grddlmZddlmZddlmZddlmZmZm 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$ZFd%ZGd&ZHe!d'ZId(S)))randint)Function)Mul)IRationaloo)Eq)S)Dummysymbols)explog)tanh)sqrt)sin)Poly)ratsimp) checkodesol)slow)riccati_normalriccati_inverse_normalriccati_reduced match_riccatiinverse_transform_poly limit_at_infcheck_necessary_conds val_at_infconstruct_c_case_1construct_c_case_2construct_c_case_3construct_d_case_4construct_d_case_5construct_d_case_6rational_laurent_series solve_riccatifxc\tt| |td|S)N)rr)maxints `/var/www/html/ai-engine/env/lib/python3.11/site-packages/sympy/solvers/ode/tests/test_riccati.py rand_rationalr,s( GVGV,,ga.@.@ A AAc\tfdt|dzD|S)Nc.g|]}tS)r,).0_r*s r+ zrand_poly.."s!@@@1v&&@@@r-r))rrange)r'degreer*s `r+ rand_polyr6!s2 @@@@fQh@@@! D DDr-ctd|}td|}t|||}t|||}|td|kr%t|||}|td|k%||z S)Nr)r)rr6r)r'r5r*degnumdegdennumdens r+rand_rational_functionr<%s Q  F Q  F Avv & &C Avv & &C a  66** a   9r-c|}||}t|dd}t|dd}|dkrt|dd}|dkt|||zz ||dzzz }t||||zz||dzzz}t||} t || dksJ||||fS)Nr)r)Tr)diffr<rr r) ratfuncr'yfyypq1q2q0eqsols r+find_riccati_oderJ/sA B 1a ( (B 1a ( (B '' #Aq! , , '' bdR1W$ % %B BGGIIrBrEzBr1uH, - -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]S\}}}}|t|t||ksJ|t |t||ksJTdtzdz 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]U\}}}}}|t|t||ksJ|t |t|||ksJVdS)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)testswb1b2rCbps r+test_riccati_transformationr`?s 1q5 A1 Q Aq1u 1a46AaDDF?+A--1Q37  1q1Q37 QqS1q5 ! 1acAg!a A!Gc!QUU.K.K.K#LLqRSTURUwV  AadFQJ Q1q Q!A#'AadFQJ'(AaC!Gc"a!ee6T6T6TVWXYVY W V71Q37 7,!!QUU;;;,= =  1rBqD1H 1acAg AqsQw1Q38c!RTAX&F&F&FGG1Q3QR7UWXY[\X\U\^_`a^a _ ^VAqD&!A#'"V#K$%&q!tVK- - ) E6BB 2r1N1aR000000*1aR88??AAAAAAA AAadFQqSL1$% 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$1Q3A26AadF++,qb1fq[81qb1f:E 1b!A#h1"q&1QT6**+aR!V4qs:SQBFUZ=[=[=[]^_`bc_c]c=dd  E "FF2r2qN1aR000000*1aR<<CCEEEEEEEFFr-c  ttttdzz tttzz tttdzzz tttttdzztdzztdzdz z ddtdzzz z fdtzdtzdzz tttztdzttdzztz z dtdzzdtz t dz tdzz zdzzdt dz dzzz tdt dz d dtzdzz zttdzztttzd tdztz ztt dz zz z fttdztttztdz ttzt t ddz z z z dtzdz dz ddtzdzdzzz dtzdz dtzdzdzz zttdzztttzddtzdzz z ftttttdztz z ttdztttzddtdzzz zfdtdz tz dzztdzdtzzdzz tttzttdztz zttdztttzdtdzztdzdtzzdzz dtztdzdtzzdzz zdtdzdtzzdzz z tz zddtdzzz zfdtzdtzdzz tttztdzttztz z dftttttzdtz z ttdz zttdztdzdz z zdfg}|D]&\}}|t |ttksJ'd S) z This function tests the transformation of a Riccati ODE to its normal Riccati ODE. Each test case 2 values - 1. eq - A Riccati ODE. 2. normal_eq - The normal Riccati ODE of eq. r?r>rRrWrVr)FrOrQN)r&r'r@rr r)r[rH normal_eqs r+test_riccati_reducedrds ! ! q!ta!f$q1qy0 ! ! qttQwA%1Q.AadF; !QqS1W ! ! $AqttQwq'88 1a41Q1}$q((!aR!VaK-83q Q< < < !"1q<* *,-aDD!G 467ddiill C Q >ArAvJ ' ( !a!A$$))A,,!a%1rAaDDF{!;; A#'A q!A#'A~&!A#'AaC!Ga<)?? !a A$$))A,, !"AaC!G - ! ! qttQwqy  !a!A$$))A,,AadF+ QTEAIMAqD1Q3JN+addiill:QqTT1WQYF !a!A$$))A,,!AqD&!Q$1*q."9AaCA A#BB=#1qs Q'#()*"+ +-.!Q$Z 8 !QqS1W ! ! $Aqtt|A~5  !QqTTYYq\\AaC!A$$q&(1Q447AqD1H+== G' EP66 IOB1555555566r-c tttdtdzzdtdzzz dtzz dzdtdzzd tdzzz d tdzzzd tzzd z z z dz ttdzdtzd zz z td dz tz ttztd dz dtzdz z z z ddtdzzdtdzzdtdzzz dtdzzzdtzzdz z dtdzzdtdzzdtdzzz dtdzzzdtzzdz z z dtzdtdzzdtdzzz dtdzzzdtzzdz z z dzddtdzzd tdzzz d tdzzzd tzzd z z zt dddtzz dd tzdz z d dtzd zz ftttdtzdz ztdz d z ttdzzz dtzdz d zttzdtzdzz z td!dz z d"tdzzd#tzzd$zd%tdzzd&tdzzzz z dd'tzdz td!dz zddtdzzdtdzzzz zd(d)tdzzdtzzz zd"d%tzd&zz zt dd*tzdz dd tzdzz tdz d z ftttd+td,zzd-tdzzz d.tdzzzd/tdzzz d0tzzd1z d)tdzzd&td,zzz d2tdzzzd3tdzzz d4tdzzzd1tzz z z td5d z z ttddz z ttztdz tddz z z z tdz dz ttdzzdtzz z dd+tdzzd)td,zzd&tdzzz d2tdzzzd3tdzzz d4tzzd1z z d-tdzzd)td,zzd&tdzzz d2tdzzzd3tdzzz d4tzzd1z z z d.tdzzd)td,zzd&tdzzz d2tdzzzd3tdzzz d4tzzd1z z zd6tzd7td,zzd tdzzz d8tdzzzd9tdzzz d:tzzd)z z z td5d z zd7dtdzzdtd,zzz d;tdzzzddtd,zzdtdzzz d;tdzzzdirr?N\rRiiNHr)T-i^rTbQ;i<rQrWFrOrV i iiDrS/$rYi0rMiiwillii7ihi$rUx5?(rN) r&r'r@r rrr rrrr)r[rHresb0r]r^matchfuncss r+test_match_riccatirsh  ! ! AqD3q!t8+bd2R7#ad( ad(;AX; #A;&(*;+, ,./ 023A$$'1Q372C D Q446A:qtt QqTT!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$$' 11QqSU 6 Q4451q5 bEE"H %(+AqD3q5(83(> ad(SAX (   1R!B%%(Q!Q$1a4002r!Q$w Q$84 s1us{# $ B1q5)))1Q373 !a  ! ! AqD3q!t8+c!Q$h6QTA a%1a4#ad(*SAX5AqD@ AqDq5 bEE!G $'(1Q446z1Q44&71 !Q:' Q37AaDD!G#QqS) *  AqD"QT'C1H$s1a4x/#ad(:SUB   1a4xAqD3q!t8+c!Q$h6QTA A 1a4xAqD3q!t8!3c!Q$h!> 1B " 1u"" "!eR1Wr!Q$w%6QT%A AqD&q5&& "#2q  ),.qAv1a4/? 1a40QT'0q!tG0$&(d0+,,  , /1!AqD&2ad72B QT'3q!tG3 d3#%'3(/)  ) BAaC%(((!a%0 BA&&&!,%, ! ! q1Q446{A!QK!O44q!tAaDDy@ !A$$' 1qttAv;  q!Q ! ! s1a4yy 3q66!A$$;.Q!a? q!Q ! ! tAQK(((1Q44/!Q$qttQw,> q!Q QT1Q4499Q??"QqS11%551Q44? q!Q ! ! q!tad1Q44i'1a4Q<1q*@@ q!Q ! ! QqTT1W1q1a4!84QqTT99Aqs @ =  ttQw  q!QWO E` %))CR$RA.. u||||  )B<5(((( ))r-c Ltdtdzzdtdzzzdtzz 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|ksJ#dS)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 rvr>rTr? rWirVrMrLrR r)rXrNrUrirYN)rr'r)r[r:r;vals r+test_val_at_infrs R1WqAv 1 $q (!,, SBYA !Q$ &1a4 /"QT' 9AadF BQq!tV KbQRTUQUg UXYZ[]^Z^X^ ^abcdad dgi iklmm  Q  R1WqAv 1a4 '!A# - 2A66  R1WqAv ! #a '++ R1Wr!Q$w 1 $q (!,,  R1Wr!Q$w AqD (1QT6 1AadF :QT AAqD H2a4 OQRSS SAX\1  Qq!tVa1f_r!Q$w &1a4 /!Q$ 61 r)rW)r)r?rTrUTN)rr0r-r+test_necessary_condsrFs !YYY / /5 8 8 8 8 III . .% 7 7 7 7 III . .% 7 7 7 7 mmm 4 4 < < < < < 1/x in rational functions represented using Poly. rr>rTr?rWrirMrrRPrtKrUirzrLrurodrvrkrVc8g|]}t|tSr0)rr')r1es r+r3z/test_inverse_transform_poly..gs ;;;1DAJJ;;;r-r)N)r'as_numer_denomrsubsrZ)fnsr&r:r;s 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*R1W4s1urjrrRiixii'ii!rTrLirWrsrMiXHi ivii| Tirmirtrrkrriii i i ir) iiZi_iN)rr'rr r)r[r:r;lims r+test_limit_at_infrls SAX1_r !1%% R1Wr!Q$w 1 $r )1--  T!Q$Yad "SAX -Q 6 =qAA T!Q$Yad "SU *S 0!44  T!Q$Yad "T!Q$Y .QT 9C1H Ds1a4x ORUVWYZVZRZ Z a%   T!Q$Yad "R1W ,tAqDy 83q!t8 Cd1a4i ORUVWRW WZ] ]_`aa   SAXAqD 2ad7 *R1W 4r!Q$w >AqD H3q5 PSU UWXYY SAXAqD 3q!t8 +c!Q$h 6ad BT!Q$Y NQUVWQW WZ] ]_`aa !Q SAXAqD 3q!t8 +c!e 3b 8!<< SAXAqD 419 ,s1u 4s :A>> #s 5 E>00 S#Ca((C/////00r-ctdtdzzdtdzzzdtzzdz 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||ksJ%d%S)&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. rbr>r?rRrMT extensionrTrLrVrUrWrr)iii0i0iiiei1ijizrrrz QQdomainFrO,N)rr'r rrrr)r[r:r;polecs r+test_construct_c_case_1rs R1WqAv ! #a 'd;;; Qq!tVbAg !Q$ &AqD 01QT6 9Bq!tG CQRVWWW ! A$$q&47719Q;  !A$$q&47719Q;"6!78  T!Q$Yad "SU *S 0!tDDD SAXAqD 419 ,tAqDy 83q5 @2 EqTXYYY !Q A$$q&4>>#% % &1a$x..2D)D(EF  QqS1Wa4((( R1WBtAwwJ1, ,RQZA/E EQtTUwwYXYHY YQi  7 7 7 1Q A$$q&4Aqay1}u===r$q''zBORSSTTUVV V W qTT!Vd3q!DGG)a-%@@@"T!WW*r/RUVVWWXYY Y Z \  E*#::S$!#sAt4499999::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|||ksJ'd$S)%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?rQr>rMrUrRrLYrprVrtiWrYri7i8irWrlB6i$rirwirN)rr'rr rr)r[r:r;rmulrs r+test_construct_c_case_2rs" QT""" a!eaZQ d333 1 "b1f+a-1b1f:a<.)  Qq!tVbAg !Q$ & *A>>> acAg\1q51* $a4888 !Q R55&)!uRxj!  QTAqD[1Q3 T222 a!eaZQ "A666 1 DGG)QrUU3Y4771, -b 0!DGG)A+ > DGGQrUU3Y4771, -b 0"T!WW*Q,? A  Qq!tVad]Q T222 acAg\1q5 !1555 1a d3ii-$s))C!B%%+5 6s :DIIcM J d3iic32u4 5c 9DII:c>J L  QTAXqD))) acAg\1a4!8 $a4888 !Q T"XX+Ry|affVm3 4R 7DHHT9I4PR88TW< X T"XXtBxx{QsVVF]2 3B 6$r(( 4$r((SVW Y  QTBYT*** a$q''kA qD111 Q r((AaDDFQtBxxZ' ( +T!WWQYR A r((AaDDFQtBxxZ' ( +d1ggXaZ$r((C E K+ EX#(??S$Q!#sAtS99Q>>>>>??r-c2tdggksJdS)z` This function tests the Case 3 in the step to calculate coefficients of c-vectors. r)N)r r0r-r+test_construct_c_case_3rs%   QC5 ( ( ( ( ( (r-cttdz 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]>\}}}}t ||tt |d}t||dz|ksJ?d&S)'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. rMr?rRr>TrrVrvrlrbrhrrWrLrUrXr}rYi rirziIBi ir))ru;iirrNrrTN)rr'rr rr$rr!)r[r:r;rdsers r+test_construct_d_case_4rs adUQq!tV^a1f $qs *Q .TBBB Qq!tVa1f_r!t #a 'd;;; Q$r'1Q31affSj1Q3./1 2 QrA2a41affSj1Q3./12 4  adUQq!tV^a1f $qAv -!Q$ 61 >!#sAv..!3333344r-c <tdtdzztdzztzdz 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]9\}}}t||ttdd}t ||ksJ:dS)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>TrrVrMr)r|rRZZrrLrYrlrNrN)rr'rr$rr")r[r:r;rrs r+test_construct_d_case_5rEsm Qq!tVad]Q  "A666 Qq!tVa1f_qs "Q &T::: q''!)d1ggXc\ "d1ggXaZa$=> Qq!tVad]QT !AqD (1Q3 . 2AdCCC Qq!tVa1f_qAv %!Q$ .1 4q 8!DIII q''!)RQZ] #tAwwhqj!DGG)B,%?@ QTAX\1T*** Qq!tVac\A q... q''!)RQZ\ "d1ggXaZ4771$=> 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|ksJ#dS)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. rYr?rMrrrRrvr)r>rWrVrrNrrUrTrqrLrlr."*N)rr'r rr#)r[r:r;rs r+test_construct_d_case_6rfs2 R1Wq[!D))) Qq!tVa1f_r!t #a '4888 A$$q&1Q3,!A$$q&1Q3,( R1WqAv ! #a '4888 QTAqD[1QT6 !AadF *Qq!tV 3ac 9A =qNNN qc  R1Wq!t^bd "R '4888 Qq!tVbAg 1a4 '"QT' 1Bq!tG ;bAg E1a4 O Q$t % % % qc  E"44 S!!#sA..!3333344r-c ttdzdtzz 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|||ksJ)d=S)>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>rVTrr)rWrLir)r)rrQrYrbrS@iirRri?iirTiiKiil`B l-Ri0i}')rr?rQr)rMrSrUrrbrQFrO)rRr>r?r)rrQrvrrrr})r>r?r)rrQrYi i ii)l?PTl@qslF41l@4lJ!tzlZdj m)rrQrYrbrSrGrr)rrYrNrQrbrSN)rr'r rrrr$)r[r:r;x0rnrs r+test_rational_laurent_seriesrs(& QTAaCZ!^Q$/// QTAXqD))) !a "!33  R1WtAv  $a4888 R1Wr!Q$w QT )DF 2S 8!tLLL !Q1 hKK QtWWS[a nn[6P V99U?    QT""" QTRQZ!^QT) )QtAwwY^QT,A AS1TRSWW9_VWYZVZDZ Z qay=!  !"d 4 4 4 QA QKB477Ns2rAd1ggI~PU/V/V/VXZ q''Y0QtAwwYd1gg(99c"b1TRSWW9n_d? ? ? Q.s' 4 4 4CHHR # # 4 4 4r-c3&K|] }|dV dS)rNr0)r1r's r+ z"check_dummy_sol..s&33qt333333r-c3JK|]\}}||VdS)N)dummy_eq)r1s1s2rs r+rz"check_dummy_sol..s5IIfb"r{{2y))IIIIIIr-N) isinstancer lhsrhsrr&r'r%r allrzip)rHsolserr2rsolsrs ` @r+check_dummy_solrs "b Vbf_RA&&HAu 1q )5 ) ) )D tB 4 4 4 4 4t 4 4 4D 33[T22333 3 3333 IIIID%8H8HIII I IIIIIIr-c(%td}ttttttdzzdz dtttt dtttt d gfttdztttzdttztz zdtdzz ztttd|ztz |tztdzzz gfdtdzztttztdttztttzdz zz ttdz ttzzttt|dtdzzz|tzz gfttttttdz dtdztdzz z z tttdtdztz z gftdzdtzdtz zttzz ttdzztttzttt|tztdzzdtzz|tdzzz ttttgftdztttztdzztdttdzztttzzz ttzttt|tdzztz|tdzzz ttttdzgfttdz tttzdtdzzd 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 gfttdztttzdtd zzdtd zzz dtdzzzdtdzzzd tdzzzd tzz dzdtdzzz z tttdtd zzdtdzzz tdzz dtdzzzdtzzdz dtdzzdtdzzz z gftttttd z dtdzzzdtdzzz dtdzzzdtzz dztdzdtdzzz dtdzzzdtd zzz dtdzzzdtd zzz d td zzzdtd zzz dtdzzzdtdzzz dtdzzzdtzz dzz ttdzzttztttdtd zd td zzz dtdzzzd tdzzz dtdzzzd tzz dzz gftttttttzdtzzdtzdz ttdzzdtzdzz zdtdzzd tzz d!zd"tdzzdtdzzz dzz zt ddz z tttddtzz dtzdz z gfttttd#tdzzd$tzz dz dtdzzdtdzzz dtzzdzz dttdzzd tzdzz ztttdtzdzdtzdz z gftttdtdzzdzttdzztz zd tdzztz dzttzttdz zz zdtdzzdtzz dzttdz dzzz zttt| tdzz tdzzdtzz |tz|z tdzztdzz tdzztz z tttd%tdz z gftttdtzttdzdzztz z tttt |zttdzzz|tdzzz tttt gftttttttzt 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 gfttttdDtdzzdDtdzzzt dz t ddz z ttdzzzd ztttd tzgfttttdEtdzzdz dtzdz ztdz t ddz z ttdzzzd zdtz ttztz zdtz ztttdEtzdz dz gfg}|D]\}}t|||dFS)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?rrRrYr)r>rrorLrVrWrMrtrnrjr9:rTrUrrkrurrrviirqrr0rQirrrrrrri0HiiFi4"i<ii2ҍi"0 iimi^ri i#??!A$$F AaDD2ad7Q;ad+ , ,b1q!tnn= 1q1Q4499Q<<2ad7RT>A#5Q AaCED9#  AaDD1R46AbD=2ad7*R1W4r!Q$w>AqDH Q$rT!Q$Y2a'!B$.1a47"QT'A 1a4QT'q!tG$&'c*+ , ,-/!qsQw1a4 A#BB7..  !a!A$$))A,,!AqD&1QT6/Bq!tG";a1f"D adF#T## !!Q$"( ( AaDD1QT6AadF?QT)AadF2QqS81QT ICPQSTPTH T 1H1a4x "%ad(+-/1W579!t<>?@ACDQ447KMNqTTR S S AaDD!QTAadF]R1W,r!Q$w6AqD@1Q3FJK L LM  1Q4499Q<<1Q44!A#1q!A$$'(91Q37(CC q!tVac\B AqD2ad7!2Q!6 78:;A$$q&A B B AaDD1qs7QqS1W% & &' 1Q4499Q<<#ad(RT/B.AqD2ad71BQqS1H11LM!ai1q!" # # AaDD1Q37QqS1W% & &' ! ! !Q$ AaDD!G+A--1a4!a10Eq!KH1 q!tVac\A%1q51* 5 6 AaDDB3A:1$qs*RTBYA-=1-Dq!t-K.  addBAJ'' ) ! ! qsAaDD!GaK(** AaDDA2b51QT6>BAI. / /AaDD1"> 1Q4499Q<<1Q441a!A#.!A#!Q,!a1G qSUQqTT!V^2 ttAv&),QTDF):S)@2ad7RPQSTPTWCT(UV W W AaDD1q5!)  b11b52ae8(;c!R%i(G$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/(122 3  1Q4499Q<<AqD2ad7*qbdQqTT!VmQqTT1W-DDqHII AaDD!A# 1Q4499Q<<AqDRT!V+qsQqTT!V|QqTT1W.DDq5!A$$,q.!#$Q3' ( ( AaDD"Q$q&1*  Ad EJ%%CC$$$$%%r-c td}ttttdtz ttztdz z ddtzz ttdzzdtzdz z zdtdzzdtdzzz d tzzd z d td zzd tdzzz dtdzzzdtzz dzz zdztttdtzdzdtzdz z gfg}|D]\}}t |||dS)z This function tests the computation of rational particular solutions for a Riccati ODE. 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