a
    RG5d=+                     @   s  d Z ddlmZmZmZmZ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 ddlmZmZmZmZ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" ddl#m$Z$ ddl%m&Z&m'Z'm(Z( e'd?ddZ)e'd@ddZ*e'dd Z+e'edfddZ,e'dAddZ-dd Z.dd Z/d d! Z0d"d# Z1d$d% Z2d&d' Z3dd(l4m5Z5 d)d* Z6d+d, Z7d-d. Z8d/d0 Z9d1d2 Z:d3d4 Z;d5d6 Z<d7d8 Z=d9d: Z>d;d< Z?d=d> Z@dS )BzIFunctions for generating interesting polynomials, e.g. for benchmarking.     )AddMulSymbolsympifyDummysymbols)Tuple)S)	nextprime)dmp_add_termdmp_negdmp_muldmp_sqr)dmp_zerodmp_one
dmp_grounddup_from_raw_dict	dmp_raise
dup_random)ZZ)dup_zz_cyclotomic_poly)DMP)PolyPurePoly)_analyze_gens)subsetspublic
filldedentNFc           	      C   s  | dkrt d|  |dur&t| ntd}| dkrddlm} ddlm} d	}|d	g}td	| d D ]}t|}|	|| qj|t
| ||d
S | dkr|d	 d	 }n\| d	kr|d d|d	   d }n:| dkr
|d d|d   d|d   d|d	   d }|rt||S |S )a  Generates n-th Swinnerton-Dyer polynomial in `x`.

    Parameters
    ----------
    n : int
        `n` decides the order of polynomial
    x : optional
    polys : bool, optional
        ``polys=True`` returns an expression, otherwise
        (default) returns an expression.
    r   z6Cannot generate Swinnerton-Dyer polynomial of order %sNx   )sqrt   )minimal_polynomial   )polys   
      (      i`  i  i@  )
ValueErrorr   r   (sympy.functions.elementary.miscellaneousr    numberfieldsr"   ranger
   appendr   r   )	nr   r$   r    r"   paiex r4   T/var/www/html/django/DPS/env/lib/python3.9/site-packages/sympy/polys/specialpolys.pyswinnerton_dyer_poly   s.    


0r6   c                 C   s^   | dkrt d|  ttt| tt}|dur>t||}nt|td}|rV|S |	 S )a  Generates cyclotomic polynomial of order `n` in `x`.

    Parameters
    ----------
    n : int
        `n` decides the order of polynomial
    x : optional
    polys : bool, optional
        ``polys=True`` returns an expression, otherwise
        (default) returns an expression.
    r   z1Cannot generate cyclotomic polynomial of order %sNr   )
r*   r   r   intr   r   newr   r   as_expr)r/   r   r$   polyr4   r4   r5   cyclotomic_polyA   s    r;   c                 O   s~   t |}| dk s | t|ks |s2td| |f n(| s>tj}ntdd t|t| D  }|ddsj|S t	|g|R  S dS )zGenerates symmetric polynomial of order `n`.

    Returns a Poly object when ``polys=True``, otherwise
    (default) returns an expression.
    r   z7Cannot generate symmetric polynomial of order %s for %sc                 S   s   g | ]}t | qS r4   )r   ).0sr4   r4   r5   
<listcomp>k       z"symmetric_poly.<locals>.<listcomp>r$   FN)
r   lenr*   r	   Oner   r   r7   getr   )r/   gensargsr:   r4   r4   r5   symmetric_poly\   s    rE   c                 C   s(   t t||||| |d}|r |S | S )a\  Generates a polynomial of degree ``n`` with coefficients in
    ``[inf, sup]``.

    Parameters
    ----------
    x
        `x` is the independent term of polynomial
    n : int
        `n` decides the order of polynomial
    inf
        Lower limit of range in which coefficients lie
    sup
        Upper limit of range in which coefficients lie
    domain : optional
         Decides what ring the coefficients are supposed
         to belong. Default is set to Integers.
    polys : bool, optional
        ``polys=True`` returns an expression, otherwise
        (default) returns an expression.
    )domain)r   r   r9   )r   r/   infsuprF   r$   r:   r4   r4   r5   random_polys   s    rI   r   yc           	         s   t dd}t tr(td | f  n|r>|t  j@ r>d}t|trZtd|| f }n|rp|t| j@ rpd}|sttdg }t fddt	| D  }t	| D ]>|    }t fddt	| D  }|
||  qtd	d t||D  S )
zConstruct Lagrange interpolating polynomial for ``n``
    data points. If a sequence of values are given for ``X`` and ``Y``
    then the first ``n`` values will be used.
    free_symbolsNz%s:%sFz~
            Expecting symbol for x that does not appear in X or Y.
            Use `interpolate(list(zip(X, Y)), x)` instead.c                    s   g | ]} |  qS r4   r4   r<   r2   )Xr   r4   r5   r>      r?   z&interpolating_poly.<locals>.<listcomp>c                    s$   g | ]}|kr   |  qS r4   r4   )r<   j)rM   r2   r4   r5   r>      r?   c                 S   s   g | ]\}}|| qS r4   r4   )r<   coeffrJ   r4   r4   r5   r>      r?   )getattr
isinstancestrr   r   rK   r*   r   r   r-   r.   r   zip)	r/   r   rM   YokcoeffsZnumertnumerdenomr4   )rM   r2   r   r5   interpolating_poly   s$    

rY   c           	      C   s   dd t | d D }|d |d  }}|tdd |dd D   }|d tdd |dd D   }|d |d  j| }|d d	| |d  |d  d  j| }tdg|R  }|||fS )
%Fateman's GCD benchmark: trivial GCD c                 S   s   g | ]}t d t| qS Zy_r   rR   rL   r4   r4   r5   r>      r?   z$fateman_poly_F_1.<locals>.<listcomp>r!   r   c                 S   s   g | ]}|qS r4   r4   r<   rJ   r4   r4   r5   r>      r?   Nr#   c                 S   s   g | ]}|d  qS )r#   r4   r]   r4   r4   r5   r>      r?   )r-   r   as_polyr   )	r/   rT   y_0Zy_1uvFGHr4   r4   r5   fateman_poly_F_1   s    "*rf   c                 C   s&  |d|dg}t | D ]}t|||g}q|d|d|dg}t d| D ]}t||t||g}qL| d }t|t|d|d| |}t|t|d|d| |}|d |dgg |d|d|d gg}t|t|d|d| |}	t||d|}
t||| |}t|	|
| |}t| |}|||fS )rZ   r!   r   r#   r   )r-   r   r   r   r   r   r   )r/   Kra   r2   rb   mUVfWrT   rc   rd   re   r4   r4   r5   dmp_fateman_poly_F_1   s     ,
rm   c                 C   s   dd t | d D }|d }tdd |dd D  }t|| d d g|R  }t|| d d g|R  }t|| d d g|R  }|| || |fS )7Fateman's GCD benchmark: linearly dense quartic inputs c                 S   s   g | ]}t d t| qS r[   r\   rL   r4   r4   r5   r>      r?   z$fateman_poly_F_2.<locals>.<listcomp>r!   r   c                 S   s   g | ]}|qS r4   r4   r]   r4   r4   r5   r>      r?   Nr#   r-   r   r   r/   rT   r`   ra   re   rc   rd   r4   r4   r5   fateman_poly_F_2   s    rq   c           	      C   s   |d|dg}t | d D ]}t|||g}q| d }t|t|d|d d| |}tt||t|||g| |}tt|||g| |}t|t|d |d| |}tt|||g| |}t||| |t||| ||fS )rn   r!   r   r#   )r-   r   r   r   r   r   r   )	r/   rg   ra   r2   rh   rb   rk   ghr4   r4   r5   dmp_fateman_poly_F_2   s    rt   c                    s   dd t  d D }|d }t fdd|dd D  }t| d  | d d g|R  }t| d  | d d g|R  }t| d  | d d g|R  }|| || |fS )8Fateman's GCD benchmark: sparse inputs (deg f ~ vars f) c                 S   s   g | ]}t d t| qS r[   r\   rL   r4   r4   r5   r>     r?   z$fateman_poly_F_3.<locals>.<listcomp>r!   r   c                    s   g | ]}| d   qS )r!   r4   r]   r/   r4   r5   r>   
  r?   Nr#   ro   rp   r4   rv   r5   fateman_poly_F_3  s    $$$rw   c                 C   s&  t | d |ji|}td| d D ]$}t|gt||| d |d |}q"t|t|d| d d| |}ttt|| d |gt| d || d | || |}tt|gt| d || d | || |}t|t| d |d| d |}tt|gt| d || d | || |}t||| |t||| ||fS )ru   r!   r   r#   )	r   oner-   r   r   r   r   r   r   )r/   rg   ra   r2   rb   rk   rr   rs   r4   r4   r5   dmp_fateman_poly_F_3  s    ".((ry   )ringc                  C   s   t dt\} }}}|d | |d  d|d  | |  d|d  |  d|d   d|  d|d  |d   d|d  |  d|d   ||d   d| |  | d S )Nx,y,zr#   r   r%      r)   r!   rz   r   Rr   rJ   zr4   r4   r5   _f_0+  s    r   c                  C   sr  t dt\} }}}|d | | |d |d  |d   |d |d   d|d  | |  d|d  |  |d |d   d|d  |  ||d  |  d| |d  |  d| |d   || |d   d| | |d   || |  d| |  d| |d   d| |  d	|  |d |d   d|d  |  d| |d   d
| |  d|  d|  d S )Nr{   r   r#         r&   ib     i,  i@     iX  ip  r}   r~   r4   r4   r5   _f_1/  s    r   c                  C   s  t dt\} }}}|d |d  |d |d  |  |d | |d   |d |d   |d |d   |d | |  d|d  |  d|d  |  |d |d  |  d|d  |d   |d |d   d|d  |d   ||  d|  d|  d S )Nr{   r|   r   r#   Z      i  r}   r~   r4   r4   r5   _f_23  s    r   c                  C   s  t dt\} }}}|d |d  |d |d   |d  |d |d  |  |d |  |d |d   |d |d  |d   |d | |d   |d | |  ||d  |d   ||d   || |d   || |d   || |d   |d |  ||d   S )Nr{   r|   r#   r%   r      r}   r~   r4   r4   r5   _f_37  s    r   c                  C   sT  t dt\} }}}|d  |d  | |d |d  |d   |d |d  |d   d|d  |d   |d	 |d  |d
   |d	 |d  |d   d|d	  |d  |  d|d	  |d  |d   |d	 |d
  |d   |d |d
  |d   d|d  |d
  |d   |d | |d   |d
 |d  |d
   d|d
  |d  |d   |d
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  |d   |d |d  |d	   d|d  |d  |d
   |d |d  |d	   d|d  |d
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  |d   |d |d  |d   d|d  |d  |d   d|d  | |d
   |d |d   d|d  |d   ||d  |d	   d| |d  |d
   d| |d  |d
   d| |d  |d   |d
 |d   d|d
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   S )Nr{   	   r'   r|   r   r      r#   r)   r%   r      r}   r~   r4   r4   r5   _f_4;  s    r   c                  C   s   t dt\} }}}|d  d|d  |  d|d  |  d| |d   d| | |  d| |d   |d  d|d  |  d| |d   |d  S )Nr{   r   r#   r)   r}   r~   r4   r4   r5   _f_5?  s    r   c                  C   s@  t dt\} }}}}d|d  | d|d  |d  |d   d|d  |d   d| |d   d| |d   d	| | |d   d
| | | |  d|d  |d  |d   d|d  |d   |d |d  |d   |d |d   d|d  |d   d|d  |d   d|d  |d   d| |d   S )Nzx,y,z,tiC  r%   -   r   r#   i  /      ^   r   r)   r}   )r   r   rJ   r   tr4   r4   r5   _f_6C  s    r   c                  C   s  t dt\} }}}d|d  |d  |d  d|d  |d  |d   d|d  |d  |d   d|d  | |d   |d |d  |d   d|d  |d  |  |d |d  |d   d|d  |d  |d   d|d  | |d   d|d  |d   d|d  |d   d|d  |d  |d   d|d  |d  |  d|d  |d  |d   d|d  |d  |d   d|d  |d  |d   d|d  | |d   d|d  | |d   d|d  | |d   d|d  |d  |  |d |d  |d   |d |d  |d   d|d  |d  |d   d	|d  |d  |  d|d  | |d   d|d  | |d   d|d  |d   d|d  |d   d|d  |d   d|d  |d  |d   d|d  |d  |  d|d  | |d   d|d  | |d   d|d  | |d   d| |d  |  d| |d  |d   d| | |  d| |d   d|d   d| |d   S )
Nr{   r%   r)   r#   r   r|   r   r'   r   r}   r~   r4   r4   r5   _w_1G  s    r   c                  C   sx  t dt\} }}d|d  |d  d|d  |d   d|d  |d   d	|d  |d   d
|d  |d   d|d  |d   d|d  |  d|d   |d |d   |d |d   d|d   |d |d   |d |d   d|d  |d   d|d  |d   |d |d   d|d  |d   |d |d   d|d  |d   d|d   d|d   S )Nzx,y   r'   r   0   r#   r   r|   H      r)   r%   r   i$  r}   )r   r   rJ   r4   r4   r5   _w_2K  s    r   c                   C   s    t  t t t t t t fS N)r   r   r   r   r   r   r   r4   r4   r4   r5   f_polysO  s    r   c                   C   s   t  t fS r   )r   r   r4   r4   r4   r5   w_polysR  s    r   )NF)NF)r   rJ   )A__doc__
sympy.corer   r   r   r   r   r   sympy.core.containersr   Zsympy.core.singletonr	   sympy.ntheoryr
   sympy.polys.densearithr   r   r   r   sympy.polys.densebasicr   r   r   r   r   r   sympy.polys.domainsr   sympy.polys.factortoolsr   sympy.polys.polyclassesr   sympy.polys.polytoolsr   r   sympy.polys.polyutilsr   sympy.utilitiesr   r   r   r6   r;   rE   rI   rY   rf   rm   rq   rt   rw   ry   sympy.polys.ringsrz   r   r   r   r   r   r   r   r   r   r   r   r4   r4   r4   r5   <module>   sP     )
!