wgrddlmZddlmZddlmZddlmZmZm Z ddl m Z ddl m Z ddlmZmZddlmZmZdd lmZdd lmZdd lmZdd lmZdd lmZddlmZddl m!Z!ddl"m#Z#m$Z$m%Z%m&Z&m'Z'm(Z(m)Z)m*Z*m+Z+m,Z,m-Z-m.Z.m/Z/m0Z0m1Z1m2Z2edZ3edZ4dZ5dZ6dZ7dZ8dZ9dZ:dZ;dZdZ?dZ@dZAd ZBd!ZCd"ZDd#ZEd$ZFd%ZGd&ZHe!d'ZIy()))randint)Function)Mul)IRationaloo)Eq)S)Dummysymbols)explog)tanh)sqrt)sin)Poly)ratsimp) checkodesol)slow)riccati_normalriccati_inverse_normalriccati_reduced match_riccatiinverse_transform_poly limit_at_infcheck_necessary_conds val_at_infconstruct_c_case_1construct_c_case_2construct_c_case_3construct_d_case_4construct_d_case_5construct_d_case_6rational_laurent_series solve_riccatifxcDtt| |td|SN)rr)maxints i/home/mcse/projects/flask/flask-venv/lib/python3.12/site-packages/sympy/solvers/ode/tests/test_riccati.py rand_rationalr-s GVGV,ga.@ AAcjtt|dzDcgc] }t|c}|Scc}wr))rranger-)r'degreer+_s r, rand_polyr3!s* fQh@1v&@! DD@s0ctd|}td|}t|||}t|||}|td|k(rt|||}|td|k(r||z S)Nr*r)rr3r)r'r1r+degnumdegdennumdens r,rand_rational_functionr9%sl Q F Q F Avv &C Avv &C a 66* a  9r.cX|}|j|}t|dd}t|dd}|dk(rt|dd}|dk(rt|||zz ||dzzz }t|j|||zz||dzzz}t||} t || dk(sJ||||fS)Nr*r)Tr)diffr9rr r) ratfuncr'yfyypq1q2q0eqsols r,find_riccati_oderG/sA B 1a (B 1a (B ' #Aq! , ' bdR1W$ %B BGGIrBrEzBr1uH, -B R)C r3 9 ,, , r2r>r.c  ttdz z tdzdzdz tzttdz tdz z ttdzdz tddz zzdz z ddtzz z fdtzdzdtzdzz ddtzz tdzz dtzdtzdtzdzzdtzdzz ddtzz tdtdzdz z ddtzz z fd dtdzzdz z d dtz d tzdz z dtz d tzdz dtdzzdz zz d tzdz td dtz dd tzdz dzz dd tzdz z z ztddtz dz z ftd tzdz dtzdzz tdzdtzdz z td z dtzdz z d tzdz tddtzdzdz z dtzdz dtdzzdtzdz dzz dtdzzdtzdz z zzdtdzzz z fg}|D]C\}}}}|t|t||k(sJ|t |t||j k(rCJdtzdz dtdzzdtzzdz z dtz t dz d tzz d tdzzdd tzz t dz d tdzzz zzt dz dzz d t dz z zdtzd d tzz t dz d tdzzz z zt dz z dtzdz t dz zd tzdtdzzdtzzdz zz z dtz zfddtdzzz dtz t dz d tzz d tdzzdd tzz t dz d tdzzz zzt dz dzz d t dz z zdtzd d tzz t dz d tdzzz z zt dz z dtz ztdt dz dd tdzzz z fg}|D]E\}}}}}|t|t||k(sJ|t |t|||j k(rEJy)ar 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]?s` 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$%&q!tVK- - ) E6B 2r1N1aR0000*1aR8??AAAAB AAadFQqSL1$% 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$1Q3A26AadF++,qb1fq[81qb1f:E 1b!A#h1"q&1QT6**+aR!V4qs:SQBFUZ=[]^_`bc_c]c=dd  E "F2r2qN1aR0000*1aR<CCEEEEFr.c  ttjttdzz tttzz tttdzzz ttjtttdzztdzztdzdz z ddtdzzz z fdtzdtzdzz ttjtztdzttdzztz z dtdzzdtz t dz tdzz zdzzdt dz dzzz tdt dz d dtzdzz zttdzzttjtzd tdztz ztt dz zz z fttdzttjtztdz ttzt t ddz z z z dtzdz dz ddtzdzdzzz dtzdz dtzdzdzz zttdzzttjtzddtzdzz z fttjtttdztz z ttdzttjtzddtdzzz zfdtdz tz dzztdzdtzzdzz ttjtzttdztz zttdzttjtzdtdzztdzdtzzdzz dtztdzdtzzdzz zdtdzdtzzdzz z tz zddtdzzz zfdtzdtzdzz ttjtztdzttztz z dfttttjtzdtz z ttdz zttdztdzdz z zdfg}|D]\}}|t |ttk(rJy ) z This function tests the transformation of a Riccati ODE to its normal Riccati ODE. Each test case 2 values - 1. eq - A Riccati ODE. 2. normal_eq - The normal Riccati ODE of eq. r<r;rOrTrSr*FrLrNN)r&r'r=rr r)rXrE normal_eqs r,test_riccati_reducedrasQ ! ! q!ta!f$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' EP6 IOB155556r.c ttjtdtdzzdtdzzz dtzz dzdtdzzd tdzzz d tdzzzd tzzd z z z dz ttdzdtzd zz z td dz tz ttztd dz dtzdz z z z ddtdzzdtdzzdtdzzz dtdzzzdtzzdz z dtdzzdtdzzdtdzzz dtdzzzdtzzdz z z dtzdtdzzdtdzzz dtdzzzdtzzdz z z dzddtdzzd tdzzz d tdzzzd tzzd z z zt dddtzz dd tzdz z d dtzd zz fttjtdtzdz ztdz d z ttdzzz dtzdz d zttzdtzdzz z td!dz z d"tdzzd#tzzd$zd%tdzzd&tdzzzz z dd'tzdz td!dz zddtdzzdtdzzzz zd(d)tdzzdtzzz zd"d%tzd&zz zt dd*tzdz dd tzdzz tdz d z fttjtd+td,zzd-tdzzz d.tdzzzd/tdzzz d0tzzd1z d)tdzzd&td,zzz d2tdzzzd3tdzzz d4tdzzzd1tzz z z td5d z z ttddz z ttztdz tddz z z z tdz dz ttdzzdtzz z dd+tdzzd)td,zzd&tdzzz d2tdzzzd3tdzzz d4tzzd1z z d-tdzzd)td,zzd&tdzzz d2tdzzzd3tdzzz d4tzzd1z z z d.tdzzd)td,zzd&tdzzz d2tdzzzd3tdzzz d4tzzd1z z zd6tzd7td,zzd tdzzz d8tdzzzd9tdzzz d:tzzd)z z z td5d z zd7dtdzzdtd,zzz d;tdzzzddtd,zzdtdzzz d;tdzzzd 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$qtQw,> q!Q QT1Q499Q?"QqS11%551Q4? q!Q ! ! q!tad1Q4i'1a4Q<1q*@@ q!Q ! ! QqT1W1q1a4!84QqT99Aqs @ =  tQw  q!QWO E` %)CR$RA. u|| B<5( (( )r.c tdtdzzdtdzzzdtzz dzttdtdzztdzz d tdzzzd td zzzdtdzzzdtd zzzd td zzz d tdzzzdtdzzz d tzzdztd ftd ttdtd zzdtdzzzdtdzzzdtzz dz td ftdtdzzdtdzzz dtzzdz ttdtdzzdtdzzzdtzz dz tdftdtdzzdtd zzz d tdzzz dtd zzzdtd zzztdzz tdzzd tzz ttdtdzztztdftd tdzzdtd zzzd td zzz dtdzzz tdzzd tzz d zttdtd zzdtdzzzdtdzzz dtzzdztdfg}|D]\}}}t||t|k(rJy)a 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>0 S#Ca(C///0r.cNtdtdzzdtdzzzdtzzdz tdtdtdzzd td zzzd tdzzzd tdzzzd tdzzzd tdzzztdtdtddz td tzd z zgtddz td tzd z z ggftdtdzzdtdzzzdtzzdztdtdtdzzdtdzzz dtdzzzdtdzzz dtzzdztdtd dz tddz tddz zgtddz tddz z ggftdtzdztdtdtdzzddtdzz tdzzzddtdzz tdzzzdd tdzz tzzdtdzzd ztd tdtdzdz tddz tt ddtdzdzd!"d#tdzd$zz dzdz zgtddz tt ddtdzdzd!"d#tdzd$zz dzdz z ggfg}|D]\}}}}t ||t||k(rJy%)&ab 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$q'zBORSSTUVV V W qT!Vd3q!DG)a-%@"T!W*r/RUVVWXYY Y Z \  E*#:S$!#sAt4999:r.c. tdtdttdz dztdz ztdddt dtz zdz gtdtzzdz ggftdtdzzdtd zzz d tdzzz dztdtdtzdz dztdzdzztdtddz dtd  d z gtd  d z ggfttdztdzz d tzztdttdz d ztdzdzztddd d t dztddz d t dzd z z zdz dt dzd z gdt dztddz d t dzd z zzdz dt dzd z ggftdtdzztd zzdztdtd tzdzd ztdzztdtd d z d dt dzt d dz tddz z zdz t ddz gdt dzt ddz tddz z zdz t d dz ggfttdzdztdtdtzdz dztdzdzztdtddz ddt dzt d dz tddz z zdz dt dzdz t dd z gd!t dzt ddz tddz z zdz dt dzdz t d d z ggfttdzdztdttt dz dztdt ddt d"tdd z dt d"zz zd#z t d d z t d"gt d" tdd z dt d"zzzd#z t d  d z t d" ggfg}|D] \}}}}}t ||t|||k(r Jy$)%a This function tests the Case 2 in the step to calculate coefficients of c-vectors. Each test case has 5 values - 1. num - Numerator of the rational function a(x). 2. den - Denominator of the rational function a(x). 3. pole - Pole of a(x) for which c-vector is being calculated. 4. mul - The multiplicity of the pole. 5. c - The c-vector for the pole. r*Trr<rNr;rJrRrOrIYrmrSrqiWrVri7i8irTriB6i$rirtirN)rr'rr rr)rXr7r8rmulrs r,test_construct_c_case_2rsk" 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$r(C E K+ EX#(?S$Q!#sAtS9Q>>>?r.c&tdggk(sJy)z` 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.c~ ttdz dtdzzz dtdzzzdtzzdztdtdtdzzdtdzzz dtzzdz tdddtzd z tdz d tztd d z tdz z zdz gd tzd z t dz dtztd d z tdz zzdz ggfttdz dtdzzzdtdzzzdtdzzzdtdzzzdtzzdztdttdzdtdzzzdtdzzztz dztdddtztt dtz zdz gdtzt tdtzzdz ggftdtdzztdzz tdzz dtdzzz tdzz dtzz dz tdtdtdzzdtzzdztdddt dztzdz dt dztzdz t dtzdz t d tztddz dt dztzdz z zdz gdt dztzdz dt dztzdz t d tzdz t dtztddz dt dztzdz zzdz ggftd tdzzdtdzzz dtdzzz tdzz dz tdtdtzdz tdddtzdz dtzdz tdz t td d z tz zgd!tzdz d"tzdz t dz ttd d z tzzggfttdz tdzz tdzz tdzz tz tdttdztddd#tzdz dtzt tt d$dtzz zdz gdtzdz dtztt td$dtzzzdz ggfttdz tdzz dtdzzz dtdzzz tdzz tdzz dtzzdz tdtdtzdz tdddt dztzdz dt dztzdz t dtzdz t dtzdz t d tztd d%z dt dztzdz z zdz gd t dztzdz d t dztzdz t d tzdz t d tzdz t dtztd d%z dt dztzdz zzdz ggfg}|D]3\}}}}t ||tt |d}t||dz|k(r3Jy&)'a9 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_4rsM 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 !#sAv.!3334r.c dtdtdzztdzztzdz tdtdtdzzdtdzzzdtzzdz tdtddz td dz gtd dz tddz ggftdtdzztd zztdzz tdzzdtzz dz td tdtdzzd td zzzdtdzzzdtdzzzdtzzd ztd tddz d tdzdz gtd dz dtdzdz ggfttdztz dztd tdtdzzd tzzdztd tddz dtdzdz gtd dz dtdzdz ggfg}|D].\}}}t||ttdd}t ||k(r.Jy)a 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$=> E , S!%c32q!<!#&!+++,r.c tdtdzzdz tdtdtdzzdtdzzzdtzzdztdtddz tdz zgtddz tdz z ggftdtd zzdtdzzz dtzz dz tdttd ztdzz dtdzzzdtd zzz dtdzzz dtzz d ztddgd ggftd td zztdzzdtzzdztdtd tdzzdtdzzz dtd zzz dtdzzz dtdzzz dtd zzz dtdzzz dtzz dz tddgd ggfg}|D]\}}}t ||t|k(rJy)a This function tests the Case 6 in the step to calculate coefficients of the d-vector. Each test case has 3 values - 1. num - Numerator of the rational function a(x). 2. den - Denominator of the rational function a(x). 3. d - The d-vector. rVr<rJrrrOrsr*r;rTrSrrKrrRrQrnrIrir."*N)rr'r rr#)rXr7r8rs r,test_construct_d_case_6rfs R1Wq[!D) Qq!tVa1f_r!t #a '48 A$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"4 S!!#sA.!3334r.cttdzdtzz dztdttdztz tdtddddd dd dd d ftd tdzzd tzz dztdtd tdzzdtdzzz dtdzzz dtzzdz tdtddz ddtddz tddz tddz tddz dftdtdttdzdtdzdz tdzzzdtdzd ztdzzzd!dtdzz tdzzzddtdzztzzdz tdtddddtdzd"dtdzz t d#d"dtdzz d$%d#tdzz d"dtdzz d#tdzdzz t d#d"dtdzz d$%d#tdzdzz d"dtdzz d#tdzdzz d&fttdzdtdzzz dtdzzzd'tzzd(z tdttdzdz tdt dddd)ddd*d+d,ftdtdzzdtdzzzdtzz dztdtdtdzztdzz dtdzzz dtzzdztdtddz d)dtd-d.z td/d0z td1d2z td3d4z td5d6z d7ftd8tdzzdtzzdz tdtdtdzzdtdzzzdtdzzzdtdzzzdtzzdztdt d)dd)d)td9 d:z d)d#td;dz d<fg}|D]"\}}}}}}|t ||t|||k(r"Jy=)>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$)rXr7r8x0rnrs r,test_rational_laurent_seriesrsa& QTAaCZ!^Q$/ 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? Qz"check_dummy_sol..s3qt3sc3HK|]\}}|j|yw)N)dummy_eq)rs1s2 dummy_syms r,rz"check_dummy_sol..s Ifb"r{{2y)Is"N) isinstancer lhsrhsrr&r'r%r rallrzip)rEsolserr2rsolsrrFs ` r,check_dummy_solrs "b VVbff_RA&HAu 1q )5 )D tB/3 4CHHR # 4D 4 3[T23 33 3 ID%8HI II I 5s0C c"td}tttj tttdzzdz dtttt dtttt d gfttdzttj tzdttztz zdtdzz ztttd|ztz |tztdzzz gfdtdzzttj tztdttzttj tzdz zz ttdz ttzzttt|dtdzzz|tzz gftttj tttdz dtdztdzz z z tttdtdztz z gftdzdtzdtz zttzz ttdzzttj tzttt|tztdzzdtzz|tdzzz ttttgftdzttj tztdzztdttdzzttj tzzz ttzttt|tdzztz|tdzzz ttttdzgfttdz ttj tzdtdzzd tzz d ztdz dzdtzdz dzzz ztttd |ztzd |zz dtd zzz dtdzzzdtdzzz dtdzzzdtzz d zd |ztdzzd |ztzz d|zzd td zzzdtd zzz dtdzzzdtdzzz dtdzzzd tzz z tttdtzdz dtdzzdtzz dzz gfttdzttj tzdtd zzdtd zzz dtdzzzdtdzzzd tdzzzd tzz dzdtdzzz z tttdtd zzdtdzzz tdzz dtdzzzdtzzdz dtdzzdtdzzz z gftttj ttd z dtdzzzdtdzzz dtdzzzdtzz dztdzdtdzzz dtdzzzdtd zzz dtdzzzdtd zzz d td zzzdtd zzz dtdzzzdtdzzz dtdzzzdtzz dzz ttdzzttztttdtd zd td zzz dtdzzzd tdzzz dtdzzzd tzz dzz gftttj ttttzdtzzdtzdz ttdzzdtzdzz zdtdzzd tzz d!zd"tdzzdtdzzz dzz zt ddz z tttddtzz dtzdz z gftttj td#tdzzd$tzz dz dtdzzdtdzzz dtzzdzz dttdzzd tzdzz ztttdtzdzdtzdz z gfttj tdtdzzdzttdzztz zd tdzztz dzttzttdz zz zdtdzzdtzz dzttdz dzzz zttt| tdzz tdzzdtzz |tz|z tdzztdzz tdzztz z tttd%tdz z gfttj tdtzttdzdzztz z tttt |zttdzzz|tdzzz tttt gftttj ttttzt ddz dtzz z tdz t ddz z ttdzzdtzdz t ddz z z zt d dz z d&tdzzd'tzz d(zd"tdzzdtdzzz z ztttd tz tz tttdtd)zzd*td+zzzd,tdzzzd-tdzzzd.tdzzzd/td zzzd0tdzzzd1td zzzd2td zzzd3td zzzd4tdzzzd5tdzzzd6tdzzzd7tzzd8zdtd)zzd9td+zzzd:tdzzzd;tdzzzdtd zzzd?td zzzd@td zzzdAtdzzzdBtdzzzdBtdzzzdCtzzz gftttj tdDtdzzdDtdzzzt dz t ddz z ttdzzzd ztttd tzgftttj tdEtdzzdz dtzdz ztdz t ddz z ttdzzzd zdtz ttztz zdtz ztttdEtzdz dz gfg}|D]\}}t|||yF)Ga This function tests the computation of rational particular solutions for a Riccati ODE. Each test case has 2 values - 1. eq - Riccati ODE to be solved. 2. sol - Expected solution to the equation. Some examples have been taken from the paper - "Statistical Investigation of First-Order Algebraic ODEs and their Rational General Solutions" by Georg Grasegger, N. Thieu Vo, Franz Winkler https://www3.risc.jku.at/publications/download/risc_5197/RISCReport15-19.pdf C0r<rrOrVr*r;rrlrIrSrTrJrqrkrgr9:rQrRrrhrrrrrsiirnrr0rNirrrrrrri0HiiFi4"i<ii2ҍi"0 iimi^ri i#??!A$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$q&1* Ad EJ%CC$%r.c \td}tttj tdtz ttztdz z ddtzz ttdzzdtzdz z zdtdzzdtdzzz d tzzd z d td zzd tdzzz dtdzzzdtzz dzz zdztttdtzdzdtzdz z gfg}|D]\}}t |||y)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%CC$%r.N)Jsympy.core.randomrsympy.core.functionrsympy.core.mulrsympy.core.numbersrrrsympy.core.relationalr sympy.core.singletonr sympy.core.symbolr r &sympy.functions.elementary.exponentialr r%sympy.functions.elementary.hyperbolicr(sympy.functions.elementary.miscellaneousr(sympy.functions.elementary.trigonometricrsympy.polys.polytoolsrsympy.simplify.ratsimprsympy.solvers.ode.subscheckrsympy.testing.pytestrsympy.solvers.ode.riccatirrrrrrrrrrr r!r"r#r$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.