o
    8VaO+                     @   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 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 Z0dd  Z1d!d" Z2d#d$ Z3d%d& Z4d'd( Z5dd)l6m7Z7 d*d+ Z8d,d- Z9d.d/ Z:d0d1 Z;d2d3 Z<d4d5 Z=d6d7 Z>d8d9 Z?d:d; Z@d<d= ZAd>d? ZBdS )BzIFunctions for generating interesting polynomials, e.g. for benchmarking.     )AddMulSymbolsympifyDummysymbols)Tuple)S)sqrt)	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dl m} | dkrtd|  |durt| ntd}| dkrFd}tdg}td| d D ]}t|}|t| q/|t	| ||d	S | dkrQ|d d }n-| dkrb|d
 d|d   d }n| dkr~|d d|d   d|d
   d|d   d }|rt
||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.
       )minimal_polynomialr   z5can't generate Swinnerton-Dyer polynomial of order %sNx      )polys   
      (      i`  i  i@  )Znumberfieldsr    
ValueErrorr   r   r
   ranger   appendr   r   )nr!   r$   r    paiex r2   :/usr/lib/python3/dist-packages/sympy/polys/specialpolys.pyswinnerton_dyer_poly   s,   

0r4   c                 C   s^   | dkr
t d|  ttt| tt}|durt||}nt|td}|r+|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   z0can't generate cyclotomic polynomial of order %sNr!   )
r*   r   r   intr   r   newr   r   as_expr)r-   r!   r$   polyr2   r2   r3   cyclotomic_polyA   s   r9   c                 O   sx   t |}| dk s| t|ks|std| |f | stj}ntdd t|t| D  }|dds4|S t	|g|R  S )zGenerates symmetric polynomial of order `n`.

    Returns a Poly object when ``polys=True``, otherwise
    (default) returns an expression.
    r   z6can't generate symmetric polynomial of order %s for %sc                 S   s   g | ]}t | qS r2   )r   ).0sr2   r2   r3   
<listcomp>k       z"symmetric_poly.<locals>.<listcomp>r$   F)
r   lenr*   r	   ZOner   r   r5   getr   )r-   Zgensargsr8   r2   r2   r3   symmetric_poly\   s   rA   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   r7   )r!   r-   infsuprB   r$   r8   r2   r2   r3   random_polys   s   rE   r!   yc           	         s   t dd}t trtd | f  n|r|t  j@ rd}t|tr-td|| f }n|r8|t| j@ r8d}|s@ttdg }t fddt	| D  }t	| D ]|    }t fddt	| D  }|
||  qTtd	d t||D  S )
zConstruct Lagrange interpolating polynomial for ``n``
    data points. If a sequence of values are given for ``X`` and ``Y``
    then the first ``n`` values will be used.
    free_symbolsNz%s:%sFz~
            Expecting symbol for x that does not appear in X or Y.
            Use `interpolate(list(zip(X, Y)), x)` instead.c                    s   g | ]} |  qS r2   r2   r:   r0   )Xr!   r2   r3   r<          z&interpolating_poly.<locals>.<listcomp>c                    s$   g | ]}|kr   |  qS r2   r2   )r:   j)rI   r0   r2   r3   r<      s   $ c                 S   s   g | ]\}}|| qS r2   r2   )r:   ZcoeffrF   r2   r2   r3   r<      rJ   )getattr
isinstancestrr   r   rG   r*   r   r   r+   r,   r   zip)	r-   r!   rI   YokZcoeffsZnumertZnumerZdenomr2   )rI   r0   r!   r3   interpolating_poly   s$   

rR   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      g | ]
}t d t| qS Zy_r   rN   rH   r2   r2   r3   r<          z$fateman_poly_F_1.<locals>.<listcomp>r   r   c                 S      g | ]}|qS r2   r2   r:   rF   r2   r2   r3   r<          Nr#   c                 S   s   g | ]}|d  qS )r#   r2   rY   r2   r2   r3   r<      r=   )r+   r   Zas_polyr   )	r-   rP   y_0Zy_1uvFGHr2   r2   r3   fateman_poly_F_1   s   "*
rb   c                 C   s&  |d|dg}t | D ]	}t|||g}q|d|d|dg}t d| D ]}t||t||g}q&| d }t|t|d|d| |}t|t|d|d| |}|d |dgg |d|d|d gg}t|t|d|d| |}	t||d|}
t||| |}t|	|
| |}t| |}|||fS )rS   r   r   r#   r"   )r+   r   r   r   r   r   r   )r-   Kr]   r0   r^   mUVfWrP   r_   r`   ra   r2   r2   r3   dmp_fateman_poly_F_1   s    ,

ri   c                 C   s   dd t | d D }|d }tdd |dd D  }t|| d d g|R  }t|| d d g|R  }t|| d d g|R  }|| || |fS )7Fateman's GCD benchmark: linearly dense quartic inputs c                 S   rT   rU   rV   rH   r2   r2   r3   r<      rW   z$fateman_poly_F_2.<locals>.<listcomp>r   r   c                 S   rX   r2   r2   rY   r2   r2   r3   r<      rZ   Nr#   r+   r   r   r-   rP   r\   r]   ra   r_   r`   r2   r2   r3   fateman_poly_F_2   s   rm   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 )rj   r   r   r#   )r+   r   r   r   r   r   r   )	r-   rc   r]   r0   rd   r^   rg   ghr2   r2   r3   dmp_fateman_poly_F_2   s   rp   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   rT   rU   rV   rH   r2   r2   r3   r<     rW   z$fateman_poly_F_3.<locals>.<listcomp>r   r   c                    s   g | ]}| d   qS )r   r2   rY   r-   r2   r3   r<   
  rJ   Nr#   rk   rl   r2   rr   r3   fateman_poly_F_3  s   $$$rs   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 )rq   r   r   r#   )	r   Zoner+   r   r   r   r   r   r   )r-   rc   r]   r0   r^   rg   rn   ro   r2   r2   r3   dmp_fateman_poly_F_3  s   ".((rt   )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   ru   r   Rr!   rF   zr2   r2   r3   _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 )Nrv   r"   r#         r&   ib     i,  i@     iX  ip  rx   ry   r2   r2   r3   _f_1/  s   b 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 )Nrv   rw   r"   r#   Z      i  rx   ry   r2   r2   r3   _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 )Nrv   rw   r#   r%   r"      rx   ry   r2   r2   r3   _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
 |d  |d
   d|d
  |d  |d   d|d
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  |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 )Nrv   	   r'   rw   r"   r      r#   r)   r%   r      rx   ry   r2   r2   r3   _f_4;  s
     H r   c                  C   s   t dt\} }}}|d  d|d  |  d|d  |  d| |d   d| | |  d| |d   |d  d|d  |  d| |d   |d  S )Nrv   r"   r#   r)   rx   ry   r2   r2   r3   _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)   rx   )rz   r!   rF   r{   tr2   r2   r3   _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 )
Nrv   r%   r)   r#   r"   rw   r   r'   r   rx   ry   r2   r2   r3   _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   rw   H      r)   r%   r   i$  rx   )rz   r!   rF   r2   r2   r3   _w_2K  s   j r   c                   C   s    t  t t t t t t fS N)r|   r   r   r   r   r   r   r2   r2   r2   r3   f_polysO  s    r   c                   C   s   t  t fS r   )r   r   r2   r2   r2   r3   w_polysR  s   r   )NF)r!   rF   )C__doc__Z
sympy.corer   r   r   r   r   r   Zsympy.core.containersr   Zsympy.core.singletonr	   Z(sympy.functions.elementary.miscellaneousr
   Zsympy.ntheoryr   Zsympy.polys.densearithr   r   r   r   Zsympy.polys.densebasicr   r   r   r   r   r   Zsympy.polys.domainsr   Zsympy.polys.factortoolsr   Zsympy.polys.polyclassesr   Zsympy.polys.polytoolsr   r   Zsympy.polys.polyutilsr   Zsympy.utilitiesr   r   r   r4   r9   rA   rE   rR   rb   ri   rm   rp   rs   rt   Zsympy.polys.ringsru   r|   r   r   r   r   r   r   r   r   r   r   r2   r2   r2   r3   <module>   sT      (
!