U
    ZÊ¹jÂX  ã                   @   s|  d dl mZ d dl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mZ d dlmZmZ d dlmZ d d	l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! d dl"m#Z# dd„ Z$dd„ Z%edd„ ƒZ&dd„ Z'dd„ Z(dd„ Z)dd„ Z*e!dd„ ƒZ+d d!„ Z,d"d#„ Z-d$d%„ Z.d&d'„ Z/d(d)„ Z0d*d+„ Z1d,d-„ Z2d.d/„ Z3d0d1„ Z4d2d3„ Z5d4d5„ Z6d6d7„ Z7d8S )9é    )ÚN)ÚFloatÚIÚRational)ÚSymbolÚsymbols)ÚAbs)Úsqrt)ÚcosÚsin)ÚeyeÚMatrix)ÚS)ÚraisesÚXFAIL)ÚNonSquareMatrixErrorÚMatrixError)ÚDFT©Úsimplify)ÚImmutableMatrix)Úslow)Úallclosec                     s:  t } t d¡}|jdd�tjdiks(t‚|jdd�dddgksBt‚| ¡ ddtdddgƒtdddgƒtdddgƒgfgks~t‚| ¡ ddtdddggƒtdddggƒtdddggƒgfgksÀt‚tdddgdddgdddggƒ}| ¡ dtj dtj dtj	dik�s
t‚| ¡ ddtdddgƒgfddtdddgƒgfddt| ddƒ| ddƒdgƒgfgk�sdt‚| ¡ ddtd	ddggƒgfddtdddggƒgfddtdddggƒgfgk�s¸t‚t
d
ƒ}t|dgddggƒ}| ¡ |dtjdik�sðt‚tddgddggƒ}| ¡ ddtddddgƒgfgk�s,t‚| ¡ ddtddggƒgfgk�sRt‚tdddgdddgdddggƒ}| ddƒ}dd| ddƒ  }| ddƒ}| ddƒd| d  }| ddƒd| d  }dd„ ‰ ||d  dtdd||d    ||d  |  d||d    dd||d    | dgƒgfddtdd	dgƒgf||d  dtdd||d    ||d  |  d||d    dd||d    | dgƒgfg‰‡ ‡fdd„ttˆƒƒD ƒ}| ¡ ‰‡ ‡fdd„ttˆƒƒD ƒ}t|ƒt|ƒk�sút‚t
ddd�}	tt|	ƒt|	 gt |	 t|	ƒggƒ}| ¡ ddtt |	 t|	ƒ gdggƒgfdt|	ƒ dtt|	 t|	ƒ gdggƒgfgk�sŒt‚| ¡ ddtt|	 t|	ƒ dggƒgfdt|	ƒ dtt |	 t|	ƒ dggƒgfgk�sèt‚tdddddddddddg	ƒ}|jdd�|_tdd „ |jd d d D ƒƒdk�s>t‚|jdd�|_td!d „ |jd d d D ƒƒdk�svt‚tt ddƒdgddggƒ}| ¡ t ddƒtd"ƒd  dttd"ƒ d t ddƒ gdggƒgft ddƒtd"ƒd  dtt d#dƒtd"ƒd  gdggƒgfgk�st‚tg ƒ ¡ i k�s.t‚tg ƒjdd�g k�sHt‚tg ƒ ¡ g k�s^t‚ttd$d%„ ƒ ttd&d%„ ƒ ttd'd%„ ƒ ttd(d%„ ƒ ttd)d%„ ƒ ttd*d%„ ƒ tddgddggƒ}
t|
jddd+�tƒ�sàt‚t|
jddd+�tƒ�sút‚t|
jd,d%„ dd+�tƒ�st‚t|
jd-d%„ dd+�tƒ�s6t‚d S ).Né   F©ÚmultipleTé   r   é   éÿÿÿÿéþÿÿÿÚaé   é   é   é   é   é	   é   é!   é   c                 S   s   t t| |ƒƒS ©N)Ústrr   )ÚeÚn© r.   ú^/var/www/html/bwave/python/venv/lib/python3.8/site-packages/sympy/matrices/tests/test_eigen.pyÚNSH   s    ztest_eigen.<locals>.NSé   é   c                    sL   g | ]D}ˆ ˆ| d  dƒˆ ˆ| d dƒ‡ fdd„ˆ| d d  D ƒf‘qS )r   r   r   c                    s   g | ]}ˆ |d ƒ‘qS ©r   r.   ©Ú.0Új©r0   r.   r/   Ú
<listcomp>R   s     ú)test_eigen.<locals>.<listcomp>.<listcomp>r.   ©r5   Úi©r0   Úrr.   r/   r8   Q   s   ÿ ÿztest_eigen.<locals>.<listcomp>c                    sL   g | ]D}ˆ ˆ| d  dƒˆ ˆ| d dƒ‡ fdd„ˆ| d d  D ƒf‘qS )r   r   r   c                    s   g | ]}ˆ |d ƒ‘qS r3   r.   r4   r7   r.   r/   r8   U   s     r9   r.   r:   r<   r.   r/   r8   T   s   ÿ ÿÚeps©Úrealéüÿÿÿr   c                 s   s   | ]}|j V  qd S r*   ©Úqr:   r.   r.   r/   Ú	<genexpr>k   s     ztest_eigen.<locals>.<genexpr>c                 s   s   | ]}|j V  qd S r*   rB   r:   r.   r.   r/   rD   m   s     éI   éýÿÿÿc                   S   s   t ddgddgddggƒ ¡ S )Nr   r   r   r!   ©r   Ú	eigenvalsr.   r.   r.   r/   Ú<lambda>{   ó    ztest_eigen.<locals>.<lambda>c                   S   s   t ddgddgddggƒ ¡ S )Nr   r   r   r!   r"   r#   rG   r.   r.   r.   r/   rI   }   rJ   c                   S   s   t dddgdddggƒ ¡ S )Nr   r   r   r   r"   r#   rG   r.   r.   r.   r/   rI      rJ   c                   S   s   t dddgdddggƒ ¡ S )Nr   r   r!   r"   rG   r.   r.   r.   r/   rI   �   rJ   c                   S   s    t dddgdddggƒjdd�S )	Nr   r   r   r   r"   r#   F©Zerror_when_incompleterG   r.   r.   r.   r/   rI   ƒ   s   ÿc                   S   s    t dddgdddggƒjdd�S )Nr   r   r!   r"   FrK   rG   r.   r.   r.   r/   rI   †   s   ÿ)r   r   c                 S   s   | S r*   r.   ©Úxr.   r.   r/   rI   Œ   rJ   c                 S   s   | S r*   r.   rL   r.   r.   r/   rI   �   rJ   )r   r   r   rH   r   ÚOneÚAssertionErrorÚ
eigenvectsÚleft_eigenvectsÚZeror   ÚrangeÚlenÚsortedÚabsr   r   Z_eigenvectsÚmaxr	   r   r   Ú
isinstanceÚdictÚlist)ÚRÚMr    ÚbÚcÚdr,   Úr1Úr2r>   Úmr.   r<   r/   Ú
test_eigen   sì    
þÿþÿ
þ* ýÿ
ýÿ
ÿÿ(& 

@ ÿ@ ÿü
ÿ
ÿÿ$*þÿ
 *þÿ
**<:þÿÿÿÿÿÿrc   c                  C   s¨   t dddgdddgdddggƒ} tddƒtdƒd  tdƒd tddƒ tjg}| jddd	�}t|ƒ}d
d„ |D ƒ}t|ƒ}t||ƒD ]\}}t|| ƒdk s†t	‚q†d S )Nr   g333333ã?gÍÌÌÌÌÌì?r"   r!   i�  é   T)Zrationalr   c                 S   s   g | ]}|  ¡ ‘qS r.   )Zevalf©r5   rM   r.   r.   r/   r8   ™   s     z(test_float_eigenvals.<locals>.<listcomp>g•Ö&è.>)
r   r   r	   r   rR   rH   rU   ÚziprV   rO   )rb   ZevalsZn_evalsZs_evalsrM   Úyr.   r.   r/   Útest_float_eigenvals�   s     ýrh   c                     sX   t dddddtgƒ‰ tt‡ fdd„ƒ ˆ  d¡r4t‚tt‡ fdd„ƒ ˆ  d¡\} }d S )Nr   r   r   c                      s
   ˆ   d¡S ©NT)Úis_diagonalizabler.   ©rb   r.   r/   rI   £   rJ   z"test_eigen_vects.<locals>.<lambda>Tc                      s
   ˆ   d¡S ri   ©Údiagonalizer.   rk   r.   r/   rI   §   rJ   )r   r   r   ÚNotImplementedErrorrj   rO   r   rm   )ÚPÚDr.   rk   r/   Útest_eigen_vects    s
    rq   c                     s>  t dƒ\} }d‰ dd„ tˆ ƒD ƒ}‡ fdd„tˆ ƒD ƒ}tˆ ƒD ]}|| || |< q@t|ƒ}| ¡ }t|ƒˆ ksvt‚tˆ ƒD ]}|||  dks~t‚q~|jdd�}t|ƒt|ƒks¸t‚t| d	d	gd|d	gd
d| ggƒ}| ¡ }|| d
|diksôt‚|jdd�}t|ƒdk�st‚| | ¡d
k�s&t‚| |¡dk�s:t‚d S )Nzx yéÈ   c                 S   s   g | ]}t d | ƒ‘qS )zx%s)r   r:   r.   r.   r/   r8   ¯   s     z#test_issue_8240.<locals>.<listcomp>c                    s   g | ]}d d„ t ˆ ƒD ƒ‘qS )c                 S   s   g | ]}d ‘qS )r   r.   r:   r.   r.   r/   r8   °   s     z.test_issue_8240.<locals>.<listcomp>.<listcomp>)rS   r4   ©r-   r.   r/   r8   °   s     r   Tr   r   r   r   )r   rS   r   rH   rT   rO   ÚsetÚcount)rM   rg   Zdiagonal_variablesr\   r;   rH   r.   rs   r/   Útest_issue_8240ª   s(     rv   c               
   C   sœ   t dddgdddgdddggƒ} |  ¡ dtj dtj dtjdiksHt‚t dddddgdddddgdddddgdddddgdddddggƒ}| ¡ s˜t‚d S )Nr   r   r   r   rF   )r   rH   r   rN   rR   rO   )r\   rb   r.   r.   r/   Útest_eigenvalsÈ   s    
þ(û	rw   c                  C   sh   t dddgdddgdddggƒ} |  ¡ }|D ]6\}}}t|ƒdksFt‚| |d  ||d  ks,t‚q,d S ©Nr   r   )r   rP   rT   rO   ©r\   ZvecsÚvalZmultZvec_listr.   r.   r/   Útest_eigenvectsÚ   s    
þr{   c                  C   sh   t dddgdddgdddggƒ} |  ¡ }|D ]6\}}}t|ƒdksFt‚|d |  ||d  ks,t‚q,d S rx   )r   rQ   rT   rO   ry   r.   r.   r/   Útest_left_eigenvectsä   s    
þr|   c            %      C   sÆ  t dddgdddgdddggƒ} |  ¡ | ks0t‚| jdd�| ksDt‚|  ¡ | ksTt‚|  ¡ | | | fksjt‚| jdd�| | | fks„t‚|  ¡ | ks”t‚dd l}tdƒD �]²}g }d}d}t|| ƒD ]}| dd¡}||g }qÂt |||ƒ} t| ƒ}| ¡ \}	}
}|  ¡ \}}}|  ¡ }| ¡ }|	|
 | }|| | }|
 ¡  | ¡  | ¡  | ¡  | ¡  | ¡  | jdd�}| jdd�\}}}|jdd�}|jdd�\}}}|| | }|| | }| ¡  | ¡  | ¡  | ¡  | ¡  | ¡  | |k�søt‚||k�st‚||k�st‚|
|k�s"t‚| |k�s0t‚||k�s>t‚||k�sLt‚||ks¤t‚q¤tdƒD �]Ì}g }d}t|| ƒD ]4}| dd¡}| dd¡} || t	  }||g }�q|t |||ƒ} t| ƒ}| ¡ \}	}
}|  ¡ \}}}|  ¡ }| ¡ }|	|
 | }|| | }|
 ¡  | ¡  | ¡  | ¡  | ¡  | ¡  | jdd�}| jdd�\}}}|jdd�}|jdd�\}}}|| | }|| | }| ¡  | ¡  | ¡  | ¡  | ¡  | ¡  | |k�sÌt‚||k�sÚt‚||k�sèt‚|
|k�söt‚| |k�st‚||k�st‚||k�s t‚||k�sbt‚�qbt dd	tdd
ƒƒ} |  
dd„ ¡} |  ¡ d |  ¡ k�slt‚| jdd�d | jdd�k�sŽt‚|  ¡ \}!}"}#|!|" |# |  }$tt|$ƒƒdk �sÂt‚d S )Nr   r   F)Úupperr   i 6eÄi Êš;é   r%   é‘   c                 S   s   t | ƒS r*   )r   )r;   r.   r.   r/   rI   p  rJ   z$test_bidiagonalize.<locals>.<lambda>gê-�™—q=)r   ZbidiagonalizerO   Zbidiagonal_decompositionÚrandomrS   Úrandintr   r   r   Z	applyfuncrV   rW   )%r\   r€   Z	real_testZtest_valuesÚrowÚcolÚ_Úvaluer   ZN1ZN2ZN3ZM1ZM2ZM3ZM0ZN0ZN4ZM4ZLM0ZLM1ZLM2ZLM3ZLN0ZLN1ZLN2ZLN3ZLN4ZLM4Zcomplex_testÚsizer@   Úcompr    r]   r^   Údiffr.   r.   r/   Útest_bidiagonalizeî   sÊ    
þ"r‰   c                     s$  t ddddddgƒ‰ tt‡ fdd„ƒ ˆ  ¡ \} }| ¡ s>t‚|t t dgdtggƒks\t‚t ddddddgƒ‰ ˆ  ¡ \} }tdd	„ | ¡ D ƒƒs–t‚td
d	„ |  ¡ D ƒƒs°t‚ˆ jdd�\}}||ksÌt‚t ddddgddddgddddgddddggƒ‰ ˆ  ¡ \} }t	| | ˆ |  ƒ�s t‚d S )Nr   r   r   r   c                      s   ˆ j dd�S )NT©Z
reals_onlyrl   r.   rk   r.   r/   rI   z  rJ   z"test_diagonalize.<locals>.<lambda>g      à?c                 s   s   | ]}t |tƒV  qd S r*   ©rX   r   ©r5   r,   r.   r.   r/   rD   „  s     z#test_diagonalize.<locals>.<genexpr>c                 s   s   | ]}t |tƒV  qd S r*   r‹   rŒ   r.   r.   r/   rD   …  s     TrŠ   gü©ñÒMb`?)
r   r   r   rm   Zis_diagonalrO   r   ÚallÚvaluesr   )ro   rp   r„   ZD2r.   rk   r/   Útest_diagonalizex  s&    þ*ÿr�   c                  C   sŠ   t dƒ\} }}tdd| |||gƒ}| ¡ s.t‚| ¡ s:t‚tddddddgƒ ¡ rVt‚tddddddgƒ}| ¡ svt‚|jdd�r†t‚d S )Nza b cr   r   r   r   TrŠ   )r   r   Zis_symmetricrO   rj   )r    r]   r^   rb   r.   r.   r/   Útest_is_diagonalizable�  s    r�   c                     s   t ddddddddgƒ‰ tt‡ fdd„ƒ t d	d	ddd
d
d
ddd
d
d
dd
d
d
d
dgƒ‰ ˆ  ¡ \} }ˆ |ksnt‚t d	d	ddd
d
d
dd
d
d
d
ddd
d
d
dgƒ‰ ˆ  ¡ \} }ˆ |ks²t‚t dd	dd
gddd
dgd
d
dd	gd
d
ddggƒ}| ¡ \} }t| | |  ¡  ƒ|k�st‚t dddgƒ ¡ t dgƒt dgƒfk�s8t‚t dddgƒjdd�t dgƒk�s^t‚t dd
d
d
dgd
ddd
dgd
dd
dd
gd
d
dd
dgdd
d
dd
ggƒ‰ tt‡ fdd„ƒ t dddtdƒ  gddtdƒ  dggƒ‰ ˆ  ¡ \} }t	dd„ | D ƒƒ�st‚t	dd„ |D ƒƒ�st‚d S )Nr   r   rF   r   rd   é
   c                      s   ˆ   ¡ S r*   ©Újordan_formr.   rk   r.   r/   rI   ž  rJ   z"test_jordan_form.<locals>.<lambda>r!   r   rA   F)Zcalc_transformc                      s   ˆ   ¡ S r*   r’   r.   rk   r.   r/   rI   Á  rJ   g      æ?g      À?g      È?g      Ô?c                 s   s    | ]}t |tƒp|d kV  qdS ©r   Nr‹   re   r.   r.   r/   rD   È  s     z#test_jordan_form.<locals>.<genexpr>c                 s   s    | ]}t |tƒp|d kV  qdS r”   r‹   re   r.   r.   r/   rD   É  s     )
r   r   r   r“   rO   r   Úinvr   r	   r�   )ro   ÚJÚAr.   rk   r/   Útest_jordan_formœ  sh             ý         ý


ý,&Dþr˜   c                  C   sD  t ddd�} tddt gddggƒ}| ¡ ddgks8t‚tdƒ}| |d< d	|d
< | ¡ }t|ƒd	dt| ƒhksrt‚tt| ƒt	| ƒgt	| ƒ t| ƒggƒ}dd„ | ¡ D ƒ}|t
jt
jgks¾t‚tddgddgddgddggƒ}| ¡ ttdƒd ƒtdtdƒ ƒgk�st‚|j ¡ ttdƒd ƒtdtdƒ ƒddgk�s@t‚d S )NrM   Tr?   r   r   r   r   )r   r   r"   )r   r   c                 S   s   g | ]}|  ¡ ‘qS r.   )Ztrigsimp)r5   Úsvr.   r.   r/   r8   Û  s     z(test_singular_values.<locals>.<listcomp>r!   éÝ   r'   )r   r   r   Zsingular_valuesrO   r   rt   r   r   r
   r   rN   r	   ÚT)rM   r—   Úvalsr.   r.   r/   Útest_singular_valuesÌ  s.    &üÿ
"ÿr�   c                   C   s0   t dddgdddgdddggƒi kdks,t‚d S )Nr   r   F)r   rO   r.   r.   r.   r/   Útest___eq__é  s    þÿýýrž   c                  C   sV  t dddgdddgdddggƒ} | jdks.t‚| jdks<t‚| jdksJt‚| jdksXt‚| jdksft‚t ddgddggƒ} | jdksˆt‚| jdks–t‚| jdks¤t‚| jdks²t‚| jdksÀt‚t dddgdddgdddggƒ} | jdksît‚| jdksüt‚| jdk�st‚| jdk�st‚| jdk�s,t‚t ddgddggƒ} | jdk�sPt‚| jdk�s`t‚| jdk�spt‚| jdk�s€t‚| jdk�s�t‚t dd	gdd
ggƒ} | jdk�s´t‚| jdk�sÄt‚| jdk�sÔt‚| jdk�sät‚| jdk�sôt‚t ddt gt dggƒ} | jdk�st‚| jdk�s.t‚| jdk�s>t‚| jdk�sNt‚| jdk�s^t‚tddd�}tddd�}t |ddgd|dgdd|ggƒ} | jdk�s¦t‚| jdk�s¶t‚| jdk�sÆt‚| jdk�sÖt‚| jdk�sæt‚t |ddgd|dgdd|ggƒ} | jdk�st‚| jdk�s&t‚| jdk�s6t‚| jdk�sFt‚| jdk�sVt‚t |dgd|ggƒ} | jdk�szt‚| jdk�sŠt‚| jdk�sšt‚| jdk�sªt‚| jdk�sºt‚t ddddgddddgddddgddddggƒ} | jdk�sút‚| jdk�s
t‚| jdk�st‚t dddgdddgdddggƒ} | j�rFt‚| j�rRt‚d S )Nr   r   r   TFr"   r!   r   r   r%   r    )Zpositiver]   )Únegativega~‹¢2^—?g=6 0z?u?g1Ú¿"�³¿g‹ÅŸ¤Ò}¿gEáß¡?g�Ö¯m’Aµ?g)ä¨Ewc?gºí’Âœ‰ò?gÿ}uqÈÕ?g!p�°b³í?)	r   Zis_positive_definiterO   Úis_positive_semidefiniteZis_negative_definiteZis_negative_semidefiniteZis_indefiniter   r   )rb   r    r]   r.   r.   r/   Útest_definiteð  s¨         ÿ ÿ ÿ ÿù
ýr¡   c                  C   s’  ddl m}  tdddgdddgdddggƒ}| |ƒdks<t‚tdddgdddt gddt dggƒ}| |ƒdkstt‚tdddgdddgddd	ggƒ}| |ƒdks¤t‚tddgddggƒ}| |ƒdksÈt‚td
ddgdddgdddggƒ}| |ƒdksøt‚tdddgdddgdddggƒ}| |ƒdk�s*t‚td
ddgddd	gdd	dggƒ}| |ƒdk�s\t‚tddd	gdddgd	ddggƒ}| |ƒdk�sŽt‚d S )Nr   )Ú"_is_positive_semidefinite_choleskyTr"   éöÿÿÿr‘   Fr   r   r!   r   éúÿÿÿr&   é   rF   r   r   r#   )Zsympy.matrices.eigenr¢   r   rO   r   )r¢   rb   r.   r.   r/   Ú#test_positive_semidefinite_choleskyQ  s"     (     r¦   c               
   C   sT   t dddddgdddddgddddd	gd
ddddgdddddggƒ} |  ¡ sPt‚d S )Nr"   éûÿÿÿrF   r   iùÿÿÿr   r   r   r¤   r$   r‘   r   r&   r!   r£   iøÿÿÿrA   )r   rP   rO   )r—   r.   r.   r/   Útest_issue_20582h  s    û	r¨   c                  C   s  t dƒ} tddddgddddgddddgddddggƒ}t | |   ¡ }|tddddgddddtd	ƒ  ddtd	ƒ  gddddgddddggƒtdt |  dddgdd
t |  ddgdd| d
t d td	ƒt d   dgddd| td	ƒ t d dt d   ggƒfk�st‚d S )NÚtr   r   r   r   r!   rA   r   é   rF   )r   r   r   r“   r	   rO   )r©   ÚHr—   r.   r.   r/   Útest_issue_19210t  s     0
"

ü&(üür¬   c               
   C   s0  t dƒ ¡ jdd�} |  ¡ }|d ddtdtdƒ gdgdggƒgfksLt‚|d ddtdtdƒ gdgdggƒgfks|t‚|d t dtdgdgdggƒgfks¦t‚t dƒ ¡ jdd�} |  ¡ }|d ddtdgdgdgdggƒgfksît‚|d ddtdgdgdgdggƒtdgdgdgdggƒgfk�s2t‚|d t dtdgdgdgdggƒgfk�sbt‚t d	ƒ ¡ jdd�} |  ¡ }|d ddtdtd	ƒ gdgdgdgdggƒgfk�s¸t‚|d ddttdƒd td	ƒd  gdgdgdgdggƒttdƒd td	ƒd  gdgdgdgdggƒgfk�s,t‚d S )
Nr   T)Úcomplexr   r   r   r   r!   r"   )	r   Zas_explicitÚexpandrP   r   r	   rO   r   r   )r—   rP   r.   r.   r/   Útest_issue_20275ƒ  s`     þ
 þ
 þ
 þ
 .þ þ $þ ..ÿþr¯   c                  C   s>   t ddd�} tdddgd| dgdd| ggƒ}|jd ks:t‚d S )Nr]   T)Znonzeror   )r   r   r    rO   )r]   rb   r.   r.   r/   Útest_issue_20752³  s     r°   c                  C   s¼   dgd dg  } }ddddddddddddg}|  ¡ }d|d< dd„ }g }tdƒD ]}| |||ƒ|||ƒ ¡ qTtdƒD ]}| |||ƒ|| |ƒ ¡ q|tt|ƒ ¡  ¡ ƒd	ks¸t‚d S )
Nr   é   r   r   r%   c                 S   s   | |d … | d |…  S r*   r.   )rM   r;   r.   r.   r/   Úrotate¿  s    z test_issue_25282.<locals>.rotater1   r2   )ÚcopyrS   ÚappendÚsumr   rH   rŽ   rO   )ÚddÚsdZdsÚssr²   Zmatr;   r.   r.   r/   Útest_issue_25282¹  s    r¹   N)8Zsympy.core.evalfr   Zsympy.core.numbersr   r   r   Zsympy.core.symbolr   r   Z$sympy.functions.elementary.complexesr   Z(sympy.functions.elementary.miscellaneousr	   Z(sympy.functions.elementary.trigonometricr
   r   Zsympy.matricesr   r   Zsympy.core.singletonr   Zsympy.testing.pytestr   r   Zsympy.matrices.exceptionsr   r   Z"sympy.matrices.expressions.fourierr   Zsympy.simplify.simplifyr   Zsympy.matrices.immutabler   r   Zsympy.testing.matricesr   rc   rh   rq   rv   rw   r{   r|   r‰   r�   r�   r˜   r�   rž   r¡   r¦   r¨   r¬   r¯   r°   r¹   r.   r.   r.   r/   Ú<module>   sJ   ~
	


 
0a0