r+J,r,J-r-J.r.J/r/J0r0J1r1J2r2 \"S5r3\"S5r4Sr5Sr6Sr7Sr8Sr9Sr:Sr;SrSr?Sr@SrAS rBS!rCS"rDS#rES$rFS%rGS&rH\!S'5rIg()))randint)Function)Mul)IRationaloo)Eq)S)Dummysymbols)explog)tanh)sqrt)sin)Poly)ratsimp) checkodesol)slow)riccati_normalriccati_inverse_normalriccati_reduced match_riccatiinverse_transform_poly limit_at_infcheck_necessary_conds val_at_infconstruct_c_case_1construct_c_case_2construct_c_case_3construct_d_case_4construct_d_case_5construct_d_case_6rational_laurent_series solve_riccatifxcD[[U*U5[SU55$N)rr)maxints \/var/www/auris/envauris/lib/python3.13/site-packages/sympy/solvers/ode/tests/test_riccati.py rand_rationalr-s GVGV,ga.@ AAcp[[US-5Vs/sHn[U5PM snU5$s snfr))rranger-)r'degreer+_s r, rand_polyr3!s. fQh@1v&@! DD@s3c[SU5n[SU5n[XU5n[XU5nU[SU5:Xa[XU5nU[SU5:XaMXV- $)Nr*r)rr3r)r'r1r+degnumdegdennumdens r,rand_rational_functionr9%sd Q F Q F Av &C Av &C a 6* a  9r.cPUnURU5n[USS5n[USS5nUS:Xa[USS5nUS:XaM[XEU-- XcS--- 5n[UR5XuU--XbS---5n[X#5n [ X5S:XdeXXV4$)Nr*r)Tr)diffr9rr r) ratfuncr'yfyypq1q2q0eqsols r,find_riccati_oderG/sA B 1a (B 1a (B ' #Aq! , ' dR1W$ %B BGGIrrEzB1uH, -B R)C r 9 ,, , 2>r.c [[S- - [S-S-S- [-[[S-*[S- - [[S-S- [S5S- --S- - SS[-- - 4S[-S-S[-S-- SS[-- [S-- S[-S[-S[-S--S[-S-- SS[-- [S[S-SS9- - SS[-- - 4S S[S--S- - S S[- S [-S- - S[- S [-S- S[S--S- -- S [-S- [S S[- SS9S [-S- S-- SS [-S- - - -[SS[- SS9- - 4[S [-S- S[-S-- [S-S[-S- - [S -*S[-S- - S [-S- [SS[-S-SS9- - S[-S- S[S--S[-S- S-- S[S--S[-S- - --S[S--- - 4/nUHBupp4U[U[X#5:XdeU[ U[X#5R 5:XaMBe S[-S- S[S--S[--S- - S[- [*S- S [-- S [S--SS [-- [*S- S [S--- --[*S- S-- S [*S- - -S[-S S [-- [*S- S [S--- - -[*S- - S[-S- [*S- -S [-S[S--S[--S- -- - S[- -4SS[S--- S[- [*S- S [-- S [S--SS [-- [*S- S [S--- --[*S- S-- S [*S- - -S[-S S [-- [*S- S [S--- - -[*S- - S[- -[S[*S- SS9S [S--- - 4/nUHDupp5nU[U[X#5:XdeU[ U[X#U5R 5:XaMDe g)aN This function tests the transformation of the solution of a Riccati ODE to the solution of its corresponding normal Riccati ODE. Each test case 4 values - 1. w - The solution to be transformed 2. b1 - The coefficient of f(x) in the ODE. 3. b2 - The coefficient of f(x)**2 in the ODE. 4. y - The solution to the normal Riccati ODE. r*r<r;Fevaluater N)r'r rrrcancel)testswb1b2r@bps r,test_riccati_transformationr]?sB 1q5 A1 Q Aq1u 1a46AaDF?+A--1Q37  1q1Q37 QqS1q5 ! 1acAg!a A!Gc!QUU.K#LLqRSTURUwV  AadFQJ Q1q Q!A#'AadFQJ'(AaC!Gc"a!ee6TVWXYVY W V71Q37 7,!!QUU;,= =  1rBqD1H 1acAg AqsQw1Q38c!RTAX&FGG1Q3QR7UWXY[\X\U\^_`a^a _ ^VAqD&!A#'"V#K$%&q!tVK- - ) E6 rN1a0000*1a8??AAAA AAadFQqSL1$% 1 a!A# !Q$1Q3A26AadF++,qb1fq[81qb1f:E 1b!A#h1"q&1QT6**+aR!V41qA267JAaCQRSTVWSWQWZ[\]Z]Q] R M 8 !   1QT6  1 a!A# !Q$1Q3A26AadF++,qb1fq[81qb1f:E 1b!A#h1"q&1QT6**+aR!V4qs:SQBFUZ=[]^_`bc_c]c=dd  E "rqN1a0000*1a<CCEEEE"r.c  [[5R[5[S-- [[[5-- [[[5S--- [[5R[5[[5S--[S--[S-S- - SS[S--- - 4S[-S[-S-- [[5R[5-[S-[[5S--[- - S[S--S[- [*S- [S-- -S--S[*S- S--- [S[*S- SS 9S[-S-- -[[5S--[[5R[5-S [S-[- -[[*S- -- - 4[[5S-[[5R[5-[S- [[5-[*[ S5S- - - - S[-S- S-*SS[-S-S--- S[-S- S[-S-S-- -[[5S--[[5R[5-SS[-S-- - 4[[5R[5[[5S-[- - [[5S-[[5R[5-SS[S--- -4S[S-*[- S--[S-S[--S-- [[5R[5-[[5S-[- -[[5S-[[5R[5-S[S--[S-S[--S-- S[-[S-S[--S-- -S[S-S[--S-- - [- -SS[S--- -4S[-S[-S-- [[5R[5-[S-[[5-[- - S4[[5[[5R[5-S[- - [[5S- -[[5S-[S-S- - -S4/nUH upU[ U[[5:XaM e g ) z This function tests the transformation of a Riccati ODE to its normal Riccati ODE. Each test case 2 values - 1. eq - A Riccati ODE. 2. normal_eq - The normal Riccati ODE of eq. r<r;rOrTrSr*FrLrNN)r&r'r=rr r)rXrE normal_eqs r,test_riccati_reducedrasJ ! ! q!ta!f$q1qy0 ! ! qtQwA%1Q.AadF; !QqS1W ! ! $AqtQwq'88 1a41Q1}$q((!aR!VaK-83q Q< !"1q<* *,-aD!G 467diil C Q >ArAvJ ' ( !a!A$))A,!a%1rAaDF{!;; A#'A q!A#'A~&!A#'AaC!Ga<)?? !a A$))A, !"AaC!G - ! ! qtQwqy  !a!A$))A,AadF+ QTEAIMAqD1Q3JN+adiil:QqT1WQYF !a!A$))A,!AqD&!Q$1*q."9AaCA A#BB=#1qs Q'#()*"+ +-.!Q$Z 8 !QqS1W ! ! $Aqt|A~5  !QqTYYq\AaC!A$q&(1Q47AqD1H+== G' EP OB15555r.c [[5R[5S[S--S[S--- S[-- S-S[S--S [S--- S [S---S [--S - - - S- [[5S-S[-S -- - [S 5S- [- [[5-[S 5S- S[-S- - - - SS[S--S[S--S[S--- S[S---S[--S- - S[S--S[S--S[S--- S[S---S[--S- - - S[-S[S--S[S--- S[S---S[--S- - - S-SS[S--S [S--- S [S---S [--S - - -[ SSS[-- SS9S [-S- - S S[-S -- 4[[5R[5S[-S- -[S- S - [[5S--- S[-S- S -[[5-S[-S-- - [S!5S- - S"[S--S#[--S$-S%[S--S&[S---- - SS'[-S- [S!5S- -SS[S--S[S---- -S(S)[S--S[--- -S"S%[-S&-- -[ SS*[-S- SS9S [-S-- [S- S - 4[[5R[5S+[S,--S-[S--- S.[S---S/[S--- S0[--S1- S)[S--S&[S,--- S2[S---S3[S--- S4[S---S1[-- - - [S55S - - [[S5S- - [[5-[S- [S5S- - - - [S- S- [[5S--S[-- - SS+[S--S)[S,--S&[S--- S2[S---S3[S--- S4[--S1- - S-[S--S)[S,--S&[S--- S2[S---S3[S--- S4[--S1- - - S.[S--S)[S,--S&[S--- S2[S---S3[S--- S4[--S1- - -S6[-S7[S,--S [S--- S8[S---S9[S--- S:[--S)- - - [S55S - -S7S[S--S[S,--- S;[S---S<[S--- S=[S---S7[-- - - S>S[S,--S[S--- S;[S---S<[S--- S=[--S7- - -[ SSS[-- SS9[S- - [ SS [- SS9S [-- 4[[5R[5[[S5S- -[[S 5S- -S- - - [S-[[5--[[[5S--[[S5S- -- -SS?S?S?4[[5R[5[ [S-5- [ [5[[5--[[5[[5S---SS?S?S?4[[5R[5[[[[5-5- [[5-[S-[[5S---SS?S?S?4S [S-- [[5R[S5-S[-[[5R[5-- S[[5--SS?S?S?4[[5R[5[S-- [S-[[5--[S-[S -- [[5S---SS?S?S?4[[5R[5[[5S--[S-S - [S-S -- [[5--S S[-S-- -[[5S--SS?S?S?4/ nUH8upp4n[U[[5upgXb:XdeU(dM.X4U/U:XaM8e g@)Az This function tests if an ODE is Riccati or not. Each test case has 5 values - 1. eq - The Riccati ODE. 2. match - Boolean indicating if eq is a Riccati ODE. 3. b0 - 4. b1 - Coefficient of f(x) in eq. 5. b2 - Coefficient of f(x)**2 in eq. ir;irr<N\rOiiNHr*T-i^rQbQ;i<rNrTFrLrS i iiDrP/$rVi0rJiiwillii7ihi$rRx5?(rN) r&r'r=r rrr rrrr)rXrEresb0rZr[matchfuncss r,test_match_riccatirs  ! ! AqD3q!t8+bd2R7#ad( ad(;AX; #A;&(*;+, ,./ 023A$'1Q372C D Q46A:qt QqT!Vac!e^ , -  1a4AqD3q!t8#bAg-14q89Bq!tG AqD3q!t8 bAg %1 ,q 0=2 246qD"QT' AqD;q!t8; d;#%';(5) )+, -/13q!t8c!Q$h3F ad(4U44 0! ! BAaC%(!A#'2 1Q37 " ! ! qs2v1q!A$' 11QqSU 6 Q451q5 bE"H %(+AqD3q5(83(> ad(SAX (   1R!B%(Q!Q$1a4002r!Q$w Q$84 s1us{# $ B1q5)1Q373 !a  ! ! AqD3q!t8+c!Q$h6QTA a%1a4#ad(*SAX5AqD@ AqDq5 bE!G $'(1Q46z1Q4&71 !Q:' Q37AaD!G#QqS) *  AqD"QT'C1H$s1a4x/#ad(:SUB   1a4xAqD3q!t8+c!Q$h6QTA A 1a4xAqD3q!t8!3c!Q$h!> 1B " 1u"" "!eR1Wr!Q$w%6QT%A AqD&q5&& "#2q  ),.qAv1a4/? 1a40QT'0q!tG0$&(d0+,,  , /1!AqD&2ad72B QT'3q!tG3 d3#%'3(/)  ) BAaC%(!a%0 BA&!,%, ! ! q1Q46{A!QK!O44q!tAaDy@ !A$' 1qtAv;  q!Q ! ! s1a4y 3q6!A$;.Q!a? q!Q ! ! tAQK((1Q4/!Q$qtQw,> q!Q QT1Q499Q?"QqS11%551Q4? q!Q ! ! q!tad1Q4i'1a4Q<1q*@@ q!Q ! ! QqT1W1q1a4!84QqT99Aqs @ =  tQw  q!QWO E` %$RA. || 3B<5( (( %r.c [S[S--S[S---S[-- S-[5[S[S--[S-- S [S---S [S ---S[S---S[S ---S [S --- S [S---S[S--- S [--S-[5S 4[S [5[S[S --S[S---S[S---S[-- S- [5S 4[S[S--S[S--- S[--S- [5[S[S--S[S---S[-- S- [5S4[S[S--S[S --- S [S--- S[S ---S[S ---[S-- [S--S [-- [5[S[S--[-[5S4[S [S--S[S ---S [S --- S[S--- [S--S [-- S -[5[S[S --S[S---S[S--- S[--S-[5S4/nUHupn[X[5U:XaMe g)z This function tests the valuation of rational function at oo. Each test case has 3 values - 1. num - Numerator of rational function. 2. den - Denominator of rational function. 3. val_inf - Valuation of rational function at oo rsr;rQr< rTirSrJrIrO r*rUrKrRrirVN)rr'r)rXr7r8vals r,test_val_at_infrs R1WqAv 1 $q (!, SBYA !Q$ &1a4 /"QT' 9AadF BQq!tV KbQRTUQUg UXYZ[]^Z^X^ ^abcdad dgi iklm  Q  R1WqAv 1a4 '!A# - 2A6  R1WqAv ! #a '+ R1Wr!Q$w 1 $q (!,  R1Wr!Q$w AqD (1QT6 1AadF :QT AAqD H2a4 OQRS SAX\1  Qq!tVa1f_r!Q$w &1a4 /!Q$ 61 1/x in rational functions represented using Poly. rr;rQr<rTrfrJrrOPrqKrRirwrIrrrldrsrhrSr*N)r'as_numer_denomrrsubsrW)fnsr&er7r8s r,test_inverse_transform_polyrUs 1WqAv!a"Q$(+AX1a4"QT'!Bq!tG+bd2R7#ad(SAX:MPRSTPT:TWY:YZAX1a4"QT'!Bq!tG+bd2R7"QT'Bq!tG:KbQRTUQUg:UXZ[\X\:\_a:ab1Wr!Q$wAqD 2ad7*R1W4s1uAqD H3q5 PSU UWXY SAXAqD 3q!t8 +c!Q$h 6ad BT!Q$Y NQUVWQW WZ] ]_`a !Q SAXAqD 3q!t8 +c!e 3b 8!< SAXAqD 419 ,s1u 4s :A> #s 5 E> #Ca(C///r.c[S[S--S[S---S[--S- [SS9[S[S--S [S ---S [S---S [S---S [S---S [S---[SS9[S5[S5S- [S 5[-S - -/[S5S- [S 5[-S - - //4[S[S--S[S---S[--S-[SS9[S[S--S[S--- S[S---S[S--- S[--S-[SS9[S 5S- [S5S- [S5S- -/[S5S- [S5S- - //4[S[-S-[SS9[S[S--SS[S5-- [S---SS[S5-- [S---SS [S5-- [--S[S5--S -[SS 9[S5[S5-S- [S5S- [[ SS[S5-S-S!S"9S#[S5-S$-- S-5S- -/[S5S- [[ SS[S5-S-S!S"9S#[S5-S$-- S-5S- - //4/nUHupp4[ X[U5U:XaMe g%)&a> This function tests the Case 1 in the step to calculate coefficients of c-vectors. Each test case has 4 values - 1. num - Numerator of the rational function a(x). 2. den - Denominator of the rational function a(x). 3. pole - Pole of a(x) for which c-vector is being calculated. 4. c - The c-vector for the pole. r_r;r<rOrJT extensionrQrIrSrRrTrr*iii0i0iiiei1ijizrrrz QQdomainFrL,N)rr'r rrrr)rXr7r8polecs r,test_construct_c_case_1rs  R1WqAv ! #a 'd; Qq!tVbAg !Q$ &AqD 01QT6 9Bq!tG CQRVW ! A$q&4719Q;  !A$q&4719Q;"6!78  T!Q$Yad "SU *S 0!tD SAXAqD 419 ,tAqDy 83q5 @2 EqTXY !Q A$q&4>#% % &1a$x.2D)D(EF  QqS1Wa4( R1WBtAwJ1, ,RQZA/E EQtTUwYXYHY YQi  7 1Q A$q&4Aqay1}u=r$q'zBORSSTUVV V W qT!Vd3q!DG)a-%@"T!W*r/RUVVWXYY Y Z \  E*#$!#At4999#r.c[S[SS9[[S- S-[S- -[SS9SS[*S[- -S- /[S[--S- //4[S[S--S[S --- S [S--- S-[SS9[S[-S- S-[S-S--[SS9[S5S- S[S 5*S - /[S 5*S - //4[[S-[S-- S [--[SS9[[S- S -[S-S--[SS9SS S [ S5-[S5S- S [ S5-S - - -S- S[ S5-S - /S[ S5-[S5S- S [ S5-S - --S- S[ S5-S - //4[S[S--[S --S-[SS9[S [-S-S -[S--[SS9[S5*S - S S[ S5-[ S5*S- [S5S- - -S- [ S5S- /S[ S5-[ S5S- [S5S- - -S- [ S5*S- //4[[S-S-[SS9[S[-S- S-[S-S--[SS9[S5S- SS[ S5-[ S5*S- [S5S- - -S- S[ S5-S- [ S5S - /S![ S5-[ S5S- [S5S- - -S- S[ S5-S- [ S5*S - //4[[S-S-[SS9[[[ S5- S-[SS9[ S5S[ S"5[S5S - S[ S"5-- -S#- [ S 5S - [ S"5/[ S"5*[S5S - S[ S"5---S#- [ S 5*S - [ S"5*//4/nUHupp4n[ X[X45U:XaMe g$)%ae This function tests the Case 2 in the step to calculate coefficients of c-vectors. Each test case has 5 values - 1. num - Numerator of the rational function a(x). 2. den - Denominator of the rational function a(x). 3. pole - Pole of a(x) for which c-vector is being calculated. 4. mul - The multiplicity of the pole. 5. c - The c-vector for the pole. r*Trr<rNr;rJrRrOrIYrmrSrqiWrVri7i8irTriB6i$rirtirN)rr'rr rr)rXr7r8rmulrs r,test_construct_c_case_2rs^" QT" a!eaZQ d3 1 "b1f+a-1b1f:a<.)  Qq!tVbAg !Q$ & *A> acAg\1q51* $a48 !Q R5&)!uRxj!  QTAqD[1Q3 T2 a!eaZQ "A6 1 DG)QrU3Y471, -b 0!DG)A+ > DGQrU3Y471, -b 0"T!W*Q,? A  Qq!tVad]Q T2 acAg\1q5 !15 1a d3i-$s)C!B%+5 6s :DIcM J d3ic32u4 5c 9DI:c>J L  QTAXqD) acAg\1a4!8 $a48 !Q T"X+Ry|afVm3 4R 7DHT9I4PR8TW< X T"XtBx{QsVF]2 3B 6$r( 4$r(SVW Y  QTBYT* a$q'kA qD1 Q r(AaDFQtBxZ' ( +T!WQYR A r(AaDFQtBxZ' ( +d1gXaZ$r(C E K+ EX#($Q!#At9Q>>>#(r.c([5S//:Xdeg)zT This function tests the Case 3 in the step to calculate coefficients of c-vectors. r*N)r rr.r,test_construct_c_case_3rs  QC5 (( (r.c4 [[S-*S[S--- S[S---S[--S-[SS9[S[S--S[S--- S[--S- [SS9SS[-S - [S- S [-[S 5S - [S- - -S- /S [-S - [*S- S[-[S 5S - [S- --S- //4[[S-*S[S---S[S---S[S---S[S---S[--S-[SS9[[S-S[S---S[S---[- S-[SS9SS[-[[*S[- -S- /S[-[*[S[--S- //4[S[S--[S-- [S-- S[S--- [S-- S[-- S- [SS9[S[S--S[--S-[SS9SS[ S5-[-S- S[ S5-[-S- [ S5[-S- [ S5*[-[S5S- S[ S5-[-S- - -S- /S[ S5-[-S- S[ S5-[-S- [ S5*[-S- [ S5[-[S5S- S[ S5-[-S- --S- //4[S [S--S[S--- S[S--- [S-- S- [SS9[S[-S- [SS9SS[-S- S[-S- [S- [*[S5*S - [- -/S![-S- S"[-S- [*S- [[S5*S - [--//4[[S-*[S-- [S-- [S-- [- [SS9[[S-[SS9SS#[-S- S[-[*[[*S$S[-- -S- /S[-S- S[-[[*[S$S[---S- //4[[S-*[S-- S[S--- S[S--- [S-- [S-- S[--S- [SS9[S[-S- [SS9SS[ S5-[-S- S[ S5-[-S- [ S5[-S- [ S5[-S- [ S5*[-[S5*S%- S[ S5-[-S- - -S- /S [ S5-[-S- S [ S5-[-S- [ S5*[-S- [ S5*[-S- [ S5[-[S5*S%- S[ S5-[-S- --S- //4/nUH1upp4[ X[[ US5n[XSS-5U:XaM1e g&)'a This function tests the Case 4 in the step to calculate coefficients of the d-vector. Each test case has 4 values - 1. num - Numerator of the rational function a(x). 2. den - Denominator of the rational function a(x). 3. mul - Multiplicity of oo as a pole. 4. d - The d-vector. rJr<rOr;TrrSrsrir_rerrTrIrRrUrzrVi rirwiIBi ir*)rr;iirrKrrQN)rr'rr rr$rr!)rXr7r8rdsers r,test_construct_d_case_4rs@ adUQq!tV^a1f $qs *Q .TB Qq!tVa1f_r!t #a 'd; Q$r'1Q31afSj1Q3./1 2 QrA2a41afSj1Q3./12 4  adUQq!tV^a1f $qAv -!Q$ 61 !#Av.!333"r.c B[S[S--[S--[-S- [SS9[S[S--S[S---S[--S- [SS9[S5S- [S5*S- /[S5*S- [S5S- //4[S[S--[S --[S-- [S--S[-- S- [S S 9[S[S--S [S ---S[S---S[S---S[--S -[S S 9[S5S- S [S5-S- /[S5*S- S[S5-S- //4[[S-[- S-[S S 9[S[S--S [--S-[S S 9[S5S- S[S5-S- /[S5*S- S[S5-S- //4/nUH.upn[X[[SS5n[ U5U:XaM.e g)z This function tests the Case 5 in the step to calculate coefficients of the d-vector. Each test case has 3 values - 1. num - Numerator of the rational function a(x). 2. den - Denominator of the rational function a(x). 3. d - The d-vector. r<r;TrrSrJr*ryrOZZrrIrVrirKrN)rr'rr$rr")rXr7r8rrs r,test_construct_d_case_5rEs Qq!tVad]Q  "A6 Qq!tVa1f_qs "Q &T: q'!)d1gXc\ "d1gXaZa$=> Qq!tVad]QT !AqD (1Q3 . 2AdC Qq!tVa1f_qAv %!Q$ .1 4q 8!DI q'!)RQZ] #tAwhqj!DG)B,%?@ QTAX\1T* Qq!tVac\A q. q'!)RQZ\ "d1gXaZ471$=> E  !%c2q!<!#&!+++r.c [S[S--S- [SS9[S[S--S[S---S[--S-[SS9[S5S- [S- -/[S5S- [S- - //4[S[S --S[S--- S[-- S- [SS9[[S -[S-- S[S---S[S --- S[S--- S[-- S -[SS9S/S //4[S [S --[S--S[--S-[SS9[S [S--S[S--- S[S --- S[S--- S[S--- S[S --- S[S--- S[-- S- [SS9S/S //4/nUHupn[ X[5U:XaMe g)z This function tests the Case 6 in the step to calculate coefficients of the d-vector. Each test case has 3 values - 1. num - Numerator of the rational function a(x). 2. den - Denominator of the rational function a(x). 3. d - The d-vector. rVr<rJrrrOrsr*r;rTrSrrKrrRrQrnrIrir."*N)rr'r rr#)rXr7r8rs r,test_construct_d_case_6rfs R1Wq[!D) Qq!tVa1f_r!t #a '48 A$q&1Q3,!A$q&1Q3,( R1WqAv ! #a '48 QTAqD[1QT6 !AadF *Qq!tV 3ac 9A =qN qc  R1Wq!t^bd "R '48 Qq!tVbAg 1a4 '"QT' 1Bq!tG ;bAg E1a4 O Q$t % qc  E" !!#A.!333r.cv[[S-S[-- S-[SS9[[S-[- [SS9[S5SSSS SS SS S .4[S [S--S [-- S-[SS9[S [S--S[S--- S[S--- S[--S- [SS9[S5S- SS[S5S- [S5S- [S5S- [S5S- S.4[S[SS9[[S-S[S5-S- [S---S[S5-S -[S---S!S[S5-- [S---SS[S5--[--S- [SS9[S5SSS[S5-S"S[S5-- [ S#S"S[S5-- S$S%9S#[S5-- S"S[S5-- S#[S5-S-- [ S#S"S[S5-- S$S%9S#[S5-S-- S"S[S5-- S#[S5-S-- S&.4[[S-S[S--- S[S---S'[--S(- [SS9[[S-S- [SS9[ SSSS)SSS*S+S,.4[S[S--S[S---S[-- S-[SS9[S[S--[S-- S[S--- S[--S-[SS9[S5S- S)S[S-5S.- [S/5S0- [S15S2- [S35S4- [S55S6- S7.4[S8[S--S[--S- [SS9[S[S--S[S---S[S---S[S---S[--S-[SS9[ S)SS)S)[S95*S:- S)S#[S;5S- S<.4/nUHupp4pVU[ X[X4U5:XaMe g=)>a This function tests the computation of coefficients of Laurent series of a rational function. Each test case has 5 values - 1. num - Numerator of the rational function. 2. den - Denominator of the rational function. 3. x0 - Point about which Laurent series is to be calculated. 4. mul - Multiplicity of x0 if x0 is a pole of the rational function (0 otherwise). 5. n - Number of terms upto which the series is to be calculated. r<r;rSTrr*rTrIir)r*rrNrVr_rP@iirOri?iirQiiKiil`B l-Ri0i}')rr<rNr*rJrPrRrr_rNFrL)rOr;r<r*rrNrsrrrrz)r;r<r*rrNrVi i ii)l?PTl@qslF41l@4lJ!tzlZdj m)rrNrVr_rPrGrr)rrVrKrNr_rPN)rr'r rrrr$)rXr7r8x0rnrs r,test_rational_laurent_seriesrsP& QTAaCZ!^Q$/ QTAXqD) !a "!3  R1WtAv  $a48 R1Wr!Q$w QT )DF 2S 8!tL !Q1 hK QtWS[a n[6P V9U?   QT" QTRQZ!^QT) )QtAwY^QT,A AS1TRSW9_VWYZVZDZ Z qay=!  !"d 4 QA QKB47Ns2rAd1gI~PU/VXZ q'Y0QtAwYd1g(99c"b1TRSW9n_d? Q"check_dummy_sol..s32t2sc3L># UHupURUT5v M g7f)N)dummy_eq)rs1s2 dummy_syms r,rrs#I8Hfbr{{2y))8Hs!$N) isinstancer lhsrhsrr&r'r%r rallrzip)rEsolserr2rsolsrrFs ` r,check_dummy_solrs "b VVbff_RA&HA 1q )5 )D tB/3 4tHHR #tD 4 3[23 3 33 3 ID8HI I II I 5s4Cc"[S5n[[[5R [5[[5S--S- S5[[[5[ S55[[[5[ S5*5/4[[5S-[[5R [5-S[[5-[- -S[S-- -[[[5SU-[- U[-[S--- 5/4S[S--[[5R [5-[S[[5-[[5R [5-S- -- [[5S- [[5--[[[5US[S---U[-- 5/4[[[5R [5[[5S-*S[S-[S-- - - 5[[[5S[S-[- - 5/4[S-S[-S[- -[[5-- [[5S--[[5R [5-[[[5U[-[S--S[--U[S--- 5[[[5[5/4[S-[[5R [5-[S--[S[[5S--[[5R [5--- [[5-[[[5U[S--[-U[S--- 5[[[5[S-5/4[[5S-*[[5R [5-S[S--S [-- S -[S- S-S[-S- S--- -[[[5S U-[-S U-- S[S --- S[S---S[S--- S[S---S[-- S -S U-[S--S U-[-- SU--S [S ---S[S --- S[S---S[S--- S[S---S [-- - 5[[[5S[-S- S[S--S[-- S-- 5/4[[5S-[[5R [5-S[S --S[S --- S[S---S[S---S [S---S [-- S-S[S--- - [[[5S[S --S[S--- [S-- S[S---S[--S- S[S--S[S--- - 5/4[[[5R [5[S -*S[S---S[S--- S[S---S[-- S-[S-S[S--- S[S---S[S --- S[S---S[S --- S [S ---S[S --- S[S---S[S--- S[S---S[-- S-- [[5S--[[5-5[[[5S[S -S [S --- S[S---S [S--- S[S---S [-- S-- 5/4[[[5R [5[[[5-S[--S[-S- [[5S--S[-S-- -S[S--S [-- S!-S"[S--S[S--- S-- -[ S5S- - 5[[[5SS[-- S[-S- - 5/4[[[5R [5S#[S--S$[-- S- S[S--S[S--- S[--S-- S[[5S--S [-S-- -5[[[5S[-S-S[-S- - 5/4[[5R [5S[S--S-[[5S--[- -S [S--[- S-[[5-[[S- -- -S[S--S[-- S-[[S- S--- -[[[5U*[S-- [S--S[-- U[-U- [S--[S-- [S--[- - 5[[[5S%[S- - 5/4[[5R [5S[-[[5S-S--[- - [[[5[*U-[[S---U[S--- 5[[[5[*5/4[[[5R [5[[[5-[ S5S- S[-- - [S- [ S5S- - [[5S--S[-S- [ S5S- - - -[ S 5S- - S&[S--S'[-- S(-S"[S--S[S--- - -5[[[5S [- [- 5[[[5S[S)--S*[S+---S,[S---S-[S---S.[S---S/[S ---S0[S---S1[S ---S2[S ---S3[S ---S4[S---S5[S---S6[S---S7[--S8-S[S)--S9[S+---S:[S---S;[S---S<[S---S/[S ---S=[S---S>[S ---S?[S ---S@[S ---SA[S---SB[S---SB[S---SC[--- 5/4[[[5R [5SD[S--SD[S---[*S- [ S5S- - [[5S---S -5[[[5S [-5/4[[[5R [5SE[S--S- S[-S- -[S- [ S5S- - [[5S---S -S[- [[5-[- -S[- -5[[[5SE[-S- S- 5/4/nUHup#[X#U5 M gF)Ga This function tests the computation of rational particular solutions for a Riccati ODE. Each test case has 2 values - 1. eq - Riccati ODE to be solved. 2. sol - Expected solution to the equation. Some examples have been taken from the paper - "Statistical Investigation of First-Order Algebraic ODEs and their Rational General Solutions" by Georg Grasegger, N. Thieu Vo, Franz Winkler https://www3.risc.jku.at/publications/download/risc_5197/RISCReport15-19.pdf C0r<rrOrVr*r;rrlrIrSrTrJrqrkrgr9:rQrRrrhrrrrrsiirnrr0rNirrrrrrri0HiiFi4"i<ii2ҍi"0 iimi^ri i#??!A$F AaD2ad7Q;ad+ ,b1q!tn= 1q1Q499Q<2ad7RT>A#5Q AaCED9#  AaD1R46AbD=2ad7*R1W4r!Q$w>AqDH Q$rT!Q$Y2a'!B$.1a47"QT'A 1a4QT'q!tG$&'c*+ ,-/!qsQw1a4 A#BB7.  !a!A$))A,!AqD&1QT6/Bq!tG";a1f"D adF#T## !!Q$"( ( AaD1QT6AadF?QT)AadF2QqS81QT ICPQSTPTH T 1H1a4x "%ad(+-/1W579!t<>?@ACDQ47KMNqTR S AaD!QTAadF]R1W,r!Q$w6AqD@1Q3FJK LM  1Q499Q<1Q4!A#1q!A$'(91Q37(CC q!tVac\B AqD2ad7!2Q!6 78:;A$q&A B AaD1qs7QqS1W% &' 1Q499Q<#ad(RT/B.AqD2ad71BQqS1H11LM!ai1q!" # AaD1Q37QqS1W% &' ! ! !Q$ AaD!G+A--1a4!a10Eq!KH1 q!tVac\A%1q51* 5 6 AaDB3A:1$qs*RTBYA-=1-Dq!t-K. adBAJ' ) ! ! qsAaD!GaK(** AaDA2b51QT6>BAI. /AaD1"> 1Q499Q<1Q41a!A#.!A#!Q,!a1G qSUQqT!V^2 tAv&),QTDF):S)@2ad7RPQSTPTWCT(UV W AaD1q5!) b11b52ae8(;c!R%i(G$qRTu*(T !R%K) A+)&(.q!t )46=adl)CELQPQT\)RT\ qDU)AqD=)!#,QT>)24=adN)CENq[)QS\)]BhQU"SBY.ae;eArEkIFSTVWSWKW 1a4K!!Q$,')0A68@A FHPQRTUQU VX` TYQTM"$,QJ/(12 3  1Q499Q<AqD2ad7*qbdQqT!VmQqT1W-DDqHI AaD!A# 1Q499Q<AqDRT!V+qsQqT!V|QqT1W.DDq5!A$,q.!#$Q3' ( AaD"Q$q&1* Ad EJ$r.c \[S5n[[[5R [5S[- [[5-[S- - SS[-- [[5S--S[-S- - -S[S--S[S--- S [--S - S [S --S [S--- S[S---S[-- S-- -S-5[[[5S[-S-S[-S- - 5/4/nUHup#[ X#U5 M g)z This function tests the computation of rational particular solutions for a Riccati ODE. Each test case has 2 values - 1. eq - Riccati ODE to be solved. 2. sol - Expected solution to the equation. rr*r;r<rRrSii/ii-rQrOi*irrrTN)r r r&r'r=rrs r,test_solve_riccati_slowrXs$ tB 1Q499QORUVWYZVZRZ>Z F??  ! " AaD2a4!8acAg& '(  E$r.N)Jsympy.core.randomrsympy.core.functionrsympy.core.mulrsympy.core.numbersrrrsympy.core.relationalr sympy.core.singletonr sympy.core.symbolr r &sympy.functions.elementary.exponentialr r%sympy.functions.elementary.hyperbolicr(sympy.functions.elementary.miscellaneousr(sympy.functions.elementary.trigonometricrsympy.polys.polytoolsrsympy.simplify.ratsimprsympy.solvers.ode.subscheckrsympy.testing.pytestrsympy.solvers.ode.riccatirrrrrrrrrrr r!r"r#r$r%r&r'r-r3r9rGr]rarrrrrrrrrrrrrrrrr.r,rs%(00$".=698&*3% SM CL BE  ?FD36l`)F).X =2.+0\#:L;?|)<4~,B4@@GFJ$z%z%%r.