a
    {þdOù ã                   @   sb  d dl mZmZmZ d dlmZ d dlmZmZm	Z	 d dl
mZ d dlmZ d dlmZmZ d dlmZ d dlmZ d d	lmZ d d
lmZ d dlmZmZmZmZmZmZm Z m!Z! d dl"m#Z#m$Z$ d dl%m&Z& d dl'm(Z( d dl)m*Z* d dl+m,Z,m-Z-m.Z.m/Z/m0Z0m1Z1m2Z2m3Z3m4Z4m5Z5m6Z6m7Z7 d dlm8Z8m9Z9 d dl:m;Z; d dl<m=Z= d dl>m?Z?m@Z@mAZAmBZBmCZC ed\ZDZEZFZGZHZIZJZKZLZMZNedZOedZPedZQedZRdd ZSdd ZTdd ZUd d! ZVd"d# ZWeAd$d% ZXd&d' ZYd(d) ZZeAd*d+ Z[d,d- Z\d.d/ Z]eAd0d1 Z^d2d3 Z_d4d5 Z`eAd6d7 Zad8d9 Zbd:d; Zcd<d= ZdeAd>d? ZeeAd@dA ZfeAdBdC ZgeAdDdE ZhdFdG ZieAdHdI ZjeAdJdK ZkdLdM ZldNdO ZmeAdPdQ ZneCeAdRdS ZoeAdTdU ZpeAdVdW ZqdXdY ZrdZd[ ZseAd\d] Ztd^d_ ZueAeCd`da ZveCdbdc ZweAeCddde ZxeCdfdg Zydhdi Zzdjdk Z{eAdldm Z|dndo Z}dpdq Z~drS )sé    )Ú
DerivativeÚFunctionÚdiff)ÚMul)ÚIÚRationalÚpi)ÚEq)ÚS)ÚSymbolÚsymbols)Úsinh)Úsqrt)ÚMatrix)ÚTuple)ÚexpÚcosÚsinÚlogÚCiÚSiÚerfÚerfi)ÚdotprodsimpÚNonSquareMatrixError)Údsolve)Úconstant_renumber)Úchecksysodesol)Ú_classify_linear_systemÚlinear_ode_to_matrixÚODEOrderErrorÚODENonlinearErrorÚ_simpsolÚ_is_commutative_anti_derivativeÚlinodesolveÚcanonical_odesÚdsolve_systemÚ_component_divisionÚ	_eqs2dictÚ_dict2graph)ÚairyaiÚairybi)ÚIntegral)Úratsimp)ÚON_CIÚraisesÚslowÚskipÚXFAILzC0:11ÚxÚfÚgÚhc                     s  t dtd\} }}td| ||g|  ¡}| ¡}| ¡}|  d¡}| d¡}| d¡}t||t|| g}	tddgddggtddgddgggtdgdggf}
t|	d d d|
ksüJ t|| d|  t||t||| |  g}tg d¢g d	¢g d
¢gtg d¢g d
¢g d¢ggtdgdgdggf}t|d|ks J td| d|  | | td| d|  | | td| d|  || g}tg d¢g d¢g d¢gtg d¢g d¢g d¢ggtdgdgdggf}t|d|ksdJ t|| || td| | | | dtd| dg}tg d¢g d¢g d¢gtg d¢g d¢g d¢gtg d¢g d¢g d¢ggtdgdgdggf}t|d|ks J t||t|| | g tt fdd t|| d t|| | gtt	fdd d S )Núf, g, h©ÚclsÚté   é   r   éÿÿÿÿ)r<   r   r   )r   r<   r   )r   r   r<   )r<   r;   r   )r<   r<   r<   é   é   é   )r;   r   r>   )r   r@   r?   )r@   r?   r   )r<   r<   r   )r<   r   r<   )r   r<   r<   )r   r<   r;   ©r   r   r   )r   r=   r   )r   r   éýÿÿÿ)r   r<   r=   c                      s   t  d d dS ©Nr=   r<   ©r   © )Úeqs_5Úfuncsr:   rE   úm/var/www/html/stable-diffusion-webui/venv/lib/python3.9/site-packages/sympy/solvers/ode/tests/test_systems.pyÚ<lambda>A   ó    z+test_linear_ode_to_matrix.<locals>.<lambda>c                      s   t  d d dS rC   rD   rE   )Úeqs_6rG   r:   rE   rH   rI   D   rJ   )
r   r   r   r   r	   r   r   r/   r    r!   )r4   r5   r6   Úf1Zg1Úh1Úf2Zg2Úh2Úeqs_1Zsol_1Úeqs_2Zsol_2Úeqs_3Zsol_3Úeqs_4Zsol_4rE   )rF   rK   rG   r:   rH   Útest_linear_ode_to_matrix"   sB    8D2ÿd2ÿD0ÿÿ (rT   c            8         s)  t dtd\} }}}t d\}}t| }t|}t|}t|}	t| }
t|}| |g||g }d| d|   d|  d| d |   d	|   f}t|d u súJ d|d  d|   d|  d| d |   d	|   f}d
tt| td|  d d| d   tt|d |  d d  d	| d  gtt| td|  d d| d  tt|d |  d d  d	| d  ggd}t||ksDJ tt| td|  d d| d   tt|d |  d d  d	| d  g}t|d u sÌJ tt| td|  d d| d  tt|d |  d d  d	| d  g}t|d u sRJ d| d|   d|  d| d|   d	|  d|	 | || f}dd|  d|  dt|   d|  d	|  dt|  |dt|  |t| f| |||g| d|d|d|did
d
d
ttddtddddgtddtd	dddgg d¢ddtdddgg dd
d
}t|||ksÖJ d| d|   d|  d| d|   d	|  || |	| f}dd|  d|  dt|   d|  d	|  dt|  | t| | t| f| |||g| d|d|d|did
d
d
ttddtddddgtddtd	dddgg d¢g d¢g dd
d
}t|||ksLJ d| d|   d|  |
 d| d|   d	|  || |	| f}dd|  d|  dt|   t| df d|  d	|  dt|  | t| | t| f| |||g| d|d|d|did
d
d
dd
d	}t|||ksJ t|d	| d|  t|d| d	|   t|d|  d|  f}d	tt| d	| d|  tt|d|  d|  tt|d|  d|  f| ||g| d|d|did
d
d
tg d¢g d¢g d¢g dd
d
}t||d d |ksÜJ t||t|| f}dtt| |tt|| f| |g| d|did
d
d
tddgddgg dd
d
}t||k	sJ t|d|  t|d |  d	|  t|d|  d|  d!|  f}d	tt| d|  tt|d |  d	|  tt|d|  d|  d!|  f| ||g| d|d|did
d
d
tg d"¢g d#¢g d$¢g dd
d
}t||d d |k
sÆJ t|d|  d|  d|  t|| d%|  d!|  t|d&|  d'|  d|  f}d	tt| d|  d|  d|  tt|| d%|  d!|  tt|d&|  d'|  d|  f| ||g| d|d|did
d
d
tg d(¢g d)¢g d*¢g dd
d
}t||d d |ksJJ td	| d+||  td| d,||   td| d| |  f} d	td	t|  d+| d+|  tdt| d-|  d,|  tdt| d|  d|  f| ||g| d|d|did
d
d
tdtd.d	td+d	gtd,ddtd-dgtddtdddgg dd
d
}!t| |d d |!ksÄJ t|d	| d|  t|d| d	|   t|d|  d|  f}"d	tt| d	| d|  tt|d|  d|  tt|d|  d|  f| ||g| d|d|did
d
d
tg d¢g d¢g d¢g dd
d
}#t|"|d d |#ksJ tt| |tt|| f}$dtt| |tt|| f| |g| d|did
d
d
tddgddgg dd
d
}%t|$| |g|%ksÎJ tt| d|  tt|d |  d	|  tt|d|  d|  d!|  f}&d	tt| d|  tt|d |  d	|  tt|d|  d|  d!|  f| ||g| d|d|did
d
d
tg d"¢g d#¢g d$¢g dd
d
}'t|&| ||g|'ks:J tt| d|  d|  d|  tt|| d%|  d!|  tt|d&|  d'|  d|  f}(d	tt| d|  d|  d|  tt|| d%|  d!|  tt|d&|  d'|  d|  f| ||g| d|d|did
d
d
tg d(¢g d)¢g d*¢g dd
d
})t|(| ||g|)ksæJ td	t|  d+| d+|  tdt| d-|  d,|  tdt| d|  d|  f}*d	td	t|  d+| d+|  tdt| d-|  d,|  tdt| d|  d|  f| ||g| d|d|did
d
d
tdtd.d	td+d	gtd,ddtd-dgtddtdddgg dd
d
}+t|*| ||g|+ksJ tt| | | d! tt|d|  d|  d/ f dtt| | | d! tt|d|  d|  d/ f| |g| d|did
d
d0d
tddgd1d2gg td!gd/ggd3d4},t |,ksJ tt| d |  d|  tt|d|  d |  f dtt| d |  d|  tt|d | d|   f| |g| d|did
d0d
td2 d1gd1d2 gg tdd  d d gd dd  d ggd5d
d6},t |,ksàJ t|| |   t||  | g dtt| |  |  tt||  | g| |g| d|did
d0d0d
td g dgg td d gd d ggtgdggd7d8},t |,ksòJ t||  |   t||  |  t	 g}-dtt| |  |   tt||  |  t	 g| |g| d|did
d0d
tgt	ggtgggd0d7td d d d gd d d d ggd9}t|-|ks2J t|| | | d  t|| | |  t|| | |  g}.d	tt| | | | d  tt|| | |  tt|| | |  g| ||g| d|d|did
d0d0d
t   g   g   gg td d d d d d gd d d d d d gd d d d d d ggtgdgdggd7d8}/t|.|d d |/ks&J t|| | |  d t|| |  | d: t||  | |  g}0d	tt| | |  | d tt|| |  | d: tt||   | | g| ||g| d|d|did
d0d0d
tdd gd dg ddgg td d gd d gd d ggtdgd:gggd7d8}1t|0|d d |1ksîJ | | | |  }2t||2t||2t||2d t|	|2g}3dtt| ||  | |  tt|||  | |  tt|||  | |  d tt|||  | |  g| |||g| d|d|d|did
d0d0d
t    g    g    g    gg td d d d d d d d gd d d d d d d d gd d d d d d d d gd d d d d d d d ggtdgdgdgdggd7d8}4t|3||4ksJ t d;}5td!|  d|  dt|   t| df d	t|  dt
t  td	|  d|  dt|   d<t|  t|df dt
t  f dtd!|  d|  dt|   t| df d	t|  dt
t  td	|  d|  dt|   d<t|  t|df dt
t  f| |g| d|did
d0d
tdt
t  gdt
t  ggdd
d
d=},t |,k!svJ tdd  d  d d t| df d|  d>|  tdd  d  d d t|df | d!|  f}-dtdd  d  d d t| df d|  d>|  tdd  d  d d t|df | d!|  f| |g| d|did
d
d
d3ttd?dd>gdtd@dggtdd  d  d tdAttdAd<  tdd<  dA tdAttdAd<  tdd<  dA  d
|5d
d
dB}t|-|k#sdJ t|  |  t t| | t
  t| df d t|  |   t| |   t|df f}.dt|  |  t t| | t
  t| df d t|  |   t| |   t|df f| |g| d|did
d
d
dt t  t
 gd	  d  ggd
d
dC}/t|.|/k%s.J t|
||  ||  t||| ||  f}0dtt| df||  |t|  tt|df|| |t|   f| |g| d|did
d
d
dd
d
dD
}1t|0|1k&sJ t dEtd\}6}7t dF\}}5|6|7gtt|6d dt|6  d dt| |6 t|7 dg d
tt|6dtt|7||6 gtt|6d	tt|7||6 ggd},t |,k'sJ tt fdGdH tt|6d|6 |7 d  tt|7|6d|7   g}-dtt|6d|6 |7 d  tt|7|6d|7   g|6|7g|6d|7did
d0d
tdgdggtddgddggd0dI|5ttdt tgtdt ggdJ}t|-|k(s\J tt|6d|6 |7  tt|7|6d|7   g}.dtt|6d|6 |7  tt|7|6d|7   g|6|7g|6d|7did
d
d
tddgddggd0dK|5tdgdggttdt tgtdt ggdL}/t|.|/k)sJ d S )MNz
x, y, z, wr8   zt k lr@   é   é   r;   é   r>   Tiôÿÿÿ)Zis_implicitZ	canon_eqsr?   éõÿÿÿr<   éúÿÿÿr   ©r   r   r   r<   Ztype1)
Úno_of_equationÚeqÚfuncÚorderÚ	is_linearÚis_constantÚis_homogeneousÚ
func_coeffÚtype_of_equationÚ
is_general)r   r   r   r=   ©r   r   r=   r   Ztype0)	r[   r\   r]   r^   r_   ra   rd   rc   Úis_higher_orderé   rB   )r   rB   rW   )r>   r   éùÿÿÿ)rX   rg   r   r=   é   é   é	   )éëÿÿÿr   r   )iïÿÿÿrB   r   )éûÿÿÿrh   é÷ÿÿÿé   é    é)   )éüÿÿÿrm   éþÿÿÿ)r=   ióÿÿÿrn   )iàÿÿÿi×ÿÿÿrX   é   é   éñÿÿÿiìÿÿÿé   Frs   rm   Útype2)r[   r\   r]   r^   r_   r`   ra   rd   rb   Úrhsrc   Útype3)r[   r\   r]   r^   r_   r`   ra   rb   Úcommutative_antiderivativerc   rd   Útype4)r[   r\   r]   r^   r_   r`   ra   rd   rb   r{   ry   rc   )r[   r\   r]   r^   r_   ra   rd   ry   rb   r`   rc   r{   é
   Út_é   )r[   r\   r]   r^   r_   ra   rd   ry   rc   Úis_second_orderrf   é#   é5   éE   é!   )r[   r\   r]   r^   r_   ra   rd   rc   ÚA0zg(t)ÚtauZis_transformedr~   r   rf   )r[   r\   r]   r^   r_   ra   rd   rc   ÚA1r   rf   )
r[   r\   r]   r^   r_   ra   rd   rc   r   rf   úf gzy t_c                      s   t  d d S ©Nr<   )r   rE   ©Úeq1rG   r:   rE   rH   rI   x  rJ   z.test__classify_linear_system.<locals>.<lambda>Ztype6)r[   r\   r]   r^   r_   ra   rd   ry   rb   r`   rc   r~   r   r{   Ztype5)r[   r\   r]   r^   r_   ra   rd   rb   r`   rc   r~   ry   r   r{   )r   r   r   r   r	   r   r   r   r   r   r   r   r   r/   Ú
ValueError)8r3   ÚyÚzÚwÚkÚlÚx1Úy1Zz1Zw1Úx2Úy2Zfuncs_2rP   rQ   Úsol2Zeqs_2_1Zeqs_2_2rR   Zanswer_3rS   Zanswer_4rF   Zanswer_5rK   Zanswer_6Zeqs_7Zanswer_7Zeqs_8Zanswer_8Zeqs_9Zanswer_9Zeqs_10Z	answer_10Zeq11Úsol11Zeq12Úsol12Úeq13Úsol13Úeq14Úsol14Úeq15Úsol15Úsol1Úeq2Úeq3Úsol3Úeq4Úsol4Z	sum_termsÚeq5Úsol5r~   r4   r5   rE   r   rH   Útest__classify_linear_systemG   s   HL4:ÿ2:ÿþÿ4:ÿ2:ÿX((ý"üðT((ý"üðX<Nÿÿ<
ýXN&ÿ.ýù	0$þûXB2ÿ.ýù	x4`ÿ0ýø	X,Tÿ0ýø	XN&ÿ.ÿý00$þü"@2ÿ@2þ ýù(`2þ`2þ ýù(T*ÿT*þ ýù(P$*ÿ$ýXX$ ,ÿý86ÿ"ÿ"üH:ÿ"2üh0Tÿ0ÿÿ,ÿ ÿúp:ÿÿ.ÿÿÿÿÿû$(4Pÿÿÿ2þÿ&$ÿ$ÿÿ^ÿÿù^^ÿ>ÿÿ*ÿÿÿÿ ÿûFBÿHBÿ4ÿRûTPÿ2ÿÿÿ,ÿÿÿÿ 4û420ÿ
ÿý, ÿ..ÿÿT:ÿ"ÿÿ*üP6ÿ"*ür§   c                     s°  ddl m} m}m} ddlm tdtddD ]}||||ks2J q2tddD ].}||}t|| }||ks\J q\tddD ]:}| ||dd }| ||fd	d}||ksJ q| ddgd
dgg}| t	t
gt
 t	gg}||ks J | ddgddgg}| dt  t
 t t	  dt  t
 gdt  t
 dt  t
 t t	  gg}||ksÈJ | g d¢g d¢g d¢g}| d td  dtd  d  td d  d td  dtd  d  dtd  d  td d td d  gd td  td d  td d  d td  td d  dtd  d  td  d td d  gd td  d td  td gg}||ksHJ | g d¢g d¢dddtd d gddtdd tdd gg}| tt d td d  d t  dtd d   dt  d td d  d t  dtd d   dt  gdt td d  dtd d   dt  td d  d dtd d   dt  gddtd d  d td d   td d  d gddtd d  d  td d  d td d  gg}||ks@J | g d¢g d¢g d¢g d¢g}| t	t
ddgt
 t	ddgddt	t
gddt
 t	gg}||ksÈJ | g d¢g d¢g d¢g d¢g}| t	t
t	 t
 gt
 t	 t
 t	 gddt	t
gddt
 t	gg}||ksrJ | dtgtdgg}| tt d tt  d  tt d tt  d  gtt d tt  d  tt d tt  d  gg}||ks&J | g d ¢g d!¢g d"¢g | dgddggtt fd#d tt fd$d ttfd%d ttfd&d d S )'Nr   )r   ÚeyeÚzeros)Ú
matrix_expr:   r<   rg   c                 S   s   | |kr| d S dS ©Nr<   r   rE   ©ÚiÚjrE   rE   rH   rI     rJ   z!test_matrix_exp.<locals>.<lambda>c                    s   | |krt | d   S dS r«   )r   r¬   ©r:   rE   rH   rI     rJ   r=   r;   rm   rr   r>   rB   )ri   rj   rV   )rm   r=   rY   )r?   r?   é   r°   r@   r?   rs   )r<   r<   r   r   )r   r<   r<   r   r   é0   )r   r<   r   r   )r=   r   r   r   rZ   re   )r=   r   r   r<   )r<   r;   r>   )r?   r@   rV   )rg   rg   rg   c                      s    d d d df S ©Nr;   rE   rE   ©ÚMrª   r:   rE   rH   rI   ò  rJ   c                      s    d dd d f S r²   rE   rE   r³   rE   rH   rI   ó  rJ   c                      s
    S ©NrE   rE   ©ÚM1rª   r:   rE   rH   rI   ô  rJ   c                      s    d dd df S r   rE   rE   r¶   rE   rH   rI   õ  rJ   )Úsympy.matrices.denser   rš   r©   Úsympy.solvers.ode.systemsrª   r   Úranger   r   r   r
   r   r/   r   )r   rš   r©   ÚnÚAZexpAtrE   )rŽ   r·   rª   r:   rH   Útest_matrix_exp  s¢    &<<þ
0ýýVFÿtDDûüüü"&ü
BBþrœ   c            	      C   s  t dtd\} }}t d}| |||||g}t| | ||¡| |d||  t||d || |¡| | g}tt| ||df| |d||  tt|||| | || d gg}t||d d ||ksæJ t| | |¡|| |¡| | t|| |¡d | ||| t|| |¡| |g}tt| ||d| | tt|||t| |||  tt|||| |gtt| ||d| | tt|||t| ||| tt|||| |gg}t||||ks
J d S )Núf g hr8   r3   r;   r<   )r   r   r	   r   r   r%   r   )	r4   r5   r6   r3   rG   Úeqs1r   Úeqs2r   rE   rE   rH   Útest_canonical_odesø  s    JP`VTÿrÁ   c            5      C   s  t dtd\} }}}}t d\}}}}tt|||||tt|||||g}	t||tt| t||tt| g}
t|	|
ksJ t|	|
dddgfks¬J tt|||d|| tt|||d|| g}t||ttd|  t||ttd|  g}t||ks*J t||dddgfksFJ tt|||||| tt|||||| g}t||tt||  t||tt||  g}t||ksÄJ t||dddgfksàJ tt|||||| tt|||||| g}t||tt||  t||tt||  g}t||ks^J t||dddgfkszJ tt||||| tt|||||g}t||t t	| tt
|  t||tt
| tt	|  g}t||ksJ t||dddgfks J tt|||d|| tt|||d|| g}t||t t	d|  tt
d|   t||tt
d|  tt	d|   g}t||ksÀJ t||dddgfksÜJ tt|||t|| tt|||t|| g}t||t tt |  ttt|   t||ttt |  ttt|   g}t||ksJ t||dddgfksJ tt|||| || tt|||||| g}t||t t tt | |  tt tt| |   t||ttt | |  ttt| |   g}t||ksZJ t||dddgfksvJ tt||||||| tt||||||| g}t||td	td  ttd |  td	td  ttd|   t||tttd |  tttd|   g}t||ksHJ t||dddgfksdJ tt||||||| tt||||||| g}t||t ttd|   t||tttd|   g}t||ksôJ t||dddgfksJ tt|||d|| || tt||||| d||  g}t||ttd|  t	| ttd|  t
|  t||ttd|  t
| ttd|  t	|  g}t||ksàJ t||dddgfksüJ tt|||||d||  tt|||d|| || g}t||t t|  ttd|   t||tt|  ttd|   g} t|| ksšJ t|| dddgfksÄJ tt|||d
|| || tt||||| d||  g}!t||t| td|  tt td|   t||t td|  t| td|   g}"t|!|"k	sJ t|!|"dddgfk	sJ tt|||||| tt|||||| g}#t||t t| |  tt||   t||tt| |  tt||   g}$t|#|$k
sBJ t|#|$dddgfk
s^J tt|||||| tt|||||| g}%t||t | t| t||   t||  t| t|t||   t||   t||tt| t||   tt|t||    g}&t|%|&ksBJ t|%|&dddgfks^J tt|||||| |||  tt|||||| g}'t||dt | t||t|d d
| |    d  |t|d d
| |    dt | t||t|d d
| |    d  |t|d d
| |     t||tt||t|d d
| |    d  tt||t|d d
| |    d   g}(t|'|(ksÀJ t|'|(dddgfksÜJ tt|||||| || tt|||||| || g})t||t| t|t|d d	   t|d d	 d	  t| t| t|d d	   t|d d	 d	   t||tt|t|d d	   tt| t|d d	    g}*t|)|*ksîJ t|)|*dddgfks
J tt|||dtt|||dg}+t||tt||tg},t|+|,ks`J t|+|,dddgfks|J tt|||d|| || tt|||||g}-t||t| t| tt t|  t||tt| t| t|  g}.t|-|.ksJ t|-|.dddgfks6J tt|||||tt||||||| g}/t||tt| t||t| t| tt|  g}0t|/|0ksŒJ t|/|0dddgfksØJ tt|||d|| tt||||||| g}1t||dt td|  t||ttd|  tt|  g}2t|1|2ksjJ t|1|2dddgfksJ tt|||d|| tt|||||g}3t||ttd|  t||tt| g}4t|3|4ksüJ t|3|4dddgfksJ d S )Nz	f g x y hr8   za b c tTr   r;   r>   rs   r<   r?   )r   r   r	   r   ÚC1r   ÚC2r   r   r   r   r   r   )5r4   r5   r3   r   r6   ÚaÚbÚcr:   r¿   r   rÀ   r   Úeqs3r¢   Úeqs4r€   Úeqs5rŠ   Úeqs6Úsol6Úeqs7Úsol7Úeqs8Úsol8Úeqs9Úsol9Úeqs10Úsol10Úeqs11r   Úeqs12r   Úeqs13r   Úeqs14r   Úeqs15r   Zeqs16Úsol16Úeqs17Úsol17Úeqs18Úsol18Úeqs19Úsol19Úeqs20Úsol20Zeqs21Úsol21Úeqs22Úsol22rE   rE   rH   Ú#test_sysode_linear_neq_order1_type1  sx   ÿÿÿÿÿÿÿÿÿ" ÿÿ*(ÿÿ,*ÿÿ<2ÿÿJ2ÿÿÿ"$ÿ88ÿ""ÿ(&ÿ"$ÿ0.ÿÿ,*ÿÿ\:ÿ&ÿfÿÿÿ`þ""ÿbÿÿBþÿÿ"ÿ($ÿÿ$ÿÿ$ÿÿÿrå   c            g      C   sd,  t d} td}td}td}td}td\}}}}}	}
tt|| | | ||  |||   |||   |
||   tt|| | |||  |||   |||   tt|| | | | |	 ||  tt|| | |	||  |
||   g}t|| t| | t||
  t|
 |   || |
   t||| |
  |d  |||	 |
   |	|
   t|  || |	   |	| | | | |	    t	t|  ||    t|| tt| t|
 |   || |
   t| || |
  ||  |d  ||	|
    t|  || |	   |	| | | | |	    t	t|  ||    t|| t || |	 |
  t|  || |	   |	 t|| tt|
 |   tt|  || |	    g}t
||ksØJ t||dg d	¢fksôJ td
td\}}}}}}td\}}tddd\}}tt|| | |||  tt|| | |||  tt|| | | | ||  g}t|| tt| t|  ||   ||   t|| t | t|  ||   ||  t t|| tt|  ||   g}t
||ksJ t||dg d¢fks.J d||  ||  d||   t|| |  d||  ||  d||   t|| |  d||  ||  d||   t|| |  g}t|| ttd|   td tdt d  ttd|    ttd|  tdt d ttdd    t|| td tdt d  ttd|   ttd|  tdt d ttdd    t|| tttd|   tttd|    ttd|    g}t
||ksØJ t||dg d¢fksôJ tt|| | || tdd d||   ||  || tdd  tt|| | || tdd d||   || tdd  tt|| | || tdd || tdd  tt|| | || tdd || tdd  g}t|| ttd|   t|  td|    t|| ttd|   dt td|    t|| dt td|   t	td|   tdd  t|| ttd|   t	td|    g}t
||ks¬J t||dg d	¢fksÈJ t||  | ¡|| t||  | ¡|| t||  | ¡|| t||  | ¡|| g}t|| tt|  t|| tt|  t|| tt|  t|| t	t|  g}t
||ksJ t||dg d	¢fksŠJ tt|| | || ||  tt|| | || ||  tt|| | || tdd ||  tt|| | || d || d  g}t|| tt|  t|  t|   dt	 |  t| tdd   dt dt	  t| tdd   t|| tt|  t	|  t| tdd   tdt	  t| tdd   t|| t	|  t| tdd  d td t	 t| tdd   t|| tt| tdd  d t	|  t| tdd  d  g} t
|| k
s~J t|| dg d	¢fk
sJ tt|| | || tt|| | || tt|| | || tt|| | || tt|| | || g}!t|| tt|  t|| tt|  t|| tt|  t|| t	t|  t|| tt|  g}"t
|!|"ksJ t|!|"dg d¢fks¢J tt|| | d||  ||  tt|| | d||  tt|| | d||  tt|| | || d||   tt|| | d||  g}#t|| ttd|   t|  td|    t|| ttd|   t|| ttd|   t|| t	td|   t|  td|    t|| ttd|   g}$t
|#|$ksîJ t|#|$dg d¢fks
J tt|| | || tt|| | || tt|| | || g}%t|| tt|  t|| tt|  t|| tt|  g}&t
|%|&ks¢J t|%|&dg d¢fksŸJ tt|| | | | ||  tt|| | |||  tt|| | |||  g}'t|| t ||  t|  ||   | t|| tt||   t|| tt|  ||   t g}(t
|'|(ksJ t|'|(dg d¢fks®J tt|| | |||  tt|| | | | ||  tt|| | |||  g})t|| tt| t|  ||   |  t|| t ||  t|  ||   | t|| tt|  ||   t g}*t
|)|*ksJ t|)|*dg d¢fks°J tt|| | |||  tt|| | |||  tt|| | | | ||  g}+t|| tt| t|  ||   ||   t|| t | t|  ||   ||  t t|| tt|  ||   g},t
|+|,ksJ t|+|,dg d¢fks¶J tdtd\}-}.}/td \}0}1}2tt|-| | d|-|  |.|  tt|.| | |0|-|  g}3t|-| tt| t|0d! d!   t|0d! d!  tt|  t|0d! d!   t|0d! d!   t|.| tt| t|0d! d!   tt|  t|0d! d!    g}4t
|3|4ksÐJ t|3|4dd"d"gfksìJ tt|-| | d|.|  d|/|   tt|.| | d|-|  d|/|   tt|/| | d|-|  d|.|   g}5t|-| dt ttd#|  td#t tdd$ ttdd$    ttd#|  ttdd$ td#t tdd$    t|.| ttdd ttd#|  td#t tdd$ ttdd$    ttd#|  ttdd$ td#t tdd$    t|/| ttttd#|    tttd#|    g}6t
|5|6ks¢J t|5|6dg d¢fksŸJ tdt|-| |  d%|.|  d%|/|   tdt|.| |  d|-|  d|/|   tdt|/| |  d|-|  d|.|   g}7t|-| tttd#|  td#t tdd& ttd'd&    ttd#|  ttdd& td#t tdd&    t|.| tttd#|  td#t tdd( ttdd&    ttd#|  ttdd& td#t tdd(    t|/| ttttd#|    tttd#|    g}8t
|7|8ksrJ t|7|8dg d¢fksJ tt|| | d)||  tt|| | d*||  d||   tt|| | d||  d+||   d||   f}9t|| d,t td)|   d- t|| d.t td)|   d- dt td|   d+  t|| ttd)|   ttd|    ttd|    g}:t
|9|:ksªJ t|9|:dg d¢fksÆJ tt|| | d||  d/||   tt|| | d0||  d+||   tt|| | d/||  d+||   g};t|| ttd+d ttd1|  td1t td/d2 ttd3d2    ttd1|  ttd)d2 td1t td/d2    t|| ttd/d ttd1|  td1t td+d2 ttd4d2    ttd1|  ttd4d2 td1t td5d2    t|| ttttd1|    tttd1|    g}<t
|;|<ksJ t|;|<dg d¢fksJ tdt|| |  d6||  d6||   tdt|| |  d7||  d8||   tdt|| |  d%||  d%||   g}=t|| ttdtd |  tdt tdd t   tdtd |  ttdt tdd    t|| ttdtd |  tdt tdd t   tdtd |  ttdt td0d    t|| tttdtd |    ttdtd |    g}>t
|=|>ksBJ t|=|>dg d¢fks^J tt|| | d||  ||  tt|| | d||  d||   ||  tt|| | d||  ||  g}?t|| t| d  td|   d | dt t  td|    tt dt  td|    t|| t| d  td|   d | dt t  td|    tdt  td|    t|| t| d  td|   | dt dt   td|    dt dt  td|    g}@t
|?|@ksúJ t|?|@dg d¢fksJ tt|| | d||  ||  d||   tt|| | d||  ||  d||   tt|| | d||  d||   g}At|| ttd|   t| ttdd td    t| td ttd0d    t|| t|  ttdd td   t| td ttd0d    t|| ttd|   tt|   tt|   g}Bt
|A|Bks|J t|A|Bdg d¢fksJ tt|| | d||  tt|| | d%||  f}Ct|| td t td9td |   d tdt tdtd |   d  t|| ttd9td |   ttdtd |    g}Dt
|C|DkspJ t|C|Ddd"d"gfksJ tt|| | d||  d||   tt|| | d%||  d:||   g}Et|| td(td;  t| td;d<  d  tdd= td(td;  t| d<td;  d  tdd=  t|| tt| td;d<  d  tt| d<td;  d   g}Ft
|E|FksJ t|E|Fdd"d"gfksºJ tt|| | || ||  tt|| | d||  d||   g}Gt|| td td+t d  ttd+|  d  t| tdd  ttd+|  d td+t d ttdd   t| tdd   t|| tttd+|  d  t| tdd  tttd+|  d  t| tdd   g}Ht
|G|Hk sþJ t|G|Hdd"d"gfk!sJ t d>dd}0||  | ¡|0||   ||  | ¡|0||   g}It|| tt|0|   tt|0|    t|| tt|0|   tt|0|    g}Jt
|I|Jk!sÀJ t|I|Jdd"d"gfk!sÜJ tt|-| | d"tt|.| | |-| d|.|   ||  |1|2  tt|| | |.| d||   ||  |1|2  tt|| | |/| ||  d||   |1|2  tt|/| | d"g}Kt|-| d0t dt  t|.| dt dt  ttd|  |1|2    t	t|  tdd  |1|2    tt|  dtd  |1|2    t|| t dt  tdt	 t|  tdd  |1|2    tdt t|  dtd  |1|2    t|| tttd|  |1|2    t	t|  tdd  |1|2    tt|  dtd  |1|2    t|/| tg}Lt
|K|Lk$sJ t|K|Ldg d¢fk$s6J tt|-| | d|-|  tt|.| | d|-|  d+|.|   g}Mt|-| d?t td|   d t|.| ttd|   ttd+|    g}Nt
|M|Nk$sØJ t|M|Ndd"d"gfk$sôJ tt|-| | d@t |-|  d|.|   tt|.| | dt |.|  g}Ot|-| dt t tdt |   d ttd@t |    t|.| ttdt |   g}Pt
|O|Pk%s®J t|O|Pdd"d"gfk%sÊJ tt|-| | d@t |-|  tt|.| | dt |.|  g}Qt|-| ttd@t |   t|.| ttdt |   g}Rt
|Q|Rk&sXJ t|Q|Rdd"d"gfk&stJ tt|-| | d"tt|.| | d"g}St|-| tt|.| tg}Tt
|S|Tk&sÊJ t|S|Tdd"d"gfk&sæJ tt|-| | |-| tt|.| | d"g}Ut|-| tt|  t|.| tg}Vt
|U|Vk'sHJ t|U|Vdd"d"gfk'sdJ tt|-| | |.| tt|.| | d"g}Wt|-| tt|   t|.| tg}Xt
|W|Xk'sÆJ t|W|Xdd"d"gfk'sâJ tt|-| | d"tt|.| | |-| g}Yt|-| tt|.| t|  t g}Zt
|Y|Zk(sDJ t|Y|Zdd"d"gfk(s`J tt|-| | d"tt|.| | |.| g}[t|-| tt|.| tt|  g}\t
|[|\k(sÂJ t|[|\dd"d"gfk(sÞJ tt|-| | |-| tt|.| | t|.|  g}]t|-| tt|  t|.| ttt|   g}^t
|]|^k)sTJ t|]|^dd"d"gfk)spJ tt|-| | t|-|  tt|.| | t |.|  g}_t|-| ttt|   t|.| ttt |   g}`t
|_|`k)sòJ t|_|`dd"d"gfk*sJ tt|-| | t|.|  tt|.| | d"g}at|-| tt tt |   t|.| tg}bt
|a|bk*s|J t|a|bdd"d"gfk*sJ tt|-| | t|.|  tt|.| | t|-|  g}ct|-| t tt |   ttt|    t|.| ttt |   ttt|    g}dt
|c|dk+s<J t|c|ddd"d"gfk+sXJ tt|-| | d |.| d t|-|  t|.| |  d"g}et|-| t t|  tt|   t|.| tt|  tt|   gt|-| t t|   tt|   t|.| tt|   tt|   gg}t
|e|k,s8J |D ]"}ft|e|fdd"d"gfk,s<J ,q<d S )ANr:   ÚZ0ÚZ1ÚZ2ÚZ3zk01 k10 k20 k21 k23 k30r;   T©r   r   r   r   zx y z u v wr8   zk2 k3za_b a_c©ÚrealrA   r?   r@   r>   rs   rV   r=   rk   iøÿÿÿr°   é%   é,   rr   r   r±   )r   r   r   r   r   r7   za, b, cr<   r   é   é   rU   ro   iðÿÿÿé'   ri   rj   rg   éØ   éÑ   éÌ   rW   rB   é³   éª   rl   r   rh   rt   rv   ru   rY   rq   i±  é+   é   rÄ   rm   rn   )r   r   r   r	   r   rÂ   rÃ   r   ÚC3ÚC4r   r   r   r   r   r   r   ÚC5r   )gr:   ræ   rç   rè   ré   Zk01Zk10Zk20Zk21Zk23Zk30r¿   r   r3   r   r   ÚuÚvr   Úk2Zk3Za_bZa_crÀ   r   rÇ   r¢   rÈ   r€   r¥   rŠ   rÊ   rË   Zeq7rÍ   rÎ   rÏ   Zeq9rÑ   rÒ   rÓ   rÔ   r   rÕ   r   r4   r5   r6   rÄ   rÅ   rÆ   rÖ   r   r×   r   rØ   r   Úeq16rÙ   rÚ   rÛ   rÜ   rÝ   rÞ   rß   rà   rá   Úeq21râ   rã   rä   Zeqs23Úsol23Úeq24Úsol24Zeqs25Úsol25Úeq26Úsol26Úeq27Úsol27Zeq28Zsol28Zeq29Zsol29Zeq30Zsol30Zeq31Zsol31Zeq32Zsol32Zeq33Zsol33Zeq34Zsol34Zeq35Zsol35Zeq36Zsol36Zeq37Zsol37r   ÚsolrE   rE   rH   Ú(test_sysode_linear_neq_order1_type1_slow­  sä   @2$&ýHÿÿÿÿÿÿþN
ÿÿÿ
ÿÿ84ù þ.0þ,,,þZÿ^@ýF>22ý,,6(ý\T(&ýVÿLD@üB,ÿPÿ""ü,,üF@ þ,"þ þ*,"þ þ.0þ"ÿXÿBþ&&&þB2ÿH2ÿ4ü***þ>2ÿ>2ÿ4ü@2ÿ 88þ&&&þ<ÿ
2ÿ<ÿ
2ÿ4ü***þLÿLÿ<ü"."þ\X\þ..&þTF0þ8R8ÿ&&ÿ<ÿÿÿ@þ&ÿDBÿ\þ4T222üZÿÿ
Pÿÿ
húDLL\@<(,$,$,$,$40:60,8Z>" ÿ$"ÿþr
  c            K      C   sŽ  t dtd\} }}}t d\}}}}}}	}
t d\}}tt| ||| ||| d tt|||| | || d g}t| |tt d|d   |td   t||t d|d   |td   g}t||ksâJ t||d	d
d
gfksüJ tt| ||| ||| d tt|||| ||| d g}t| |t ttd|   | d t||tttd|   | d g}t||ks€J t||d	d
d
gfksÀJ tt| ||| |d tt|||| |d g}t| |tt| d t||tt| t d|  d g}t||ksJJ t||d	d
d
gfksfJ tt| ||| |t| tt||||t| | | || g}t| |tt| |t|  t||t| t| tt|  |d t|  g}t||ksJ t||d	d
d
gfks8J tt| ||d| | | || tt|||| ||| g}t| |tdt	d  tt	d|  tdt	d  tt	d |   |t
dd  t
dd t||ttt	d|  ttt	d |   |t
dd  g}t||ks8J t||d	d
d
gfksTJ tt| ||d| | d||  tt|||d|| tt|||||t| g}t| |ttd|  t
dd ttd|   t||ttd|  t||tt| |t|  g}t||ks6J t||d	g d¢fksRJ tt| ||d| | | tt|||||g}t| |tt| d|  d t||tt| g}t||ksÔJ t||d	d
d
gfksðJ tt| ||| |d||  tt|||d| | || dt|  g}t| |tt| td|  tt| td|   t|td| d   t|td| d   t||tt| td|  tt| td|   g}t||ksúJ t||d	d
d
gfksJ tt| |||d| |  || ||  tt|||| |d||  || ||  tt||||	|| d||  ||  g}t| |t td| ||   tt| t	dd  ||    tt| dt	d  ||    tt
ddd| |	 dd t||t	d t t| t	dd  ||   t	dt t| dt	d  ||    tt
dd||	 dd t||ttd| ||   tt| t	dd  ||    tt| dt	d  ||    tt
dd|d|	  dd g}t||k	s6J t||d	g d¢fk	sRJ tt| ||d| | | || d tt|||d| | | || d g}t| |tt d|d   d|d   |td   t||td|d   d|d   |td   g} t|| k
s$J t|| d	d
d
gfk
s@J tt| ||d dt| ||  d dt|
 | | t||| d
g}!t| |t| t||t| |
 t|
  |d |
 t
dd  gt| |td|  t||t| |
 t|
  |d |
 t
dd  gg}"t|!|"ks2J |"D ]"}#t|!|#d	d
d
gfks6J q6t dd	d}$t d}%t d}&t d}'t d}(t d}td}td}
t d })tt|
||||tt||||$|
|d  g}*t|
|ttt	|$|  |$t
dd  ttt	|$ |  |$t
dd   d t||ttt	|$|  ttt	|$ |   g}+t|*|+ksxJ t|*|+d	d
d
gfksJ t|
||)tt	|$ |(  d tt	|$ |( d  |'tt	|$ |(  dt	|$   tt	|$|  |)tt	|$|(  d tt	|$|( d  |'tt	|$|(  dt	|$   tt	|$ |   d t|||)t	|$ tt	|$ |(  d t	|$tt	|$ |(  d  |'tt	|$ |(  d  tt	|$|  |)t	|$ tt	|$|(  d t	|$tt	|$|(  d  |'tt	|$|(  d  tt	|$ |   g},t|*|
|&|)||&|%id!|,ksbJ t|*|,d	d
d
gfks~J tt| ||| ||| d" tt|||d| | d||  d# g}-t| |t dt	d  t|dt	d   d tdt	d  t|t	dd   d  t
d$d t||tt|dt	d   tt|t	dd    t
dd g}.t|-|.ksJ t|-|.d	d
d
gfksŠJ tt| ||| ||| d% tt|||d| | || d# g}/t| |t	dt t| tt	d|  d t	dt t| tt	d|  d  d&tt	d| d  d  d&tt	d| d  d  t||tt| tt	d|  tt| tt	d|   d'tt	d| d  d  d'tt	d| d  d  g}0t|/|0ksJ t|/|0d	d
d
gfks,J tt| ||| |d||  | tt|||d| | d||  | g}1t| |t dt	d(  t|dt	d(  d  d tdt	d(  t|dt	d(  d  d  d|  | t||tt|dt	d(  d  tt|dt	d(  d   tt
ddd| | dd g}2t|1|2ks`J t|1|2d	d
d
gfks|J tt| |||tt||||g}3t| |t||  t||t||  g}4t|3|4ksâJ t|3|4d	d
d
gfksþJ tt| ||d
tt||||| | | g}5t| |tt||t| |t| |   g}6t|5|6kstJ t|5|6d	d
d
gfksJ tt| |||tt|||| || g}7t| |t||  t||t||d  d  |t|   g}8t|7|8ksJ t|7|8d	d
d
gfks.J tt| |||tt|||| |d||  | g}9t| |dt ||  t||tttd|   || d  tt
dd|d|  dd g}:t|9|:ksÜJ t|9|:d	d
d
gfksøJ tt| ||| || tt||||g};t| |tt| | t||t||  g}<t|;|<ksjJ t|;|<d	d
d
gfksJ tt| ||||| tt|||d
g}=t| |t|t|   t||tg}>t|=|>ksðJ t|=|>d	d
d
gfksJ tt| ||| |d||  | tt||||g}?t| |dt tt|  | d| |  d|  t||t||  g}@t|?|@ksŠJ t|?|@d	d
d
gfksÂJ tt| ||||| tt|||d|| | g}At| |tttd|  d  |d  |d| |  d  t||ttd|  |d  g}Bt|A|BkspJ t|A|Bd	d
d
gfksJ tt| ||| || tt|||d| | | g}Ct| |tt| d | t||tt| t d|  |d| |   g}Dt|C|Dks*J t|C|Dd	d
d
gfksFJ tt| ||| |d||  | tt|||d| | d||  | g}Et| |dt ttd|  d  d| d| |  d  tt
dd)|d|  dd t||tttd|   |d| |  d  tt
dd)|d|  dd g}Ft|E|FksTJ t|E|Fd	d
d
gfkspJ tt| ||d| | || | tt|||d| | d||  d|  g}Gt| |tdt  |dt |   t||dt |dt d|    g}Ht|G|Hks&J t|G|Hd	d
d
gfksBJ tt|||d|
| d tt|
||d|| d|
|  d g}It||td dt d  t|  td|  dt d td  t|  td|   d*td| d  d  d*td| d  d  t|
|tt|  td|  tt|  td|   dtd| d   dtd| d   g}Jt|I|JksJ t|I|Jd	d
d
gfks°J d S )+Núf g h kr8   zx t a b c d yzk1 k2r@   rg   rV   r;   Tr   r>   r<   rm   rn   r?   rr   rA   rs   F©Úevaluateru   r}   rv   r=   r»   ©ZpositiveÚx_0Út_0r:   r3   r   ÚT©Úicsrk   rw   é   éQ   é:   é¹   r   é1   rj   )r   r   r	   r   rÂ   rÃ   r   r   r   r   r   rù   r   r   r   r   )Kr4   r5   r6   r   r3   r:   rÄ   rÅ   rÆ   Údr   Zk1rþ   r¿   r   rÀ   r   rÇ   r¢   rÈ   r€   rÉ   rŠ   rÊ   rË   rÌ   rÍ   rÎ   rÏ   rÐ   rÑ   rÒ   rÓ   rÔ   r   Zs11r»   Úx0Út0r  r  r  rÕ   r   Zsol12br   r   r   r   r   r   rÿ   rÙ   Zeq17rÛ   Zeq18rÝ   Zeq19rß   Zeq20rá   r   râ   Zeq22rä   Zeq23r  r  r  Zeq25r  r  r  r  r  rE   rE   rH   Ú#test_sysode_linear_neq_order1_type2  s`   "$ÿ(&ÿ""ÿ&$ÿÿ$ÿ*ÿ 4ÿ&ÿVÿ@þ&þ2 þÿ ÿ".ÿ6ÿÿ8þ.2.þdÿ\ÿbÿü**ÿ40ÿ, ÿDHÿ ÿR2ÿ6 ÿÿ*ÿÿþýFÿÿ:ÿÿþû
&"*ÿXÿBþ"&ÿ.&ÿþþ>ÿÿý&*ÿ4*ÿÿÿ>ÿþÿÿÿ ÿÿ(ÿ&ÿDÿÿÿÿÿ&ÿ4ÿÿ< ÿÿ0ÿ&*ÿ:ÿ.ÿþ&.ÿ$$ÿL0(ÿþþ:ÿÿýr  c            !      C   s¬  t dtd\} }}}}}t d\}}}t ddd}	tt| |	|	|	||	 | |	 tt||	|	|	 | |	 ||	 g}
t| |	tt|	 t|	d d  tt|	 t|	d d   t||	tt|	 t|	d d  tt|	 t|	d d   g}t	|
|ksJ t
|
|dddgfks.J tt| |||d || || |  tt|||d|d  | | d	|d  | ||  g}t| |td
t d
 td
d	td
   d  t|d	 d	td
  d |d d   t|d	 d	td
  d |d d  td
t d
 td	td
 d
  tdd    t||t|d	 d	td
  d |d d  td
d	td
   d td
t tdd
   t|d	 d	td
  d |d d  td	td
 d
  d td
t tdd
    g}t	||ksJ t
||dddgfks*J t| | |¡|| | || t|| |¡| | |||  g}t| |td tt d  t|d d t|   t|d d t|  td tt tdd    t||tt d td  t|d d t|   t|d d t|  tt d ttdd    g}t	||ksVJ t
||dddgfksrJ t| | |¡|| ||| ||  t|| |¡|| ||| ||  t|| |¡|| ||| ||  g}t| |t d	 td	  dt d	  td	 td	  td	  td	|d  d   t||dt d	 td	  td	  td	 td	  td	  td	|d  d   t||t d	 dt d	  td	  td	 td	  td	  td	|d  d   g}t	||ksþJ t
||dg d¢fksJ t| | |¡|d | ||| ||  t|| |¡|d | ||| ||  t|| |¡|d | ||| ||  g}t| |t d	 td	  dt d	  td	 td	  td	  t|d	   t||dt d	 td	  td	  td	 td	  td	  t|d	   t||t d	 dt d	  td	  td	 td	  td	  t|d	   g}t	||ksJ t
||dg d¢fks¶J tt| |||| ||| || ||  tt||||| ||| || ||  tt||||| ||| || ||  tt||||| ||| || ||  g}t| |t d td  td  d	t d  td td  td  td  td|d    t||d	t d td  td  td  td td  td  td  td|d    t||t d d	t d  td  td  td td  td  td  td|d    t||t d td  d	t d  td  td td  td  td  td|d    g}t	||k
sJ t
||dg d¢fk
s*J t ddd}tt| |||| | || tt|||||| | | g}t| |tt|d d  t| tt|d d  t|  t||tt|d d  t| tt|d d  t|  g}t	||ksJ t
||dddgfks0J tt| ||d| | | d||  tt|||d| | d| ||  g}t| |tdtd td  ddtd|d  d d|   td td  td|d  d d|    t||td td  td|d  d d|   td td  td|d  d d|    g}t	||kshJ t
||dddgfksJ tt| ||d| | | |d ||  tt||||d  | | d| ||  g}t| |td tt d  tt|d	  d	 d|d  d   td tt d  tt |d	  d	 d|d  d    t||tdtt d td  ddtt |d	  d	 d|d  d   tt d td  tt|d	  d	 d|d  d    g}t	||ksúJ t
||dddgfksJ tt| |||d || d| | |  tt||||d  | | d|d  d|  ||  g}t| |tddtd   d tdt d  t|d	 tdd  d d|d  d   tddtd   d tdt d  t|d	 dtd  d d|d  d    t||tdt d tddtd   d  t|d	 dtd  d d|d  d   tdt d tddtd   d  t|d	 tdd  d d|d  d    g}t	||ksúJ t
||dddgfksJ tt| ||d| | | |d ||  tt|||d|d  | | d| d|d   ||  g}t| |t|| t|| t|d ttd| | ttd|d  d|  | ||d  |  t||t|| t||t|d ttd| | ttd|d  d|  | ||d  | ttd| |ttd|d  d|  | ||    g} t	|| ksšJ d S )Nzf g h k x0 y0r8   úx t aÚrTrë   r;   r   r>   rj   é"   rV   r=   rs   rA   r?   rê   r   r@   Fr  rk   éM   é   r<   )r   r   r	   r   rÂ   r   r   rÃ   r   r   r   r   r   r   r   rù   rú   r   r,   )!r4   r5   r6   r   r  Úy0r3   r:   rÄ   r  r¿   r   rÀ   r   rÇ   r¢   rÈ   r€   rÉ   rŠ   rÊ   rË   r   rÌ   rÍ   rÎ   rÏ   rÐ   rÑ   rÒ   rÓ   rÔ   r   rE   rE   rH   Ú#test_sysode_linear_neq_order1_type3  sÎ    "$ÿ@@ÿ*:ÿVTÿRÿTÿþ"$ÿffÿT*ÿRPRþ\.ÿJHJþ2222ý^\^^ý""ÿ@@ÿ**ÿj`ÿ.0ÿzÿ.<ÿ°°ÿ.>ÿj¢ÿr#  c                  C   sÔ  t dtd\} }}}t d\}}}t ddd}tt| ||| ||||  |d  tt|||| | | || | g}t| |tt| t|d d  tt| t|d d   t|t|d d  t	|d t|  t|d d  |t|  t|d d   |  t|t|d d  t	|d t|  t|d d  |t|  t|d d   |  t||tt| t|d d  tt| t|d d   t|t|d d  t	|d t|  t|d d  |t|  t|d d   |  t|t|d d  t	|d t|  t|d d  |t|  t|d d   |  g}	t
||	ksJ t||	dddgfks®J tt| ||| ||||  | tt|||| | | || t| g}
t| |tt| t|d d  tt| t|d d   t|t|d d  t	|t|  t|d d  t| t| t|d d   |  t|t|d d  t	|t|  t|d d  t| t| t|d d   |  t||tt| t|d d  tt| t|d d   t|t|d d  t	|t|  t|d d  t| t| t|d d   |  t|t|d d  t	|t|  t|d d  t| t| t|d d   |  g}t|
d	d	d
|gksJ t|
|dddgfks6J tt| |||| ||| ||  | tt||||| ||| ||  | tt||||| ||| ||  d g}t| |ttdd ttdd  ttdd  |d d  |tdd  td td  td  t|d tdd   tdtt ttd| d  t|d tdd  d  tdd t||ttdd ttdd  ttdd  |d d  |tdd  td td  td  t|d tdd   tdtt ttd| d  t|d tdd  d  tdd t||ttdd ttdd  ttdd  |d tdd  |tdd  td td  td  t|d tdd   tdtt ttd| d  t|d tdd  d  tdd g}t
||ks&J t||dg d¢fksBJ tt| |||| ||| ||  t| tt||||| ||| ||  t| tt||||| ||| ||  t| g}t| |ttdd ttdd  ttdd  td td  td  t|d tdd   t	t|t|d tdd  |t|d tdd   t||ttdd ttdd  ttdd  td td  td  t|d tdd   t	t|t|d tdd  |t|d tdd   t||ttdd ttdd  ttdd  td td  td  t|d tdd   t	t|t|d tdd  |t|d tdd   g}t
||k
sÀJ t||dg d¢fk
sÜJ tt| |||| ||| || || d  tt||||| ||| || || d  tt||||| ||| || || d  tt||||| ||| || || d  g}t| |ttdd ttdd  ttdd  ttdd  td td  td  td  td|d    tdd t||ttdd ttdd  ttdd  ttdd  td td  td  td  td|d    tdd t||ttdd ttdd  ttdd  ttdd  td td  td  td  td|d    tdd t||ttdd ttdd  ttdd  ttdd  td td  td  td  td|d    tdd g}t
||ks¶J t||dg d¢fksÒJ tt| |||d | ||| || || d  tt||||d | ||| || || d  tt||||d | ||| || || d  tt||||d | ||| || || d  g}t| |ttdd ttdd  ttdd  ttdd  td td  td  td  t|d tdd   tdd t||ttdd ttdd  ttdd  ttdd  td td  td  td  t|d tdd   tdd t||ttdd ttdd  ttdd  ttdd  td td  td  td  t|d tdd   tdd t||ttdd ttdd  ttdd  ttdd  td td  td  td  t|d tdd   tdd g}t
||ksÔJ t||dg d¢fksðJ tt| ||| ||| || t| t| tt|||| ||| || t| t| tt|||| ||| || t| t| g}t| |t d td  dt d  td td  td  t|dt| d    t|dt| d  t	td| td| t|  t| |  t||dt d td  td  td td  td  t|dt| d    t|dt| d  t	td| td| t|  t| |  t||t d dt d  td  td td  td  t|dt| d    t|dt| d  t	td| td| t|  t| |  g}td( t
|d	d	d
|ksJ W d    n1 s€0    Y  t||dg d¢fksÊJ tt| ||| ||| || || t| t| tt|||| ||| || || t| t| tt|||| ||| || || t| t| tt|||| ||| || || t| t| g}t| |t d td  td  dt d  td td  td  td  t|dt| d    t|dt| d  t	td| td| t|  t| |  t||dt d td  td  td  td td  td  td  t|dt| d    t|dt| d  t	td| td| t|  t| |  t||t d dt d  td  td  td td  td  td  t|dt| d    t|dt| d  t	td| td| t|  t| |  t||t d td  dt d  td  td td  td  td  t|dt| d    t|dt| d  t	td| td| t|  t| |  g}td" t
||ksJ W d    n1 sª0    Y  t||dg d¢fksÐJ d S )Nr  r8   r  r  Trë   r;   r   F©ZsimplifyÚdoitr<   r=   r>   rV   é   rs   rk   rA   rB   r?   rê   rr   )r   r   r	   r   rÂ   r   r   rÃ   r   r,   r   r   r   r&   r   r   rù   r   r   r   rú   r   )r4   r5   r6   r   r3   r:   rÄ   r  r¿   r   rÀ   r   rÇ   r¢   rÈ   r€   rÉ   rŠ   rÊ   rË   rÌ   rÍ   rÎ   rÏ   rE   rE   rH   Ú#test_sysode_linear_neq_order1_type4d  s   Vvÿÿ\ÿvÿÿ\ÿþVr ÿÿF
ÿÿÿr ÿÿF
ÿÿÿý...þJ,ÿ<þþJ,ÿ<þþBÿ,ÿ<þþú	222þ>ÿÿ.ÿÿ>ÿÿ.ÿÿ>ÿÿ.ÿÿú	6666ýDÿÿÿÿÿDÿÿÿÿÿDÿÿÿÿÿDÿÿÿÿÿú::::ýDÿÿÿÿÿDÿÿÿÿÿDÿÿÿÿÿDÿÿÿÿÿúRÿÿÿ6ÿ2ÿÿÿ,ÿÿ0ÿÿÿ,ÿÿ2ÿÿÿ,ÿÿú	
8Rÿÿÿÿÿ4ÿ>þBÿÿÿ,ÿÿ@ÿÿÿ,ÿÿBÿÿÿ,ÿÿBÿÿÿ,ÿÿ÷
2r'  c               	   C   sè  t dtd\} }t d\}}tt| ||d| | || | tt|||| |d||  | g}t| |t | t|d   t||t| t|d   g}t||ksºJ t||dddgfksÔJ tt| ||d| | || d | tt||||| | d||  | g}t| |t|d  |ttdd	   |t	| td
d  tdd t||t|d  |t	| d  |ttd
d	   tdd g}t||ksÈJ t||dddgfksäJ d S )Nr   r8   zx x_r;   r>   Tr   r<   r?   r=   rs   )
r   r   r	   r   rÂ   rÃ   r   r   r   r   )r4   r5   r3   Úx_r¿   r   rÀ   r   rE   rE   rH   Ú)test_sysode_linear_neq_order1_type5_type6Ý  s    P>**ÿF@ÿr)  c                   C   s~  t dtd\} }t d}tt| ||dfd| | || tt|||df| | g}t| |t | t|  t| t|  tt t|   tt t|  t||t| t|  t| t|  tt t|   tt t|  g}t	||ksJ t
||dddgfks*J t| | |d¡dt|| |d¡| |g}t| |tt|  t||t|d  d t|d  d  t t|  g}t	||ksžJ t
||dddgfksÔJ tt| ||dfd| | tt|||df| | d||  g}t| |d	t ttd |  d	t ttd|   t||tdt | ttd |  tdt | ttd|   ttdt  ttd |   ttdt  ttd|   g}t	||ksúJ t
||dddgfksJ tt| ||dfd| | || tt|||dfd|| g}	t| |t| ttd|  d	 t| ttd |  d	  td	 tdt d
  ttd |   ttd| tdt d
 ttdd	    t||tdt ttd|  d tdt ttd |  tdd  g}
t	|	|
ks^J t
|	|
dddgfkszJ t| | |d¡| |t|| |d¡| |g}t| |t t|  tt|  t||t t|  tt|  t t|  g}t	||ksJ t
||dddgfks4J tt| ||df| ||| tt|||df| | || g}t| |tt|d  d  t t|d  d  |tt   t||t t|d  tdd  t|  t|d  tdd  g}t	||ksJ t
||dddgfks.J tt| ||df| ||| d tt|||df| ||| d g}t| |t t|  tdt ttd|  d  tdt ttd |  tdd  tdd t||tt|  tdt ttd|  d  tdt ttd |  tdd  tdd g}t	||ksbJ t
||dddgfks~J tt| ||df| ||| d tt|||df| | || d g}t| |tt t|d  d  |d	 d  |d td tdd   |tt   t||t t|  t|d  tdd  |d	 tdd  |d td tdd   g}t	||ksJ t
||dddgfks¬J t dtd\}}t d\}}tt|||dfd|| d||  tt|||df||d||  g}t||tddtd   ttdd  t| ttdd   d ttdtd  ddtd   t| tdtd   d  tddtd   ttdd  t|ttdd   tdd  ttdtd  ddtd   t|tdtd   tdd  t||tdtd  ttdd  t| ttdd   tdd ttdtd  tdd  t| tdtd   tdd  tdtd  ttdd  t|ttdd   d  ttdtd  tdd  t|tdtd   d  g}t	||ksRJ t
||dddgfksnJ td|| t|||df dt|||  dtd|| dt|||  t|||df dg}t||tdtd  ttd| tdtd d  d  d tdtd  ttd| tdtd d  d  tdd  tdtd  ttd| tddtd   d  d  tdtd  ttd| tddtd   d  tdd  t||tdtd  ttd| tdtd d  d  tdd tdtd  ttd| tdtd d  d  tdd  tdtd  ttd| tddtd   d  d  tdtd  ttd| tddtd   d  d  g}t	||ksøJ t
||dddgfksJ tt| ||dft| ||| | ||d   |||d   tt|||df|||d  g}t| |ttdd  tdd |tddtdtdd    t|tddtdd    dtd  tdd  t|dtd   dtd   t|dtd   dtd   t||tdtd  tdd |tddtdtdd    t|tddtdd    dtd  tdd  g}t	||ksÀJ t
||dddgfksÜJ tt| ||df|| | tt|||df||| g}t| |tt| tt|  t||tt| tt|  g}t	||ksrJ t
||dddgfksJ tt|||df|d	t||| d
t|||   tt|||df|dt||| dt|||   g}t||tttd| tdtt t d tdtt t tdd    ttd| tdtt t tdd tdtt t td	d    t||tttd| tdtt t tdd tdtt t tdd    ttd| tdtt t tdd tdtt t tdd    g}t	||ks^J t
||dddgfkszJ d S )Nr   r8   r3   r;   Tr   r>   rV   r?   r   r=   r<   rU   úx, yút, lr@   r÷   rk   é=   é/   rg   ém   é_   é   r   rî   é7   r  én   )r   r   r	   r   rÃ   r   rù   rÂ   rú   r   r   r   r   r   r   r   r*   r+   r   r   r   ) r4   r5   r3   r¿   r   rÀ   r   rÇ   r¢   rÈ   r€   rÉ   rŠ   rÊ   rË   rÌ   rÍ   rÎ   rÏ   r   r:   r   rÒ   rÓ   rÔ   r   rÖ   r   r×   r   rØ   r   rE   rE   rH   Ú test_higher_order_to_first_orderð  sà   &ÿNLÿ0L(ÿ:J
ÿÿÿÿ&ÿj,ÿPþ4X"$ÿ<Bÿ&&ÿ\ÿZÿþ&(ÿRLÿ
ÿ*&ÿNÿÿÿÿÿÿÿÿ*ÿ
ÿþPÿ
ÿÿÿÿÿÿÿÿÿÿÿþü44ÿFÿ&ÿÿÿ&ÿÿÿ
ÿÿÿþLÿ&ÿÿÿ&ÿÿÿ
ÿÿÿþüD"ÿ*ÿ
ÿ
ÿÿÿÿ
ÿþý*ÿ
ÿ
ÿÿÿüÿ  ÿ::ÿPJÿJÿ
Jÿþr3  c               	   C   sP  t dtd\} }t d}| ||| dtt|   dt| ||  t| ||df | ||| dtt|   dt|||  t|||df g}t| |t ttd|  d  td t	d  t|  t
|  td t	d  t|  t|  dddt  tt|  t|d  d   dddt  tt|  t
|d  d   t||tttd|  d  td t	d  t|  t
|  td t	d  t|  t|  dddt  tt|  t|d  d   dddt  tt|  t
|d  d   g}t||ks0J t||dd	d	gfksLJ d S )
Nr   r8   r3   r;   rs   r<   r@   Tr   )r   r   r   r   r   r	   rÂ   rú   rÃ   rù   r   r   r   r   )r4   r5   r3   rÐ   rÑ   rE   rE   rH   Ú"test_higher_order_to_first_order_9k  s,    BBÿB ÿ*ÿ*þ@ ÿ*ÿ*þýr4  c                  C   s  t dtd\} }t d}t dtd\}}t d\}}td|| t|||df dt|||  d	td|| dt|||  t|||df d	g}t||tdtd
  td| tdtd
 d   tdd tdtd
  t	d| tdtd
 d   d  t
dtd
  td| tddtd
    tdd  tdtd
  t	d| tddtd
    d  t||tdtd
  t	d| tdtd
 d   tdd tdtd
  td| tdtd
 d   tdd  t
dtd
  t	d| tddtd
    d  tdtd
  td| tddtd
    d  g}t||ks~J t||dd	d	gfksJ d S )Nr   r8   r3   r*  r+  r?   r;   r   r   r@   rk   r=   T)r   r   r	   r   rÂ   r   r   r   rÃ   r   rù   rú   r   r   )r4   r5   r3   r   r:   r   rÕ   r   rE   rE   rH   Ú#test_higher_order_to_first_order_12|  sL    44ÿDÿÿÿ2ÿÿ2þDÿÿÿ
ÿ
ÿÿ2þýr5  c                  C   sŽ  t dtd\} }t d\}}}}}}}t| | |d¡d||| |¡ ||  t|| |d¡d|| | |¡ | |  g}	t| |t| |ttt|  tttd|   d tt|  tt td|   d  tt	 t|  tttd|   d  tt	 t|  tt td|   d  |t
|f  t||t| |ttt t|  tttd|   d tt t|  tt td|   d  t	t|  tttd|   d  t	t|  tt td|   d  |t
|f  g}
t|	ddd|
gks0J t|	|
dd	d	gfksLJ tt| |||d
| t||| d
||  tt||||d| t| || d| |  f}t| |t| |ttt|  tdtd td|  d  d tt|  tdtd td|  d  d  dtd t	 t|  tdtd td|  d  d  dtd t	 t|  tdtd td|  d  d  |t
|f  t||t| |ttdt t|  tdtd td|  d  d tdt t|  tdtd td|  d  d  t	t|  tdtd td|  d  d  t	t|  tdtd td|  d  d  |t
|f  g}t|ddd|gksÎJ t||dd	d	gfksêJ tt|||df|| tt| ||dfd	g}t||tt|  || |d  d  t| |t	t|  g}t||kspJ t||dd	d	gfksJ tt| ||df| ||d  tt|||df||| |d  g}t| |tt|d  td|  tt|d  td|   t||t	t|d  tt| |  |tdd  tt|d  tt||  |tdd   g}t||ksJ t||dd	d	gfks°J d S )Nr   r8   úx t x_ t_ d a mr;   rs   Fr$  Tr   rk   rg   r>   rB   r0  rV   r?   r=   r<   )r   r   r	   r   rÂ   r,   rÃ   r   r   rù   r   rú   r&   r   r   r   r   r   )r4   r5   r3   r:   r(  r~   r  rÄ   ÚmrÀ   r   rÇ   r¢   rÉ   rŠ   rÊ   rË   rE   rE   rH   Ú"test_second_order_to_first_order_2  s    ..ÿ^(ÿ*ÿ
ÿf$ÿ&ÿ
ÿþhNÿÿ<ÿ<þ
þ^ÿÿ0ÿ
ÿÿÿÿ
þýÿ(ÿ"&ÿ@6,ÿÿr8  c                  C   s  t dtd\} }t d\}}}}}}}t| | |d¡d| ||| |¡ ||  t|| |d¡d| || | |¡ | |  g}	t| |t| dt | td|   ttd|   dt | t	d|   tt
d|   t||dt | t	d|  tt
d|   dt | td|   ttd|   t|  g}
t|	|
ksdJ t|	|
dddgfksJ d S )Nr   r8   r6  r;   rs   Tr   )r   r   r	   r   rÂ   rÃ   r   r   rù   r   r   rú   r   r   )r4   r5   r3   r:   r(  r~   r  rÄ   r7  r¿   r   rE   rE   rH   Ú&test_second_order_to_first_order_slow1œ  s    22ÿ``ÿr9  c                  C   s  t dtd\} }t d\}}}}}}}tt| ||df|t| t||| ||t|  tt|||df|t| t| || | |t|  g}	t| |t| |ttt|  tt|t	t|  ttt|  d tt|  tt| t	t|  ttt| d  t
t|  tt|t	t|  ttt|  d  t
t|  tt| t	t|  ttt| d  |t|f  t||t| |tt t|  tt|t	t|  ttt|  d tt|  tt| t	t|  ttt| d  t
t|  tt|t	t|  ttt|  d  t
t|  tt| t	t|  ttt| d  |t|f  g}
t|	ddd|
gksæJ t|	|
dddgfksJ d S )	Nr   r8   r6  r;   Fr$  Tr   )r   r   r	   r   r   rÂ   r,   rÃ   r   r   rù   r   rú   r&   r   )r4   r5   r3   r:   r(  r~   r  rÄ   r7  rÈ   r€   rE   rE   rH   Ú&test_second_order_to_first_order_slow4Ì  sD    <<ÿL:ÿ$ÿÿÿ:ý
ýN:ÿ$ÿÿÿ:þ
þür:  c                  C   s  t dtd\} }}}t d}| |||||||g}tt| ||d| | tt|||| |tt|||||tt|||||d || g}t| |dt td|  t||ttd|  t t||tt| t||td td|  d tt|  g}t	||ks2J t
||dg d¢fksNJ tt| ||d| | ftt|||| |fftt|||||ftt|||||d || ffh}| |t ||| |h||t ||||hi| |tt| ||d| | ||tt|||| |||tt|||||||tt|||||d || if}	| |||||||h||| |f||||fhg}
d	d
 t|||D |ksÚJ t|||	ksîJ dd t|	d D |
ksJ tt| ||d| | tt|||| |tt|||||tt|||| |d || g}t| |ttd|  t||ttd|  d t t||tt| t||td td|  d tt|  g}t	||ksJ t
||dg d¢fks$J ttt| ||d| | ftt|||| |ftt|||| |d || fgttt|||||fgh}| |t ||| |h||t ||| |hi| |tt| ||d| | ||tt|||| |||tt|||||||tt|||| |d || if}| |||||||h||| |f||| |fhg}dd
 t|||D |ksžJ t|||ksÌJ dd t|d D |ksìJ tt| ||d| | tt||||| | tt|||||tt|||| |d || g}t| |ttd|  t||ttd|  d t |d d  t||tt| t||td td|  d tt|  g}t	||ksöJ t
||dg d¢fksJ ttt| ||d| | ftt||||| | ftt|||| |d || fgttt|||||fgh}| |t ||| |h||t ||| |hi| |tt| ||d| | ||tt||||| | ||tt|||||||tt|||| |d || if}| |||||||h||| |f||| |fhg}dd
 t|||D |ks®J t|||ksÂJ dd t|d D |ksâJ tt| |||| | d||  tt|||| ||||  | tt|||||tt|||| |d || g}t| |td tdt d  tdt|t|d  d td|   d |t|d  d td|   d  | d  ttd| t|d  d td|   d td| t|d  d td|   d  |d  t|d d td|   td tdt d  tdt|t|d  d td|   d |t|d  d td|   d  | d  ttd| t|d  d td|   d td| t|d  d td|   d  |d  t|d d td|    t||td t d td  t|t|d  d td|   d |t|d  d td|   d  |d  tdttd| t|d  d td|   d td| t|d  d td|   d  | d  t|d d td|   tdt d td  t|t|d  d td|   d |t|d  d td|   d  |d  tdttd| t|d  d td|   d td| t|d  d td|   d  | d  t|d d td|    t||tt| t||tt| t|ttt|d d td|   d tt|d d td|   d  tdt t|d d td|   d  tdt t|d d td|   d  tdt|d d td|   t|t|d  d td|   d |t|d  d td|   d  | d  t|d d td|  ttd| t|d  d td|   d td| t|d  d td|   d  | d  tdt|d d td|   t|t|d  d td|   d |t|d  d td|   d  | d  t|d d td|  ttd| t|d  d td|   d td| t|d  d td|   d  | d  d t|  |  g}ttt| |||| | d||  tt|||||| | | | gttt|||| |d || gfttt|||||gfh}| |||h||| |h||t ||| |hi| |tt| |||| | d||  ||tt|||||| | | | ||tt|||||||tt|||| |d || if}| |||||||h| |||f||| |f||| |fhg}dd
 t|||D |ksJ t|||ks0J dd t|d D |ksPJ t|ddd|gksjJ t
||dg d¢fksJ tt| |||| | d||  tt|||||| | | tt|||||tt|||| |d || g}t| |td tdt d  t|d d td|   td tdt d  t|d d td|    t||td t d td  t|d d td|   tdt d td  t|d d td|    t||tt| t||tt| t|ttt|d d td|   d tt|d d td|   d  tdt t|d d td|   d  tdt t|d d td|   d  d t|  |  g}ttt| |||| | d||  tt|||||| | | gttt|||| |d || gfttt|||||gfh}| |||h||| |h||t ||| |hi| |tt| |||| | d||  ||tt|||||| | | ||tt|||||||tt|||| |d || if}| |||||||h| |||f||| |f||| |fhg}dd
 t|||D |ks¬J t|||ksÀJ dd t|d D |ksàJ t|ddd|gksúJ t
||dg d¢fksJ d S )Nr  r8   r3   r;   r?   r>   Trê   c                 S   s   h | ]}t d d |D qS )c                 s   s   | ]}t |V  qd S rµ   ©Útuple©Ú.0ZsccrE   rE   rH   Ú	<genexpr>ö  rJ   ú4test_component_division.<locals>.<setcomp>.<genexpr>r;  ©r>  ZwccrE   rE   rH   Ú	<setcomp>ö  rJ   z*test_component_division.<locals>.<setcomp>c                 S   s   g | ]}t |qS rE   ©Úset©r>  ÚelementrE   rE   rH   Ú
<listcomp>ø  rJ   z+test_component_division.<locals>.<listcomp>r   r   rg   c                 S   s   h | ]}t d d |D qS )c                 s   s   | ]}t |V  qd S rµ   r;  r=  rE   rE   rH   r?    rJ   r@  ©Ú	frozensetrA  rE   rE   rH   rB    rJ   c                 S   s   g | ]}t |qS rE   rC  rE  rE   rE   rH   rG    rJ   c                 S   s   h | ]}t d d |D qS )c                 s   s   | ]}t |V  qd S rµ   r;  r=  rE   rE   rH   r?  (  rJ   r@  rH  rA  rE   rE   rH   rB  (  rJ   c                 S   s   g | ]}t |qS rE   rC  )r>  r   rE   rE   rH   rG  *  rJ   c                 S   s   h | ]}t d d |D qS )c                 s   s   | ]}t |V  qd S rµ   rH  r=  rE   rE   rH   r?  P  rJ   r@  r;  rA  rE   rE   rH   rB  P  rJ   c                 S   s   g | ]}t |qS rE   rC  rE  rE   rE   rH   rG  R  rJ   Fr$  c                 S   s   h | ]}t d d |D qS )c                 s   s   | ]}t |V  qd S rµ   rH  r=  rE   rE   rH   r?  j  rJ   r@  r;  rA  rE   rE   rH   rB  j  rJ   c                 S   s   g | ]}t |qS rE   rC  rE  rE   rE   rH   rG  l  rJ   )r   r   r	   r   rÂ   r   rÃ   rù   rú   r   r   rD  r'   r(   r)   rI  r   r,   r&   )r4   r5   r6   r   r3   rG   r¿   r   Zcomponents1Zeqsdict1Zgraph1rÀ   r   Zcomponents2Zeqsdict2Zgraph2rÇ   r¢   Zcomponents3Zeqsdict3Zgraph3rÈ   r€   Zcomponents4Zeqsdict4Zgraph4rÉ   rŠ   Zcomponents5Zeqsdict5Zgraph5rE   rE   rH   Útest_component_divisionÞ  s   "ý,ý6>ÿ2 (ýÿ<  "ý ,ý$þý2 (ýÿ<  "ý,,ý$þý2  (ýÿ<  &&"ýZ
ÿÿÿÿF
ÿÿÿÿÿþ.
ÿÿ$ÿÿÿ.,ÿÿÿÿþþvbÿÿÿþ<$ÿÿÿ4,ÿÿÿÿþþ`*ÿ*ÿ
ÿPÿÿÿ
ÿ\ÿÿþþX
ÿÿÿÿüH,ÿÿÿûúúúõ(&ÿ(þý6,,(ýÿJ  &""ýpr`*ÿ*ÿÿÿÿý("ÿ(þý6,((ýÿJ  rJ  c               
      sp  t d\} t dtd\}}}ttfdd ttfdd tddgdd	gd	d
gg tt fdd tg d¢g d¢gttfdd ||g}t| ¡| ¡ |t| ¡|g}t||}t	|d |d\\ }}|}	t
tdd tdd   ttdd tdd    ttd d tdd  ttd d tdd    t
ttdd tdd    tttd d tdd    g}
tt|	t| jd|
ksöJ ttfdd ttfdd ttfdd ttfdd ttfdd ttfdd tddgdd	gd	d
gg tdd gtt fdd tg d¢g d¢gtd gttfdd tt fdd tt fdd tddgddgg td d d d gt
t  t dd    dd   dd  d    d  t
t  dd    dd   dd  d    d  g}tt d!d"d#gd|ksòJ ||g}t| ¡| ¡ | t| ¡|g}t||}t	|d |d\\ }}|}	t
 t d td d   d tdt
 t d td d   d  tdt ttd  d d   d  tttd  d d   d  t d td d  tttd  d d   d$td  td ttd  d d   d$td    d  tdt d td d   tttd  d d   d$td  td ttd  d d   d$td    d  tdttd  d d   ttd  td td d   d  d  ttd  d d  ttd  td td d   d  d  t
t d td d   tttd  d d    t d td d  tttd  d d   d$td  td ttd  d d   d$td     ttd  d d  ttd  td td d   d   g}
tt|	|d%gd|
ks,J t
 t d td d   d tdt
 t d td d   d  tdt ttd  d d   d  tttd  d d   d  t
t d td d   tttd  d d    g}tt|	d!d&gd|k	s<J ttfd'd ttfd(d tdg dgg t \}tt fd)d tt fd*d tggggt\}ttfd+d ttfd,d ttfd-d tt fd.d ttfd/d ||g}t| ¡||  t| ¡ | | g}t||}t	|d |d\\ }}|}	t
d tt d  ttd  d   t
d tt d  tt d  d    t t
 d td  tt d  d   tt
 d td  ttd  d    g}
tt|	t| jd|
ksŠJ tt|	d0d&t| jd|
ksÌJ tdgdgg tt fd1d ttfd2d ttfd3d ttfd4d ttfd5d ttfd6d ttfd7d tdg dgg td gt \}tt fd8d tggggtdd gt\}ttfd9d tt fd:d ttfd;d tt fd<d ttfd=d || || || g}t||  | ¡| || ||  ||   |  t||  | ¡| || ||  ||   |  t||  | ¡| || ||  ||   d g}t||| }t	|d || d\\ }}|}	td>| d  d }td | d  d }td| |  d  |d   | d   tdd  | }d| | d  }td| |  d  |d   d|  d   tdd  | }t
| d  t
d   t| d   td   t| d   dt d   || d   |d   | |d   t
| d  dt
 d   t| d   td   t| d   td   || d   |d   | |d   t
| d  t
d   t| d   dt d   t| d   td   || d   d| d   | d| d   g}
tt|	| |d%t| jd|
ksJ tt|	| |d?d@t| jd|
ks@J tdgdgg tt fdAd t
 d  td   dt d   t
d  td   td   td | d  d   dt
 d  td   td   t
d  td   td   td | d  d   t
 d  dt d   td   t
d  td   td   td | d  d   g}tt|	| d?d"dBt| jd|kslJ d S )CNzt x arŸ   r8   c                      s
   t d S r   ©r$   rE   r¯   rE   rH   rI   w  rJ   z"test_linodesolve.<locals>.<lambda>c                      s
   t  S rµ   rK  rE   ©rÄ   r:   rE   rH   rI   x  rJ   r<   r;   r?   rV   c                      s
   t  S rµ   rK  rE   ©r   r:   rE   rH   rI   {  rJ   )r<   r;   r<   )r>   r<   r;   c                      s
   t  S rµ   rK  rE   ©ÚA2r:   rE   rH   rI   ~  rJ   r   r@   )Ú	variablesc                      s   t d t d gdS ©Nr<   ©rÅ   ©r$   r   rE   r¯   rE   rH   rI     rJ   c                      s    t  ttt gdS ©NrR  ©r$   r   r   r   rE   rL  rE   rH   rI     rJ   c                      s   t tdg  d dS ©Nrg   r;   rR  rS  rE   r¯   rE   rH   rI     rJ   c                      s$   t t d gtt dS ©Nr}   rR  ©r$   r   r   r   rE   rL  rE   rH   rI     rJ   c                      s   t d  d dS rV  rK  rE   r¯   rE   rH   rI     rJ   c                      s   t  tt dS rT  ©r$   r   r   rE   rL  rE   rH   rI     rJ   c                      s   t  dS rT  rK  rE   ©r   Úb1r:   rE   rH   rI     rJ   c                      s   t  dS rT  rK  rE   ©rO  Úb2r:   rE   rH   rI     rJ   c                      s   t  d dd d f dS ©Nr;   rR  rK  rE   rZ  rE   rH   rI     rJ   c                      s&   t  d dd d f d d dS )Nr;   r<   rR  rK  rE   rZ  rE   rH   rI     rJ   r=   ru   r}   rv   r>   rx   T)rÅ   Útyper%  rm   rR  )r_  c                      s   t  d  S )Nr}   rK  rE   r¯   rE   rH   rI   Á  rJ   c                      s   t   S rµ   rK  rE   rL  rE   rH   rI   Â  rJ   c                      s   t  d d d df dS ©Nr<   ©ÚBrK  rE   )r   ÚB1r:   rE   rH   rI   Æ  rJ   c                      s   t  ddS r`  rK  rE   rM  rE   rH   rI   Ç  rJ   c                      s   t  d dd d f dS ©Nr;   ra  rK  rE   ©rO  ÚB2r:   rE   rH   rI   Ë  rJ   c                      s   t  ddS rd  rK  rE   rN  rE   rH   rI   Ì  rJ   c                      s   t  ddS )NZtype31)rb  r_  rK  rE   re  rE   rH   rI   Í  rJ   c                      s   t  dS ©Nra  rK  rE   )r   rf  r:   rE   rH   rI   Ï  rJ   c                      s   t  dS rg  rK  rE   )rO  rc  r:   rE   rH   rI   Ð  rJ   rz   c                      s
   t  S rµ   rK  rE   rM  rE   rH   rI   Þ  rJ   c                      s   t  d  t d gdS )Nr}   r<   rR  rS  rE   r¯   rE   rH   rI   á  rJ   c                      s$   t   ttt gdS rT  rU  rE   rL  rE   rH   rI   â  rJ   c                      s   t td  g  d dS rV  rS  rE   r¯   rE   rH   rI   ä  rJ   c                      s,   t t dt  gtt dS rW  rX  rE   rL  rE   rH   rI   å  rJ   c                      s   t d    d dS rV  rK  rE   r¯   rE   rH   rI   ç  rJ   c                      s"   t  d  tt dS r^  rY  rE   rL  rE   rH   rI   è  rJ   c                      s   t  d d d df dS rQ  rK  rE   rZ  rE   rH   rI   í  rJ   c                      s   t  d dd d f dS r^  rK  rE   r\  rE   rH   rI   ò  rJ   c                      s   t  dS rT  rK  rE   )r   r]  r:   rE   rH   rI   ô  rJ   c                      s   t  dS rT  rK  rE   )rO  r[  r:   rE   rH   rI   õ  rJ   c                      s   t  dS ©N)rÅ   rb  rK  rE   )r   rf  r[  r:   rE   rH   rI   ÷  rJ   c                      s   t  dS rh  rK  rE   )rO  rc  r]  r:   rE   rH   rI   ø  rJ   rB   r|   )rÅ   r_  c                      s   t  dS rT  rK  rE   rZ  rE   rH   rI     rJ   )r_  r%  )r   r   r/   r   r   r   r	   r   r%   r   rÂ   r   r   r   rÃ   r   r$   r   Zfree_symbolsr,   r#   r   ÚNotImplementedErrorrù   )r3   r4   r5   r6   r]   r\   Zceqr   rÅ   rŒ   r	  r   Ú_Z_x1Z_x2Z_x3Z_x4Z_x5rE   )r   rO  rc  rf  rÄ   r[  r]  r:   rH   Útest_linodesolver  s   >TÿFþ"FBÿÿÿ
Bpÿÿ$ÿ2ÿ
ÿ4ÿÿÿÿÿJÿ
ÿÿÿþþ:ÿÿÿÿüTûLÿJÿ
ÿÿÿÿ2ÿþú
"pÿÿ$ÿBþ" JXZÿ"& ...þ04^^fý&ÿÿ
Hÿrk  c                  C   s.  t dtd\} }}td}|d t| }|d t| }tt| ||d| | | | |||  d| ||  tt|||d| | | || ||  d| ||  tt|||d| | | |||  d| | ||  f}td	 t| W d    n1 s 0    Y  d S )
Núx, y, zr8   r:   r>   r;   r?   r@   rB   T)	r   r   r   r   r   r	   r   r   r   )r3   r   r   r:   r4   r5   r   rE   rE   rH   Ú!test_linear_3eq_order1_type4_slow  s    >^ÿÿÿ
rm  c                  C   sÄ   t d\} }}}}}td}td}|| t||| || || ||   }|| t||| || t|||  || ||   }	||	g}
t|
ddd\}t|
|dddgfksÀJ d S )Nzi, r1, c1, r2, c2, tr   r   Fr$  Tr   )r   r   r   r&   r   )r­   Zr1Zc1Zr2Úc2r:   r   r   r   r    r\   r	  rE   rE   rH   Ú"test_linear_neq_order1_type2_slow1'  s    .<ro  c                  C   s:  t rtd tddd} tddd}tddd}tddd\}}}tddd\}}}td	tdd
\}	}
}| | | t|	||  || || ||
|    | | | t|
||  |||| ||	|    | | | t|||  || |
| ||	|    g}t|}t||dg d¢fks6J d S )NúToo slow for CI.r7  Trë   Úqr:   ze1:4zb1:4zv1:4)r9   rì   rA   )r.   r1   r   r   r   r   r   r   )r7  rq  r:   Úe1Úe2Ze3r[  r]  Zb3Zv1Zv2Zv3Úeqsr	  rE   rE   rH   Ú2test_linear_new_order1_type2_de_lorentz_slow_check8  s    868ýru  c                  C   sŒ   t d\} }}}}}}}td}td}	t|| |¡d|  || |	||  t|	| |¡| | |	| d| ||  ||  g}
t|
ddd\}t|
|dd	d	gfksžJ d S )
NzRC t C Vs L R1 V0 I0ÚVr   r=   r<   Fr$  Tr   )r   r   r	   r   r&   r   )ÚRCr:   ÚCZVsÚLZR1ZV0ZI0rv  r   Úsystemr	  rE   rE   rH   Ú'test_linear_neq_order1_type2_slow_checkP  s    fr{  c               	   C   sâ  t dtd\} }}td}|d t| }|d t| }tt| ||d| | | | |||  d| ||  tt|||d| | | || ||  d| ||  tt|||d| | | |||  d| | ||  f}t|}d	d
 |D }t|d  d|d  t|  d|d   dt|d   dt|  d }	td}
dt	 |	 |
 dt
 |	 |
  dt |	 |
  }dd|d  d|	 |
  dt|  d  }t|d d ||	 |
 d  t| }t|d d ||	 |
 d  t| }t|d d |d d  d| t|  d|  t| }dt	 |	 |
 dt
 |	 |
  dt |	 |
  }dd|d  d|	 |
  dt|  d  }dt	 d dt
 d  dt d  ||  }dt	 d dt
 d  dt d  ||  }t| ||| ||  |t	t
   t|||| ||  t|||dt	 d dt
 d  d t d  ||   |dt	 d d t
 d  dt d  ||    |t	t
   g}||ksÂJ t||d!g d"¢fksÞJ d S )#Nrl  r8   r:   r>   r;   r?   r@   rB   c                 S   s   g | ]}t |qS rE   ©r"   ©r>  r	  rE   rE   rH   rG  g  rJ   z1_linear_3eq_order1_type4_long.<locals>.<listcomp>rV   r   r°   rp   l   €XW` l   Rl+p r<   istql   +RH l   ,N£ ifd i iÅÅ  i ieb  i  i;:ÿÿi°1  iXþÿÿi  é§   iš  TrA   )r   r   r   r   r   r	   r   r   r   rÂ   rÃ   rù   r   r   r   )r3   r   r   r:   r4   r5   r   Ú
dsolve_solÚdsolve_sol1Zx_1Zx_2Zx_3Zx_4Zx_5Zx_6Zx_7Zx_8Zx_9Zx_10Zx_11r	  rE   rE   rH   Ú_linear_3eq_order1_type4_long[  sJ    >^ÿÿÿN0,((<0,,,F*ÿ
ÿÿÿ
þr  c                  C   s®   t dtd\} }t d}tt| |||| | |d ||  | tt|||d|d  | | |d|d   ||  d g}t|}t||dddgfksªJ d S )	Nr   r8   r3   r;   r>   r<   Tr   )r   r   r	   r   r   r   ©r4   r5   r3   rt  r	  rE   rE   rH   Ú!test_neq_order1_type4_slow_check1|  s    .>ÿr  c                  C   sö   t dtd\} }}td}tt| ||||| | | || d tt|||||| | | || d tt||||| | | || || g}td t|}W d    n1 sÎ0    Y  t||dg d¢fksòJ d S )Nr7   r8   r3   r<   r}   TrA   )r   r   r   r	   r   r   r   r   )r4   r5   r6   r3   rt  r	  rE   rE   rH   Ú!test_neq_order1_type4_slow_check2  s    ...ý
&r  c                  C   sŠ  t dtd\} }t d}tt| |||| | || t| tt||||d |||  | | g}t| |td tt d  tt|d t	|d  d t|   d |d t	|d  d t|   d  tt	|d  d t|   t| d  tt	|d  d t|   t| d  | d  tt |d  t	|d  d t|   d t|d  t	|d  d t|   d  t	|d  d t|  t| d  t	|d  d t|  t| d  |d  t	|d d t|   td tt d  tt|d t	|d  d t|   d |d t	|d  d t|   d  tt	|d  d t|   t| d  tt	|d  d t|   t| d  | d  tt |d  t	|d  d t|   d t|d  t	|d  d t|   d  t	|d  d t|  t| d  t	|d  d t|  t| d  |d  t	|d d t|    t||t t d td  t|d t	|d  d t|   d |d t	|d  d t|   d  tt	|d  d t|   t| d  tt	|d  d t|   t| d  |d  ttt |d  t	|d  d t|   d t|d  t	|d  d t|   d  t	|d  d t|  t| d  t	|d  d t|  t| d  | d  t	|d d t|   tt d td  t|d t	|d  d t|   d |d t	|d  d t|   d  tt	|d  d t|   t| d  tt	|d  d t|   t| d  |d  ttt |d  t	|d  d t|   d t|d  t	|d  d t|   d  t	|d  d t|  t| d  t	|d  d t|  t| d  | d  t	|d d t|    g}||fS )Nr   r8   r3   r;   )
r   r   r	   r   r   rÂ   r   rÃ   r,   r   r  rE   rE   rH   Ú_neq_order1_type4_slow3  s   **þB$ÿ(ÿ(þþþÿÿ(ÿÿÿÿÿ$þþþþÿü:$ÿ(ÿ(þþþÿÿ(ÿÿÿÿÿ$þþþþÿüûB$ÿ(ÿ(þþþFÿÿ$ÿÿÿÿÿþþýûL(ÿ(ÿþþÿ.(ÿ$ÿ$þþþýûûôr  c                  C   s&   t  \} }t| ddd|gks"J d S )NFr$  )r  r&   ©rt  r	  rE   rE   rH   Útest_neq_order1_type4_slow3º  s    
r  c                  C   s(   t  \} }t| |dddgfks$J d S )NTr   )r  r   r  rE   rE   rH   Ú!test_neq_order1_type4_slow_check3Â  s    
r  c                  C   s<   t rtd t \} }t| }dd |D }||ks8J d S )Nrp  c                 S   s   g | ]}t |qS rE   r|  r}  rE   rE   rH   rG  Ñ  rJ   zGtest_linear_3eq_order1_type4_long_dsolve_slow_xfail.<locals>.<listcomp>)r.   r1   r  r   ©r\   r	  r  r  rE   rE   rH   Ú3test_linear_3eq_order1_type4_long_dsolve_slow_xfailÈ  s    
r  c                  C   sd   t rtd t \} }td t| }W d    n1 s<0    Y  dd |D }||ks`J d S )Nrp  Tc                 S   s   g | ]}t |qS rE   r|  r}  rE   rE   rH   rG  â  rJ   zHtest_linear_3eq_order1_type4_long_dsolve_dotprodsimp.<locals>.<listcomp>)r.   r1   r  r   r   r  rE   rE   rH   Ú4test_linear_3eq_order1_type4_long_dsolve_dotprodsimpÖ  s    

&r  c                  C   s4   t rtd t \} }t| |dg d¢fks0J d S )Nrp  TrA   )r.   r1   r  r   )r\   r	  rE   rE   rH   Ú'test_linear_3eq_order1_type4_long_checkæ  s    
r  c                     s  t dtd\t dt ¡ t ¡ ggtt ttd   ttttd   gg} tdd| ksÀJ tt	dd  tt	fd	d tt	fd
d tt	fdd t ¡d d  t ¡d d d  f tt
 fdd tt
 fdd tt
 fdd tt
 fdd tt
 fdd tt
 fdd tt
 fdd d S )Nr   r8   r3   r;   T)rG   r:   r%  c                   S   s   t dS r   ©r&   rE   rE   rE   rH   rI   ø  rJ   z$test_dsolve_system.<locals>.<lambda>c                      s
   t  dS r   r  rE   ©rt  rE   rH   rI   ù  rJ   c                      s   t  dS r   r  rE   )rt  rG   rE   rH   rI   ú  rJ   c                      s   t  d d S r   r  rE   )rt  rG   r3   rE   rH   rI   û  rJ   rU   rV   rW   r>   c                      s   t  g g fkS rµ   r  rE   )r\   rE   rH   rI   þ  rJ   c                      s    t  gdg g fkS )N)rG   r  rE   ©r\   r4   r5   r3   rE   rH   rI    	  rJ   c                      s"   t  gdg g fkS )N)rG   r:   r  rE   r  rE   rH   rI   	  rJ   c                      s4   t  gddddidg g fkS )Nr   r<   )rG   r:   r  r  rE   r  rE   rH   rI   	  rJ   c                      s&   t  ddddidg g fkS )Nr   r<   )r:   r  r  rE   r  rE   rH   rI   	  rJ   c                      s$   t  ddddidg g fkS )Nr   r<   r  r  rE   )r\   r4   r5   rE   rH   rI   	  rJ   c                      s2   t  gddddidg g fkS )Nr   r<   )rG   r  r  rE   r  rE   rH   rI   	  rJ   )r   r   r	   r   rÂ   rÃ   r   r&   r/   r   ri  )r	  rE   )r\   rt  r4   rG   r5   r3   rH   Útest_dsolve_systemï  s$    @@Tr  c                  C   s  t dtd\} }t d\}}| | |¡| | | |¡| g}tt t| W d    n1 sf0    Y  | || |¡g}tt t| W d    n1 sš0    Y  | || |¡|| |¡ || |¡g}tt t| W d    n1 s0    Y  d S )Nr   r8   úx y)r   r   r   r/   r   r   )r4   r5   r3   r   rt  rE   rE   rH   Útest_dsolve	  s    $
&
&,
r  c                  C   sB  t dtd\} }t d}tt| |||t||d  t| || t||d  d t|||  tt||||t||d  d t| || t||d  d t|||  g}t|ddd\}|D ]h}|j|j  |d	 j|d	 j|d
 j|d
 ji¡}| dd | 	t
¡D ¡}t| |d
¡d	ksÔJ qÔd S )Nr  r8   r:   r;   r>   rk   Fr$  r   r<   c                 S   s   i | ]}||j d dqS )F)Údeep)r%  )r>  r  rE   rE   rH   Ú
<dictcomp>'	  rJ   z,test_higher_order1_slow1.<locals>.<dictcomp>)r   r   r	   r   r   r&   Úlhsry   ÚsubsZatomsr   r-   )r3   r   r:   r\   r	  ÚeÚresrE   rE   rH   Útest_higher_order1_slow1	  s    PTþ2r  c               
   C   sÎ  t dtd\} }}t d\}}tt| ||df|d| | ||  tt|||df|| | d||   g}t| |tt t|dt tdd    tt t	|dt tdd     tt
 t|dt tdd     tt t	|dt tdd     t||tt|dt tdd    tt	|dt tdd     t
t|dt tdd     tt	|dt tdd     g}t||ks®J t||dddgfksÊJ d S )	Nrl  r8   r+  r;   r<   r>   Tr   )r   r   r	   r   r   rÂ   r*   r
   rÃ   r+   rù   rú   r   r   )r3   r   r   r:   r   r¿   r   rE   rE   rH   Útest_second_order_type2_slow1+	  s$    *,ÿZÿ
ÿ
"ÿfÿþr  c                  C   sŠ   t rtd td\} }}| tt t¡ || tt tt  |tt t¡ ||  tt tt  |tt t¡ | | tt tt  g}t|s¢J d S )Nrp  za b c)	r.   r1   r   r4   r3   r   r5   r6   r   )rÄ   rÅ   rÆ   rt  rE   rE   rH   Útest_nonlinear_3eq_order1_type19	  s    (((ýr  c                  C   s²   t tt t¡dtt tt dtt tt  t tt t¡dtt tt dtt tt  t tt t¡dtt tt dtt tt  g} t|  d S ©Nr;   r>   r?   )r	   r4   r3   r   r6   r5   r   r  rE   rE   rH   Útest_nonlinear_3eq_order1_type4I	  s
    666ýr  c                  C   s   t rtd ttt t¡dttd  d ttt t¡ddtt  ttt t¡dtt dttd   g} t|  d S )Nrp  r;   r>   r?   )	r.   r1   r	   r4   r3   r   r5   r6   r   r  rE   rE   rH   Útest_nonlinear_3eq_order1_type3V	  s    "*ýr  c               	   C   s   t tt t¡ttdtt dtt   t tt t¡ttdtt dtt   t tt t¡ttdtt dtt   g} t|  d S r  )r	   r4   r3   r   r5   r6   r   r  rE   rE   rH   Útest_nonlinear_3eq_order1_type5f	  s
    ...ýr  c                  C   s  t dtd\} }}t ddd\}}}}td}t dtd\}}	tt| ||| ||| d tt|||d	| | d
||  d f}
t| |tt|tdd   tt|td d    t	dd t||td	td  t|tdd   ttd d	  t|td d    t	d
d g}t
|
|dddgfksPJ tt| ||| ||| d tt|||d| | || d f}t| |tttd	|  tttd	|   t| t	dd t||td	 t ttd	|  td	t ttd	|   t| t	dd g}t
||dddgfksRJ tt| ||d
| | | d	||  tt|||d	| | d
| ||  f}t| |ttd	|  ttd|   tt	d
d	|d	   t||ttd	|  ttd|   tt	d
d	|d	   g}t
||dddgfksFJ tt| ||d
| | | |d	 ||  tt||||d	  | | d
| ||  f}t| |tt|d d  tt|d d   tt	d
d	|d	   t||t t|d d  tt|d d   tt	d
d	|d	   g}t
||dddgfksVJ tt| ||d
| | | |d	 ||  tt||||d	  | | d
| d|d	   ||  f}t| |tttdd	 t	dd	 |d  d  tttd d	 t	dd	 |d  d   tt	d
d	|d	   t||ttdd	 t	dd	  ttdd	 t	dd	 |d  d  ttd d	 t	dd	  ttd d	 t	dd	 |d  d   tt	d
d	|d	   g}t
||dddgfksúJ tt| ||d
| | | |d	 ||  tt|||d|d	  | | d
| d|d	   ||  f}t| |t|| t|| t|d	 ttd
| | ttd|d	  d
|  | ||d	  |  t||t|	| t|	|t|d	 ttd
| | ttd|d	  d
|  | ||d	  | ttd
| |ttd|d	  d
|  | ||    g}t|}||ksJ d S )Nrl  r8   z
k, l, m, nT)ZIntegerr:   zx0, y0rk   r;   r@   rw   rV   r>   r  r   r  rs   r  r  r   r<   )r   r   r   r	   r   rÂ   r   r   rÃ   r   r   r   r   r,   r   )r3   r   r   r   r   r7  r»   r:   r  r"  r   r   r    r   r¡   r¢   r£   r€   r¥   rŠ   Úeq6rË   ÚsrE   rE   rH   Útest_linear_2eq_order1s	  sZ    PD^ÿLBTÿX>>ÿbFHÿn0*ÿÿFBÿÿþpjd6ÿÿr¢  c                  C   s2  t dtd\} }}td}tt| ||| |||d  tt|||||d f}t| |ttddt d|   tdd  t||ddt d|   tdd  t| |ttdddt d|   tdd   t||ddt d|   tdd t| |ttt	 ddt d|   tdd   t||t	 ddt d|   tdd  t| |ttt	ddt d|   tdd   t||t	ddt d|   tdd  g}t
||ksÚJ t||d	d
d
gfksöJ tt| ||td| | ||d  tt|||||d f}t| |ttdddt d|   tdd    d t||ddt d|   tdd  t| |ttdddt d|   tdd    d t||ddt d|   tdd t| |ttdt	 ddt d|   tdd    d t||t	 ddt d|   tdd  t| |ttdt	 ddt d|   tdd    d t||t	ddt d|   tdd  g}t
||ksàJ t||d	d
d
gfksüJ tt| ||||| | tt|||| |d f}tdd}	t| |d|	 dtttt|  d  tt |	   t||ttttttt|  d d   d g}
t
||
ksÆJ tt| ||| ||| t|d  tt|||||d t|d  f}t| |dtt ttt | td| d   t||dt| td| d   h}t
||ksJ t| ||t| || t| ||t|||  t|||t||| t|||d  f}t| |tt t|  t||td t|  h}t
||ks4J t||d	d
d
gfksPJ tt| ||| |d ||d  tt|||||d f}t| |dtdddt d|   tdd    t||ddt d|   tdd  t| |dtddt d|   tdd   t||ddt d|   tdd t| |dtt	ddt d|   tdd    t||t	 ddt d|   tdd  t| |dtt	ddt d|   tdd    t||t	ddt d|   tdd  g}t
||ksJ t||d	d
d
gfks.J d S )Nrl  r8   r:   r>   r@   r=   r?   r<   Tr   r;   rV   rs   )r   r   r   r	   r   rÂ   r   rÃ   r   r   r   r   r   r   r   r   )r3   r   r   r:   r   r   r    r   r¡   Úttr¢   r£   r€   r¥   rŠ   r   rË   rE   rE   rH   Útest_nonlinear_2eq_order1	  sf    @.(2&4,2*ø	H8(8&<,<*ø	<
:4þT\f4D2(.&2,2*ø	r€  c            	      C   sŠ  t dtd\} }}t d\}}dt| || d|| ||  dt||| ||| |  dt||| | |||  f}tdtdtd	|d  dt  t td	|d  dt  t   || |f ttd
| d
  tdtdtd|d  t dt  td|d  t dt    |||f ttd| d  tdtdtd|d  dt  t td|d  dt  t   |||f ttd| d  g}dd t	t
||D sÎJ dt| || d|| || t|  dt||| ||| | t|  dt||| | ||| t|  f}tdtdtd|d  t dt  td|d  t dt    || |f ttdt| d  tdtdtd	|d  dt  t td	|d  dt  t   |||f ttd
t| d
  tdtdtd|d  dt  t td|d  dt  t   |||f ttdt| d  g}dd t	t
||D s¢J d S )Nrl  r8   zt ur?   r;   r>   r@   r<   rr   ru   rY   r}   iöÿÿÿrV   c                 S   s   g | ]\}}|  |¡qS rE   ©Zdummy_eq©r>  r­   r®   rE   rE   rH   rG  ã	  rJ   z-test_nonlinear_3eq_order1.<locals>.<listcomp>c                 S   s   g | ]\}}|  |¡qS rE   r¥  rŠ  rE   rE   rH   rG  ë	  rJ   )r   r   r   r	   r,   r   rÂ   rÃ   rù   Úzipr   r   r   )	r3   r   r   r:   rü   r   r   r    r   rE   rE   rH   Útest_nonlinear_3eq_order1Ú	  sL    hNÿ@
ÿÿ"ÿ
ÿÿþPÿ@
ÿÿ"ÿ
ÿÿþrš  c               	   C   s  t d} td}td}t d}t d}tt|| | d||  tt|| | d||  f}t|| d| tdtd	 |   d| td
td	 |    t|| dtd	 | tdtd	 |   dtd	 | td
td	 |    g}t||dddgfksJ d S )Nr:   rÂ   rÃ   rù   rú   rk   rU   rV   r>   rY   Tr   )r   r   r	   r   r   r   r   )r:   rÂ   rÃ   rù   rú   r\   r	  rE   rE   rH   Útest_C1_function_9239ï	  s    8@Pÿr©  c                  C   sÌ   t ddd} ttttt|  tt ttttt| tt f}tttt |  t| t  t|  t	| t   tttt |  t	| t  t|  t| t   g}t
||dddgfksÈJ d S )NÚepsilonTr  r   )r   r	   r   r4   r3   r5   rÂ   r   rÃ   r   r   )Zepsr   r   rE   rE   rH   Útest_dsolve_linsystem_symbolû	  s    :22ÿr«  N)Zsympy.core.functionr   r   r   Zsympy.core.mulr   Zsympy.core.numbersr   r   r   Zsympy.core.relationalr	   Zsympy.core.singletonr
   Zsympy.core.symbolr   r   Z%sympy.functions.elementary.hyperbolicr   Z(sympy.functions.elementary.miscellaneousr   rž   r   Zsympy.core.containersr   Zsympy.functionsr   r   r   r   r   r   r   r   Zsympy.matricesr   r   Zsympy.solvers.oder   Zsympy.solvers.ode.oder   Zsympy.solvers.ode.subscheckr   r¹   r   r   r    r!   r"   r#   r$   r%   r&   r'   r(   r)   r*   r+   Zsympy.integrals.integralsr,   Zsympy.simplify.ratsimpr-   Zsympy.testing.pytestr.   r/   r0   r1   r2   ZC0rÂ   rÃ   rù   rú   rû   ZC6ZC7ZC8ZC9ZC10r3   r4   r5   r6   rT   r§   rœ   rÁ   rå   r
  r  r#  r'  r)  r3  r4  r5  r8  r9  r:  rJ  rk  rm  ro  ru  r{  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r¢  r€  rš  r©  r«  rE   rE   rE   rH   Ú<module>   sÈ   (8%  Fm '
  X  _
x{
.
  (




!


$





*=
