o *ήcr@sddlmZddlmZddlmZddlmZmZm 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/m0Z0m1Z1m2Z2edZ3edZ4ddZ5ddZ6ddZ7ddZ8ddZ9ddZ:dd Z;d!d"Zd'd(Z?d)d*Z@d+d,ZAd-d.ZBd/d0ZCd1d2ZDd3d4ZEd5d6ZFd7d8ZGd9d:ZHe!d;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_riccatifxcCstt| |td|S)N)rrmaxintr*K/tmp/pip-target-vg8gfxp4/lib/python/sympy/solvers/ode/tests/test_riccati.py rand_rationalsr,cs tfddt|dD|S)Ncsg|]}tqSr*)r,).0_r(r*r+ "zrand_poly..r')rrange)r&degreer)r*r(r+ rand_poly!s r3cCs\td|}td|}t|||}t|||}|td|kr*t|||}|td|ks||S)Nr'r)rr3r)r&r2r)degnumdegdennumdenr*r*r+rand_rational_function%s     r8c Cs|}||}t|dd}t|dd}|dkr!t|dd}|dkst|||||d}t||||||d}t||} t|| dksNJ||||fS)Nr'r)Tr)diffr8rrr) ratfuncr&yfyypq1q2q0eqsolr*r*r+find_riccati_ode/s    "  rEc Csvttdtdddtttd tdttddtdddddtfdtddtdddttddtdtdtddtdddttdtdddddtfd dtddd dtd tddtd tddtddd tdtd dtddd tdddd tdtddtddftd tddtdtddtdtd  dtdd tdtddtddddtddtddtdddtddtddtdfg}|D]!\}}}}|t|t||ks$J|t|t||ks2Jqdtddtddtddtt dd td tddd tt dd tdt ddd t ddtd d tt dd tdt ddtdt dd tdtddtddtfddtddtt dd td tddd tt dd tdt ddd t ddtd d tt dd tdt ddttdt dddd tdfg}|D]#\}}}}}|t|t||ks(J|t|t|||ks7JqdS)ar This function tests the transformation of the solution of a Riccati ODE to the solution of its corresponding normal Riccati ODE. Each test case 4 values - 1. w - The solution to be transformed 2. b1 - The coefficient of f(x) in the ODE. 3. b2 - The coefficient of f(x)**2 in the ODE. 4. y - The solution to the normal Riccati ODE. r'r:r9Fevaluater N)r&r rrrcancel)testswb1b2r>bpr*r*r+test_riccati_transformation?sz <HBP  "H^ HZ"rZc Csttttdttttttdtttttdtdtddddtdfdtdtdttttdttdtdtddtt dtdddt ddtdt ddd dtdttdtttd tdttt dfttdttttdttt tdddtdd ddtdddtddtddttdtttddtdftttttdtttdtttddtdfdtd tdtddtdtttttdtttdtttdtdtddtddttddtddtddtdtddtdfdtdtdttttdtttdftttttdtttdttdtdddfg}|D]\}}|t|ttksJqd S) z This function tests the transformation of a Riccati ODE to its normal Riccati ODE. Each test case 2 values - 1. eq - A Riccati ODE. 2. normal_eq - The normal Riccati ODE of eq. r:r9rLrQrPr'FrIrKN)r%r&r;rr r)rUrC normal_eqr*r*r+test_riccati_reducedsf0 <8>    :@  (FB4@ (r]c Cstttdtddtddtddtdd tdd tdd td dttddtd td dttttd ddtdddtddtddtddtddtddtddtddtddtddtddtdtddtddtddtddddtdd tdd tdd td tdddtddd tdd dtd ftttdtdtdd ttddtdd ttdtdtd!dd"tdd#td$d%tdd&tddd'tdtd!dddtddtdd(d)tddtd"d%td&tdd*tdddd tdtdd ftttd+td,d-tdd.tdd/tdd0td1d)tdd&td,d2tdd3tdd4tdd1ttd5d ttddtttdtddtddttddtdd+tdd)td,d&tdd2tdd3tdd4td1d-tdd)td,d&tdd2tdd3tdd4td1d.tdd)td,d&tdd2tdd3tdd4td1d6td7td,d tdd8tdd9tdd:td)td5d d7dtddtd,d;tdddtd,dtdd;tdd  :.   @8.(       @<8<@@ Prc CsXtdtddtddtdttdtdtdd tdd td dtddtd d td d tddtdd tdtd ftd ttdtd dtddtddtdtd ftdtddtddtdttdtddtddtdtdftdtddtd d tddtd dtd tdtdd tttdtdttdftd tddtd d td dtdtdd td ttdtd dtddtddtdtdfg}|D]\}}}t||t|ks(JqdS)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 rnr9rNr: rQirPrGrFrL r'rRrHrOrirSN)rr&r)rUr6r7valr*r*r+test_val_at_infs0(x4((XH4rcCs\tdgddks JtdgddksJtdgddks!Jtdgdd ks,Jd S) zt This function tests the necessary conditions for a Riccati ODE to have a rational particular solution. r[)r'r:rLFr'r:)r9r'rQ)r'r:rNrOTN)rr*r*r*r+test_necessary_condsFsrc Cs<dtddtddtddtddtdd td d tdd tdd td dtddtddtddtddtd dtdd tdd td d td d tdd tdd td d tddtddtddtd d tddtd dtddtd d tddtddtd dtddtd dtddtddtddtd dtddtddtdg}|D]&}dd|D\}}t||t\}}|tdt||ksJqdS)zi This function tests the substitution x -> 1/x in rational functions represented using Poly. rr9rNr:rQrarGr{rLPrlKrOirrrFrmrgdrnrcrPcSsg|]}t|tqSr*)rr&)r-er*r*r+r/gsz/test_inverse_transform_poly..r'N)r&as_numer_denomrsubsrT)fnsr%r6r7r*r*r+test_inverse_transform_polyUs".^j^t  $rc Cstdtddtdttdtddtddtdtdftdtd dtdd tddtd ttd tdd tddtdttftdtdd tddtddtddtd dtddtddtdttdtddtddtddtd dtdd tdd!td"tt ftd!tdd#tdd$tdd$td d%tdd&tdd'td(ttd)tddtdd*tdd*td d+tdd,tdd-td.ttd/dftd0td d1tdd.tdd2td3ttd4td dtdd5tddtdttd0d4fg}|D]\}}}t||t|ksaJqQd6S)7a This function tests the limit at oo of a rational function. Each test case has 3 values - 1. num - Numerator of rational function. 2. den - Denominator of rational function. 3. limit_at_inf - Limit of rational function at oo r:rg r9rbrrLiixii'ii!rNrFirQrkrGiXHi ivii| Tireirlrrcrxriii i i ir' iiZi_iN)rr&rr r)rUr6r7limr*r*r+test_limit_at_infls<(4(TXXX 44 rc Cstdtddtddtdtddtdtdd td d tdd tdd tdd tdtddtdtddtd td gtddtd td ggftdtddtddtdtddtdtddtddtddtddtdtddtd dtddtddgtddtddggftdtdtddtdtdddtdtdddtdtddd tdtdtdd tdd tdtddtddttddtddd!d"d#tdd$ddgtddttddtddd!d"d#tdd$ddggfg}|D]\}}}}t||t||ksWJqEd%S)&ab This function tests the Case 1 in the step to calculate coefficients of c-vectors. Each test case has 4 values - 1. num - Numerator of the rational function a(x). 2. den - Denominator of the rational function a(x). 3. pole - Pole of a(x) for which c-vector is being calculated. 4. c - The c-vector for the pole. r[r9r:rLrGT extensionrNrFrPrOrQrr'iii0i0iiiei1ijizrrrz QQdomainFrI,N)rr&r rrrr)rUr6r7polecr*r*r+test_construct_c_case_1s6,P:,D 2P BBrcCstdtddttddtdtddddt dtdgtdtdggftdtddtd d tddtddtdtddtddtddtdddtd  d gtd  d ggfttdtdd ttddttdd tddtdddd d tdtddd tdd ddtdd gdtdtddd tdd ddtdd ggftdtdtd dtddtd tdd tdtddtd d d dtdtd dtdddtddgdtdtddtdddtd dggfttddtddtdtddtddtddtddddtdtd dtddddtddtdd gd!tdtddtddddtddtd d ggfttddtddtttddtddtddtd"tdd dtd"d#td d td"gtd" tdd dtd"d#td  d td" ggfg}|D]\}}}}}t||t|||ksJqd$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:rKr9rGrOrLrFYrhrPrliWrSri7i8irQrdB6i$r|iroirN)rr&rr rr)rUr6r7rmulrr*r*r+test_construct_c_case_2sT $0$   ::  44 $ BB4:,rcCstdggks JdS)z` This function tests the Case 3 in the step to calculate coefficients of c-vectors. r'N)rr*r*r*r+test_construct_c_case_3srcCsttd dtddtddtdtddtdtddtddtdtddddtd tdd ttd d tddgd td t ddttd d tddggfttd dtddtddtddtddtdtddttddtddtdtdtddddttt dtdgdtt tdtdggftdtdtdtddtdtddtdtddtdtddtdtddddtdtddtdtdtdtdtd ttdddtdtddgdtdtddtdtdtd tdtdttdddtdtddggftd tddtddtdtddtddtdtdtddddtddtdtdt td d tgd!tdd"tdt dttd d tggfttd tdtdtdttddttdtdddd#tddtt tt d$dtdgdtddttt td$dtdggfttd tddtddtdtdtddtdtddtdtdtddddtdtddtdtdtdtdtdtdtd ttd d%dtdtddgd tdtdd tdtdtd tdtd tdtdttd d%dtdtddggfg}|D]\}}}}t||tt|d}t||d|ksJqd&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. rGr:rLr9TrrPrnrdr[r`rrQrFrOrRrurSi rirriIBi ir')rm;iirrHrrNN)rr&rr rr#rr )rUr6r7rdserr*r*r+test_construct_d_case_4sn6,02N06D H4.8 22. XNXD0.rc Cstdtdtdtdtddtdtddtddtdtddtddtd dgtd dtddggftdtdtd tdtddtdtd d tdtdd td dtddtddtd td d tddd tddgtd ddtddggfttdtdtd d tdtdd tdtd d tdddtddgtd ddtddggfg}|D]\}}}t||ttdd}t||ksJqdS)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:r9TrrPrGr'rtrLZZrrFrSrdrHrN)rr&rr#rr!)rUr6r7rrr*r*r+test_construct_d_case_5Es"$ ,28D8 8rcCstdtddtddtdtddtddtdtddtddtdgtddtdggftdtd dtddtdtddttd tddtddtd dtddtd tdddgd ggftd td tddtdtddtd tddtddtd dtddtddtd dtddtdtdddgd ggfg}|D]\}}}t||t|ksJqdS)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. rSr:rGrrrLrnr'r9rQrPrrHrrOrNrirFrdr."*N)rr&r rr")rUr6r7rr*r*r+test_construct_d_case_6fs, ,*,H (T rcCsttddtdtddttdttddtddddd dd dd d ftd tdd tdtddtd tddtddtddtdtddtddddtddtddtddtdddftdtddttddtddtddtdd tdd!dtdtdddtdtdtddtddddtdd"dtdtd#d"dtdd$d%d#tdd"dtdd#tddtd#d"dtdd$d%d#tddd"dtdd#tddd&fttddtddtdd'td(tddttddtddtdddd)ddd*d+d,ftdtddtddtdtddtdtdtddtddtdtddtddd)dtd-d.td/d0td1d2td3d4td5d6d7ftd8tddtdtddtdtddtddtddtddtdtddtd)dd)d)td9 d:d)d#td;dd<fg}|D]\}}}}}}|t||t|||ksJqd=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 calcuated. r:r9rPTrr'rQrFir)r'rrKrSr[rM@iirLri?iirNiiKiil`B l-Ri0i}')rr:rKr'rGrMrOrr[rKFrI)rLr9r:r'rrKrnrrrru)r9r:r'rrKrSi i ii)l?PTl@qslF41l@4lJ!tzlZdj m)rrKrSr[rMrGrr)rrSrHrKr[rMN)rr&r rrrr#)rUr6r7x0rnrr*r*r+test_rational_laurent_seriessf  8  P 264 , 4 D"/ rcst|tr |j|j}t|tt\}}ttttg|R}tdfdd|D}t ddt ||Ds:Jt fddt ||DsJJdS)z Helper function to check if actual solution matches expected solution if actual solution contains dummy symbols. C1csg|]}|qSr*)r)r-rDr dummy_symr*r+r/sz#check_dummy_sol..cSsg|]}|dqS)rr*)r-r&r*r*r+r/r0csg|] \}}||qSr*)dummy_eq)r-s1s2)rr*r+r/sN) isinstancerlhsrhsrr%r&r$r allrzip)rCsolserr.rsolsr*rr+check_dummy_sols  $rcCs0 td}ttttttdddttttdttttd gfttdtttdtttdtdtttd|t|ttdgfdtdttttdtttttdttdttttt|dtd|tgfttttttd dtdtdtttdtdtgftddtdtttttdtttttt|ttddt|tdttttgftdttttdtdttdtttttttt|tdt|tdttttdgfttd tttdtdd td tdddtddtttd |td |dtd dtddtddtddtd d |tdd |td|d td dtd dtddtddtdd ttttdtddtddtdgfttdtttdtd dtd dtddtdd tdd tddtdtttdtd dtdtddtddtddtddtdgftttttd  dtddtddtddtdtddtddtddtd dtddtd d td dtd dtddtddtddtdttdtttttdtd d td dtdd tddtdd tdgftttttttdtdtdttddtddtdd td!d"tddtddtddtttddtdtdgfttttd#tdd$tddtddtddtddttdd tdtttdtddtdgftttdtddttdtd tdtdttttddtddtdttddttt| tdtddt|t|tdtdtdttttd%tdgftttdtttddttttt |ttd|tdtttt gfttttttttdddttdtddttddtdtddtd dd&tdd'td(d"tddtdtttd tttttdtd)d*td+d,tdd-tdd.tdd/td d0tdd1td d2td d3td d4tdd5tdd6tdd7td8dtd)d9td+d:tdd;tddtd d?td d@td dAtddBtddBtddCtgfttttdDtddDtdt dtddttdd tttd tgfttttdEtdddtdtdtddttdd dttttdttttdEtddgfg}|D] \}}t|||qdFS)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:rrLrSr'r9rrgrFrPrQrGrlrfrbr9:rNrOr{rcrmrrrniirirr0rKirrrrrrri0HiiFi4"i<ii2ҍi"0 iimi^ri isP           H B6c,.&>?! C}