a
    RG5dÎ  ã                   @   sP   d Z ddlmZmZ dd„ Zdd„ Zdd„ Zd	d
„ Zdd„ Zdd„ Z	dd„ Z
dS )zCImplementation of matrix FGLM Groebner basis conversion algorithm. é    )Úmonomial_mulÚmonomial_divc                    s  |j ‰|j}|jˆd}t| |ƒ}t|| |ƒ}|jg‰ˆjgˆjgt|ƒd   g}g ‰ dd„ t	|ƒD ƒ}|j
‡‡fdd„dd | ¡ }	tt|ƒˆƒ}
tˆƒ‰t||	d	  ||	d  ƒ}t|
|ƒ‰t‡‡fd
d„t	ˆt|ƒƒD ƒƒrF| tˆ|	d  |	d	 ƒˆj¡}| ‡‡fdd„t	ˆƒD ƒ¡}||  |¡}|r¸ˆ  |¡ nrtˆˆ|
ƒ}
ˆ tˆ|	d  |	d	 ƒ¡ | |¡ | ‡fdd„t	|ƒD ƒ¡ tt|ƒƒ}|j
‡‡fdd„dd ‡ ‡fdd„|D ƒ}|södd„ ˆ D ƒ‰ tˆ ‡fdd„ddS | ¡ }	q˜dS )aZ  
    Converts the reduced Groebner basis ``F`` of a zero-dimensional
    ideal w.r.t. ``O_from`` to a reduced Groebner basis
    w.r.t. ``O_to``.

    References
    ==========

    .. [1] J.C. Faugere, P. Gianni, D. Lazard, T. Mora (1994). Efficient
           Computation of Zero-dimensional Groebner Bases by Change of
           Ordering
    )Úorderé   c                 S   s   g | ]}|d f‘qS ©r   © ©Ú.0Úir   r   úQ/var/www/html/django/DPS/env/lib/python3.9/site-packages/sympy/polys/fglmtools.pyÚ
<listcomp>    ó    zmatrix_fglm.<locals>.<listcomp>c                    s   ˆ t ˆ| d  | d ƒƒS ©Nr   r   ©Ú_incr_k©Zk_l©ÚO_toÚSr   r   Ú<lambda>!   r   zmatrix_fglm.<locals>.<lambda>T©ÚkeyÚreverser   c                 3   s   | ]}ˆ | ˆj kV  qd S ©N©Úzeror   )Ú_lambdaÚdomainr   r   Ú	<genexpr>+   r   zmatrix_fglm.<locals>.<genexpr>c                    s   i | ]}ˆ | ˆ| “qS r   r   r   )r   r   r   r   Ú
<dictcomp>.   r   zmatrix_fglm.<locals>.<dictcomp>c                    s   g | ]}|ˆ f‘qS r   r   r   )Úsr   r   r   9   r   c                    s   ˆ t ˆ| d  | d ƒƒS r   r   r   r   r   r   r   ;   r   c                    s2   g | ]*\‰ ‰t ‡‡ ‡fd d„ˆD ƒƒrˆ ˆf‘qS )c                 3   s(   | ] }t tˆ ˆ ˆƒ|jƒd u V  qd S r   )r   r   ÚLM©r	   Úg)r   ÚkÚlr   r   r   =   r   z)matrix_fglm.<locals>.<listcomp>.<genexpr>)Úall©r	   )ÚGr   )r$   r%   r   r   =   r   c                 S   s   g | ]}|  ¡ ‘qS r   )Úmonicr"   r   r   r   r   @   r   c                    s
   ˆ | j ƒS r   ©r!   )r#   )r   r   r   r   A   r   N)r   ÚngensÚcloneÚ_basisÚ_representing_matricesÚ
zero_monomÚoner   ÚlenÚrangeÚsortÚpopÚ_identity_matrixÚ_matrix_mulr&   Úterm_newr   Ú	from_dictÚset_ringÚappendÚ_updateÚextendÚlistÚsetÚsorted)ÚFÚringr   r+   Zring_toZ	old_basisÚMÚVÚLÚtÚPÚvÚltÚrestr#   r   )r(   r   r   r   r   r    r   Úmatrix_fglm   s@    

$ 
rJ   c                 C   s6   t t| d |… ƒ| | d g t| |d d … ƒ ƒS )Nr   )Útupler=   )Úmr$   r   r   r   r   F   s    r   c                    s8   ‡ ‡fdd„t ˆƒD ƒ}t ˆƒD ]}ˆ j|| |< q |S )Nc                    s   g | ]}ˆ j gˆ ‘qS r   r   ©r	   Ú_©r   Únr   r   r   K   r   z$_identity_matrix.<locals>.<listcomp>)r2   r0   )rP   r   rB   r
   r   rO   r   r5   J   s    r5   c                    s   ‡ fdd„| D ƒS )Nc                    s,   g | ]$‰ t ‡ ‡fd d„ttˆƒƒD ƒƒ‘qS )c                    s   g | ]}ˆ | ˆ|  ‘qS r   r   r   )ÚrowrG   r   r   r   T   r   z*_matrix_mul.<locals>.<listcomp>.<listcomp>)Úsumr2   r1   r'   ©rG   )rQ   r   r   T   r   z_matrix_mul.<locals>.<listcomp>r   )rB   rG   r   rS   r   r6   S   s    r6   c                    s¦   t ‡fdd„t| tˆƒƒD ƒƒ‰ttˆƒƒD ]4‰ˆˆkr,‡ ‡‡‡fdd„ttˆ ˆ ƒƒD ƒˆ ˆ< q,‡ ‡‡fdd„ttˆ ˆ ƒƒD ƒˆ ˆ< ˆ |  ˆ ˆ  ˆ ˆ< ˆ | < ˆ S )zE
    Update ``P`` such that for the updated `P'` `P' v = e_{s}`.
    c                    s   g | ]}ˆ | d kr|‘qS r   r   ©r	   Új)r   r   r   r   [   r   z_update.<locals>.<listcomp>c                    s4   g | ],}ˆ ˆ | ˆ ˆ | ˆˆ  ˆˆ   ‘qS r   r   rT   ©rF   r   r$   Úrr   r   r   _   r   c                    s    g | ]}ˆ ˆ | ˆˆ  ‘qS r   r   rT   )rF   r   r$   r   r   r   a   r   )Úminr2   r1   )r    r   rF   r   rV   r   r;   W   s     *&r;   c                    sJ   ˆj ‰ˆjd ‰‡fdd„‰‡ ‡‡‡fdd„‰‡‡fdd„tˆd ƒD ƒS )zn
    Compute the matrices corresponding to the linear maps `m \mapsto
    x_i m` for all variables `x_i`.
    r   c                    s"   t dg|  dg dgˆ |    ƒS )Nr   r   )rK   )r
   )Úur   r   Úvaro   s    z#_representing_matrices.<locals>.varc                    st   ‡‡fdd„t tˆƒƒD ƒ}tˆƒD ]J\}}ˆ t| |ƒˆj¡ ˆ ¡}| ¡ D ]\}}ˆ |¡}||| |< qNq$|S )Nc                    s   g | ]}ˆj gtˆ ƒ ‘qS r   )r   r1   rM   )Úbasisr   r   r   r   s   r   zG_representing_matrices.<locals>.representing_matrix.<locals>.<listcomp>)	r2   r1   Ú	enumerater7   r   r0   ÚremÚtermsÚindex)rL   rB   r
   rG   rW   ÚmonomÚcoeffrU   )r(   r[   r   rA   r   r   Úrepresenting_matrixr   s    
z3_representing_matrices.<locals>.representing_matrixc                    s   g | ]}ˆ ˆ|ƒƒ‘qS r   r   r   )rb   rZ   r   r   r   ~   r   z*_representing_matrices.<locals>.<listcomp>)r   r+   r2   )r[   r(   rA   r   )r(   r[   r   rb   rA   rY   rZ   r   r.   g   s
    
r.   c                    s‚   |j }dd„ | D ƒ‰ |jg}g }|rj| ¡ ‰| ˆ¡ ‡ ‡fdd„t|jƒD ƒ}| |¡ |j|dd q tt	|ƒƒ}t
||dS )z°
    Computes a list of monomials which are not divisible by the leading
    monomials wrt to ``O`` of ``G``. These monomials are a basis of
    `K[X_1, \ldots, X_n]/(G)`.
    c                 S   s   g | ]
}|j ‘qS r   r*   r"   r   r   r   r   ‰   r   z_basis.<locals>.<listcomp>c                    s.   g | ]&‰ t ‡ ‡fd d„ˆD ƒƒrtˆˆ ƒ‘qS )c                 3   s"   | ]}t tˆˆ ƒ|ƒd u V  qd S r   )r   r   )r	   Zlmg)r$   rE   r   r   r   ’   s   ÿz$_basis.<locals>.<listcomp>.<genexpr>)r&   r   r'   ©Zleading_monomialsrE   )r$   r   r   ‘   s   ÿÿTr   )r   )r   r/   r4   r:   r2   r+   r<   r3   r=   r>   r?   )r(   rA   r   Ú
candidatesr[   Znew_candidatesr   rc   r   r-      s    

r-   N)Ú__doc__Úsympy.polys.monomialsr   r   rJ   r   r5   r6   r;   r.   r-   r   r   r   r   Ú<module>   s   @	