a
    RG5d.e                     @   sL  d Z ddlmZmZmZmZmZmZmZm	Z	m
Z
mZmZmZmZmZmZmZmZmZmZmZ ddlmZmZmZmZmZmZmZmZmZm Z m!Z!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/m0Z1m2Z3 dd Z4d	d
 Z5dd Z6dd Z7dd Z8dd Z9dd Z:dd Z;dd Z<dd Z=dd Z>dd Z?dd  Z@d!d" ZAd#d$ ZBd%d& ZCd'd( ZDd)d* ZEd+d, ZFd-d. ZGd/d0 ZHd1d2 ZId3d4 ZJd5d6 ZKd7d8 ZLd9d: ZMd;d< ZNd=d> ZOd?d@ ZPdAdB ZQdCdD ZRdEdF ZSdGdH ZTdIdJ ZUdKdL ZVdMdN ZWdOdP ZXdQdR ZYdSdT ZZdUdV Z[dcdYdZZ\d[d\ Z]ddd]d^Z^d_d` Z_dadb Z`dWS )ezHAdvanced tools for dense recursive polynomials in ``K[x]`` or ``K[X]``.     )dup_add_termdmp_add_term
dup_lshiftdup_adddmp_adddup_subdmp_subdup_muldmp_muldup_sqrdup_divdup_remdmp_rem
dmp_expanddup_mul_grounddmp_mul_grounddup_quo_grounddmp_quo_grounddup_exquo_grounddmp_exquo_ground)	dup_strip	dmp_stripdup_convertdmp_convert
dup_degree
dmp_degreedmp_to_dictdmp_from_dictdup_LCdmp_LCdmp_ground_LCdup_TCdmp_TCdmp_zero
dmp_ground
dmp_zero_pdup_to_raw_dictdup_from_raw_dict	dmp_zeros)MultivariatePolynomialErrorDomainError)
variations)ceillogc              	   C   sv   |dks| s| S |j g| }tt| D ]H\}}|d }td|D ]}||| d 9 }qB|d|||| q(|S )a  
    Computes the indefinite integral of ``f`` in ``K[x]``.

    Examples
    ========

    >>> from sympy.polys import ring, QQ
    >>> R, x = ring("x", QQ)

    >>> R.dup_integrate(x**2 + 2*x, 1)
    1/3*x**3 + x**2
    >>> R.dup_integrate(x**2 + 2*x, 2)
    1/12*x**4 + 1/3*x**3

    r      )zero	enumeratereversedrangeinsertexquo)fmKgicnj r=   R/var/www/html/django/DPS/env/lib/python3.9/site-packages/sympy/polys/densetools.pydup_integrate'   s    r?   c           
   	   C   s   |st | ||S |dks"t| |r&| S t||d ||d  }}tt| D ]J\}}|d }td|D ]}	|||	 d 9 }qf|dt||||| qL|S )a&  
    Computes the indefinite integral of ``f`` in ``x_0`` in ``K[X]``.

    Examples
    ========

    >>> from sympy.polys import ring, QQ
    >>> R, x,y = ring("x,y", QQ)

    >>> R.dmp_integrate(x + 2*y, 1)
    1/2*x**2 + 2*x*y
    >>> R.dmp_integrate(x + 2*y, 2)
    1/6*x**3 + x**2*y

    r   r.   )r?   r%   r(   r0   r1   r2   r3   r   )
r5   r6   ur7   r8   vr9   r:   r;   r<   r=   r=   r>   dmp_integrateG   s    rB   c                    sH   krt | | S |d d  t fdd| D |S )z.Recursive helper for :func:`dmp_integrate_in`.r.   c              	      s   g | ]}t | qS r=   )_rec_integrate_in.0r:   r7   r9   r<   r6   wr=   r>   
<listcomp>q       z%_rec_integrate_in.<locals>.<listcomp>)rB   r   r8   r6   rA   r9   r<   r7   r=   rF   r>   rC   j   s    rC   c                 C   s2   |dk s||kr t d||f t| ||d||S )a+  
    Computes the indefinite integral of ``f`` in ``x_j`` in ``K[X]``.

    Examples
    ========

    >>> from sympy.polys import ring, QQ
    >>> R, x,y = ring("x,y", QQ)

    >>> R.dmp_integrate_in(x + 2*y, 1, 0)
    1/2*x**2 + 2*x*y
    >>> R.dmp_integrate_in(x + 2*y, 1, 1)
    x*y + y**2

    r   z(0 <= j <= u expected, got u = %d, j = %d)
IndexErrorrC   r5   r6   r<   r@   r7   r=   r=   r>   dmp_integrate_int   s    rM   c                 C   s   |dkr| S t | }||k r g S g }|dkr\| d|  D ]}||||  |d8 }q:nT| d|  D ]D}|}t|d || dD ]}||9 }q||||  |d8 }qjt|S )a#  
    ``m``-th order derivative of a polynomial in ``K[x]``.

    Examples
    ========

    >>> from sympy.polys import ring, ZZ
    >>> R, x = ring("x", ZZ)

    >>> R.dup_diff(x**3 + 2*x**2 + 3*x + 4, 1)
    3*x**2 + 4*x + 3
    >>> R.dup_diff(x**3 + 2*x**2 + 3*x + 4, 2)
    6*x + 4

    r   r.   N)r   appendr2   r   )r5   r6   r7   r;   derivcoeffkr9   r=   r=   r>   dup_diff   s"    

rS   c           
      C   s   |st | ||S |dkr| S t| |}||k r6t|S g |d  }}|dkr| d|  D ]$}|t||||| |d8 }qZnZ| d|  D ]J}|}t|d || dD ]}	||	9 }q|t||||| |d8 }qt||S )a3  
    ``m``-th order derivative in ``x_0`` of a polynomial in ``K[X]``.

    Examples
    ========

    >>> from sympy.polys import ring, ZZ
    >>> R, x,y = ring("x,y", ZZ)

    >>> f = x*y**2 + 2*x*y + 3*x + 2*y**2 + 3*y + 1

    >>> R.dmp_diff(f, 1)
    y**2 + 2*y + 3
    >>> R.dmp_diff(f, 2)
    0

    r   r.   NrN   )rS   r   r#   rO   r   r2   r   )
r5   r6   r@   r7   r;   rP   rA   rQ   rR   r9   r=   r=   r>   dmp_diff   s&    


rT   c                    sH   krt | | S |d d  t fdd| D |S )z)Recursive helper for :func:`dmp_diff_in`.r.   c              	      s   g | ]}t | qS r=   )_rec_diff_inrD   rF   r=   r>   rH      rI   z _rec_diff_in.<locals>.<listcomp>)rT   r   rJ   r=   rF   r>   rU      s    rU   c                 C   s2   |dk s||kr t d||f t| ||d||S )aS  
    ``m``-th order derivative in ``x_j`` of a polynomial in ``K[X]``.

    Examples
    ========

    >>> from sympy.polys import ring, ZZ
    >>> R, x,y = ring("x,y", ZZ)

    >>> f = x*y**2 + 2*x*y + 3*x + 2*y**2 + 3*y + 1

    >>> R.dmp_diff_in(f, 1, 0)
    y**2 + 2*y + 3
    >>> R.dmp_diff_in(f, 1, 1)
    2*x*y + 2*x + 4*y + 3

    r   0 <= j <= %s expected, got %s)rK   rU   rL   r=   r=   r>   dmp_diff_in   s    rW   c                 C   s8   |s| t| |S |j}| D ]}||9 }||7 }q|S )z
    Evaluate a polynomial at ``x = a`` in ``K[x]`` using Horner scheme.

    Examples
    ========

    >>> from sympy.polys import ring, ZZ
    >>> R, x = ring("x", ZZ)

    >>> R.dup_eval(x**2 + 2*x + 3, 2)
    11

    )convertr!   r/   )r5   ar7   resultr:   r=   r=   r>   dup_eval  s    
r[   c                 C   sd   |st | ||S |st| |S t| ||d  }}| dd D ] }t||||}t||||}q>|S )z
    Evaluate a polynomial at ``x_0 = a`` in ``K[X]`` using the Horner scheme.

    Examples
    ========

    >>> from sympy.polys import ring, ZZ
    >>> R, x,y = ring("x,y", ZZ)

    >>> R.dmp_eval(2*x*y + 3*x + y + 2, 2)
    5*y + 8

    r.   N)r[   r"   r   r   r   )r5   rY   r@   r7   rZ   rA   rQ   r=   r=   r>   dmp_eval   s    
r\   c                    sH   krt |  S d d  t fdd| D S )z)Recursive helper for :func:`dmp_eval_in`.r.   c              	      s   g | ]}t | qS r=   )_rec_eval_inrD   r7   rY   r9   r<   rA   r=   r>   rH   D  rI   z _rec_eval_in.<locals>.<listcomp>)r\   r   )r8   rY   rA   r9   r<   r7   r=   r^   r>   r]   =  s    r]   c                 C   s2   |dk s||kr t d||f t| ||d||S )a2  
    Evaluate a polynomial at ``x_j = a`` in ``K[X]`` using the Horner scheme.

    Examples
    ========

    >>> from sympy.polys import ring, ZZ
    >>> R, x,y = ring("x,y", ZZ)

    >>> f = 2*x*y + 3*x + y + 2

    >>> R.dmp_eval_in(f, 2, 0)
    5*y + 8
    >>> R.dmp_eval_in(f, 2, 1)
    7*x + 4

    r   rV   )rK   r]   )r5   rY   r<   r@   r7   r=   r=   r>   dmp_eval_inG  s    r_   c                    sf   krt |  d S  fdd| D }t  d k rH|S t |   d  S dS )z+Recursive helper for :func:`dmp_eval_tail`.rN   c                    s    g | ]}t |d   qS r.   )_rec_eval_tailrD   Ar7   r9   r@   r=   r>   rH   d  rI   z"_rec_eval_tail.<locals>.<listcomp>r.   N)r[   len)r8   r9   rc   r@   r7   hr=   rb   r>   ra   _  s    ra   c                 C   s\   |s| S t | |r"t|t| S t| d|||}|t|d krF|S t||t| S dS )a!  
    Evaluate a polynomial at ``x_j = a_j, ...`` in ``K[X]``.

    Examples
    ========

    >>> from sympy.polys import ring, ZZ
    >>> R, x,y = ring("x,y", ZZ)

    >>> f = 2*x*y + 3*x + y + 2

    >>> R.dmp_eval_tail(f, [2])
    7*x + 4
    >>> R.dmp_eval_tail(f, [2, 2])
    18

    r   r.   N)r%   r#   rd   ra   r   )r5   rc   r@   r7   er=   r=   r>   dmp_eval_taill  s    
rg   c                    sT   kr t t|   S d d  t fdd| D S )z+Recursive helper for :func:`dmp_diff_eval`.r.   c              
      s    g | ]}t | qS r=   )_rec_diff_evalrD   r7   rY   r9   r<   r6   rA   r=   r>   rH     rI   z"_rec_diff_eval.<locals>.<listcomp>)r\   rT   r   )r8   r6   rY   rA   r9   r<   r7   r=   ri   r>   rh     s    rh   c                 C   sJ   ||krt d|||f |s6tt| ||||||S t| |||d||S )a]  
    Differentiate and evaluate a polynomial in ``x_j`` at ``a`` in ``K[X]``.

    Examples
    ========

    >>> from sympy.polys import ring, ZZ
    >>> R, x,y = ring("x,y", ZZ)

    >>> f = x*y**2 + 2*x*y + 3*x + 2*y**2 + 3*y + 1

    >>> R.dmp_diff_eval_in(f, 1, 2, 0)
    y**2 + 2*y + 3
    >>> R.dmp_diff_eval_in(f, 1, 2, 1)
    6*x + 11

    z-%s <= j < %s expected, got %sr   )rK   r\   rT   rh   )r5   r6   rY   r<   r@   r7   r=   r=   r>   dmp_diff_eval_in  s
    rj   c                    s^   |j rDg }| D ]2}|  }| d kr6||   q|| qn fdd| D }t|S )z
    Reduce a ``K[x]`` polynomial modulo a constant ``p`` in ``K``.

    Examples
    ========

    >>> from sympy.polys import ring, ZZ
    >>> R, x = ring("x", ZZ)

    >>> R.dup_trunc(2*x**3 + 3*x**2 + 5*x + 7, ZZ(3))
    -x**3 - x + 1

       c                    s   g | ]}|  qS r=   r=   rD   pr=   r>   rH     rI   zdup_trunc.<locals>.<listcomp>)is_ZZrO   r   )r5   rm   r7   r8   r:   r=   rl   r>   	dup_trunc  s    ro   c                    s   t  fdd| D S )a9  
    Reduce a ``K[X]`` polynomial modulo a polynomial ``p`` in ``K[Y]``.

    Examples
    ========

    >>> from sympy.polys import ring, ZZ
    >>> R, x,y = ring("x,y", ZZ)

    >>> f = 3*x**2*y + 8*x**2 + 5*x*y + 6*x + 2*y + 3
    >>> g = (y - 1).drop(x)

    >>> R.dmp_trunc(f, g)
    11*x**2 + 11*x + 5

    c                    s   g | ]}t |d   qS r`   )r   rD   r7   rm   r@   r=   r>   rH     rI   zdmp_trunc.<locals>.<listcomp>)r   r5   rm   r@   r7   r=   rp   r>   	dmp_trunc  s    rr   c                    s4   |st |  S |d t fdd| D |S )a   
    Reduce a ``K[X]`` polynomial modulo a constant ``p`` in ``K``.

    Examples
    ========

    >>> from sympy.polys import ring, ZZ
    >>> R, x,y = ring("x,y", ZZ)

    >>> f = 3*x**2*y + 8*x**2 + 5*x*y + 6*x + 2*y + 3

    >>> R.dmp_ground_trunc(f, ZZ(3))
    -x**2 - x*y - y

    r.   c                    s   g | ]}t | qS r=   )dmp_ground_truncrD   r7   rm   rA   r=   r>   rH     rI   z$dmp_ground_trunc.<locals>.<listcomp>)ro   r   rq   r=   rt   r>   rs     s    rs   c                 C   s0   | s| S t | |}||r | S t| ||S dS )a7  
    Divide all coefficients by ``LC(f)`` in ``K[x]``.

    Examples
    ========

    >>> from sympy.polys import ring, ZZ, QQ

    >>> R, x = ring("x", ZZ)
    >>> R.dup_monic(3*x**2 + 6*x + 9)
    x**2 + 2*x + 3

    >>> R, x = ring("x", QQ)
    >>> R.dup_monic(3*x**2 + 4*x + 2)
    x**2 + 4/3*x + 2/3

    N)r   is_oner   )r5   r7   lcr=   r=   r>   	dup_monic  s    

rw   c                 C   sH   |st | |S t| |r| S t| ||}||r6| S t| |||S dS )a  
    Divide all coefficients by ``LC(f)`` in ``K[X]``.

    Examples
    ========

    >>> from sympy.polys import ring, ZZ, QQ

    >>> R, x,y = ring("x,y", ZZ)
    >>> f = 3*x**2*y + 6*x**2 + 3*x*y + 9*y + 3

    >>> R.dmp_ground_monic(f)
    x**2*y + 2*x**2 + x*y + 3*y + 1

    >>> R, x,y = ring("x,y", QQ)
    >>> f = 3*x**2*y + 8*x**2 + 5*x*y + 6*x + 2*y + 3

    >>> R.dmp_ground_monic(f)
    x**2*y + 8/3*x**2 + 5/3*x*y + 2*x + 2/3*y + 1

    N)rw   r%   r    ru   r   )r5   r@   r7   rv   r=   r=   r>   dmp_ground_monic  s    


rx   c                 C   sd   ddl m} | s|jS |j}||kr<| D ]}|||}q(n$| D ]}|||}||r@ q`q@|S )aA  
    Compute the GCD of coefficients of ``f`` in ``K[x]``.

    Examples
    ========

    >>> from sympy.polys import ring, ZZ, QQ

    >>> R, x = ring("x", ZZ)
    >>> f = 6*x**2 + 8*x + 12

    >>> R.dup_content(f)
    2

    >>> R, x = ring("x", QQ)
    >>> f = 6*x**2 + 8*x + 12

    >>> R.dup_content(f)
    2

    r   QQ)sympy.polys.domainsrz   r/   gcdru   )r5   r7   rz   contr:   r=   r=   r>   dup_content;  s    
r~   c                 C   s   ddl m} |st| |S t| |r*|jS |j|d  }}||krb| D ]}||t|||}qFn,| D ]&}||t|||}||rf qqf|S )aa  
    Compute the GCD of coefficients of ``f`` in ``K[X]``.

    Examples
    ========

    >>> from sympy.polys import ring, ZZ, QQ

    >>> R, x,y = ring("x,y", ZZ)
    >>> f = 2*x*y + 6*x + 4*y + 12

    >>> R.dmp_ground_content(f)
    2

    >>> R, x,y = ring("x,y", QQ)
    >>> f = 2*x*y + 6*x + 4*y + 12

    >>> R.dmp_ground_content(f)
    2

    r   ry   r.   )r{   rz   r~   r%   r/   r|   dmp_ground_contentru   )r5   r@   r7   rz   r}   rA   r:   r=   r=   r>   r   e  s    


r   c                 C   s>   | s|j | fS t| |}||r*|| fS |t| ||fS dS )at  
    Compute content and the primitive form of ``f`` in ``K[x]``.

    Examples
    ========

    >>> from sympy.polys import ring, ZZ, QQ

    >>> R, x = ring("x", ZZ)
    >>> f = 6*x**2 + 8*x + 12

    >>> R.dup_primitive(f)
    (2, 3*x**2 + 4*x + 6)

    >>> R, x = ring("x", QQ)
    >>> f = 6*x**2 + 8*x + 12

    >>> R.dup_primitive(f)
    (2, 3*x**2 + 4*x + 6)

    N)r/   r~   ru   r   )r5   r7   r}   r=   r=   r>   dup_primitive  s    


r   c                 C   sV   |st | |S t| |r"|j| fS t| ||}||r@|| fS |t| |||fS dS )a  
    Compute content and the primitive form of ``f`` in ``K[X]``.

    Examples
    ========

    >>> from sympy.polys import ring, ZZ, QQ

    >>> R, x,y = ring("x,y", ZZ)
    >>> f = 2*x*y + 6*x + 4*y + 12

    >>> R.dmp_ground_primitive(f)
    (2, x*y + 3*x + 2*y + 6)

    >>> R, x,y = ring("x,y", QQ)
    >>> f = 2*x*y + 6*x + 4*y + 12

    >>> R.dmp_ground_primitive(f)
    (2, x*y + 3*x + 2*y + 6)

    N)r   r%   r/   r   ru   r   )r5   r@   r7   r}   r=   r=   r>   dmp_ground_primitive  s    



r   c                 C   sL   t | |}t ||}|||}||sBt| ||} t|||}|| |fS )a  
    Extract common content from a pair of polynomials in ``K[x]``.

    Examples
    ========

    >>> from sympy.polys import ring, ZZ
    >>> R, x = ring("x", ZZ)

    >>> R.dup_extract(6*x**2 + 12*x + 18, 4*x**2 + 8*x + 12)
    (2, 3*x**2 + 6*x + 9, 2*x**2 + 4*x + 6)

    )r~   r|   ru   r   )r5   r8   r7   fcgcr|   r=   r=   r>   dup_extract  s    


r   c                 C   sT   t | ||}t |||}|||}||sJt| |||} t||||}|| |fS )a  
    Extract common content from a pair of polynomials in ``K[X]``.

    Examples
    ========

    >>> from sympy.polys import ring, ZZ
    >>> R, x,y = ring("x,y", ZZ)

    >>> R.dmp_ground_extract(6*x*y + 12*x + 18, 4*x*y + 8*x + 12)
    (2, 3*x*y + 6*x + 9, 2*x*y + 4*x + 6)

    )r   r|   ru   r   )r5   r8   r@   r7   r   r   r|   r=   r=   r>   dmp_ground_extract  s    
r   c           
      C   s  |j s|jstd| td}td}| s4||fS |j|jgg|jgg gg}t| d d}| dd D ](}t||d|}t|t|ddd|}qht	|}|
 D ]b\}}|d }	|	st||d|}q|	dkrt||d|}q|	dkrt||d|}qt||d|}q||fS )a4  
    Return bivariate polynomials ``f1`` and ``f2``, such that ``f = f1 + f2*I``.

    Examples
    ========

    >>> from sympy.polys import ring, ZZ
    >>> R, x,y = ring("x,y", ZZ)

    >>> R.dup_real_imag(x**3 + x**2 + x + 1)
    (x**3 + x**2 - 3*x*y**2 + x - y**2 + 1, 3*x**2*y + 2*x*y - y**3 + y)

    z;computing real and imaginary parts is not supported over %sr.   r   rk   N   )rn   is_QQr*   r#   oner/   r$   r
   r   r&   itemsr   r   )
r5   r7   f1f2r8   re   r:   HrR   r6   r=   r=   r>   dup_real_imag  s,    r   c                 C   s4   t | } tt| d ddD ]}| |  | |< q| S )z
    Evaluate efficiently the composition ``f(-x)`` in ``K[x]``.

    Examples
    ========

    >>> from sympy.polys import ring, ZZ
    >>> R, x = ring("x", ZZ)

    >>> R.dup_mirror(x**3 + 2*x**2 - 4*x + 2)
    -x**3 + 2*x**2 + 4*x + 2

    rk   rN   )listr2   rd   )r5   r7   r9   r=   r=   r>   
dup_mirror:  s    r   c                 C   sP   t | t| d |  } }}t|d ddD ]}|| |  ||  | |< }q,| S )z
    Evaluate efficiently composition ``f(a*x)`` in ``K[x]``.

    Examples
    ========

    >>> from sympy.polys import ring, ZZ
    >>> R, x = ring("x", ZZ)

    >>> R.dup_scale(x**2 - 2*x + 1, ZZ(2))
    4*x**2 - 4*x + 1

    r.   rN   r   rd   r2   )r5   rY   r7   r;   br9   r=   r=   r>   	dup_scaleP  s    r   c                 C   sX   t | t| d  } }t|ddD ]0}td|D ] }| |d   || |  7  < q0q"| S )z
    Evaluate efficiently Taylor shift ``f(x + a)`` in ``K[x]``.

    Examples
    ========

    >>> from sympy.polys import ring, ZZ
    >>> R, x = ring("x", ZZ)

    >>> R.dup_shift(x**2 - 2*x + 1, ZZ(2))
    x**2 + 2*x + 1

    r.   r   rN   r   )r5   rY   r7   r;   r9   r<   r=   r=   r>   	dup_shiftf  s
     r   c           	      C   s   | sg S t | d }| d g|jgg }}td|D ]}|t|d || q4t| dd |dd D ],\}}t|||}t|||}t|||}qj|S )a  
    Evaluate functional transformation ``q**n * f(p/q)`` in ``K[x]``.

    Examples
    ========

    >>> from sympy.polys import ring, ZZ
    >>> R, x = ring("x", ZZ)

    >>> R.dup_transform(x**2 - 2*x + 1, x**2 + 1, x - 1)
    x**4 - 2*x**3 + 5*x**2 - 4*x + 4

    r.   r   rN   N)rd   r   r2   rO   r	   zipr   r   )	r5   rm   qr7   r;   re   Qr9   r:   r=   r=   r>   dup_transform}  s    "r   c                 C   sf   t |dkr$tt| t|||gS | s,g S | d g}| dd D ]}t|||}t||d|}qB|S )z
    Evaluate functional composition ``f(g)`` in ``K[x]``.

    Examples
    ========

    >>> from sympy.polys import ring, ZZ
    >>> R, x = ring("x", ZZ)

    >>> R.dup_compose(x**2 + x, x - 1)
    x**2 - x

    r.   r   N)rd   r   r[   r   r	   r   )r5   r8   r7   re   r:   r=   r=   r>   dup_compose  s    
r   c                 C   s\   |st | ||S t| |r| S | d g}| dd D ]"}t||||}t||d||}q4|S )z
    Evaluate functional composition ``f(g)`` in ``K[X]``.

    Examples
    ========

    >>> from sympy.polys import ring, ZZ
    >>> R, x,y = ring("x,y", ZZ)

    >>> R.dmp_compose(x*y + 2*x + y, y)
    y**2 + 3*y

    r   r.   N)r   r%   r
   r   )r5   r8   r@   r7   re   r:   r=   r=   r>   dmp_compose  s    

r   c                 C   s   t | d }t| |}t| } ||ji}|| }td|D ]}|j}td|D ]Z}	||	 | | vrdqN||	 |vrrqN| ||	 |  |||	   }
}||||	  |
 | 7 }qN|||| | ||| < q:t||S )+Helper function for :func:`_dup_decompose`.r.   r   )rd   r   r&   r   r2   r/   quor'   )r5   sr7   r;   rv   r8   rr9   rQ   r<   r   r   r=   r=   r>   _dup_right_decompose  s     

r   c                 C   sV   i d }}| rLt | ||\}}t|dkr.dS t||||< ||d  } }q
t||S )r   r   Nr.   )r   r   r   r'   )r5   re   r7   r8   r9   r   r   r=   r=   r>   _dup_left_decompose  s    
r   c                 C   sb   t | d }td|D ]F}|| dkr(qt| ||}|durt| ||}|dur||f  S qdS )z*Helper function for :func:`dup_decompose`.r.   rk   r   N)rd   r2   r   r   )r5   r7   dfr   re   r8   r=   r=   r>   _dup_decompose  s    r   c                 C   s8   g }t | |}|dur.|\} }|g| }qq.q| g| S )ae  
    Computes functional decomposition of ``f`` in ``K[x]``.

    Given a univariate polynomial ``f`` with coefficients in a field of
    characteristic zero, returns list ``[f_1, f_2, ..., f_n]``, where::

              f = f_1 o f_2 o ... f_n = f_1(f_2(... f_n))

    and ``f_2, ..., f_n`` are monic and homogeneous polynomials of at
    least second degree.

    Unlike factorization, complete functional decompositions of
    polynomials are not unique, consider examples:

    1. ``f o g = f(x + b) o (g - b)``
    2. ``x**n o x**m = x**m o x**n``
    3. ``T_n o T_m = T_m o T_n``

    where ``T_n`` and ``T_m`` are Chebyshev polynomials.

    Examples
    ========

    >>> from sympy.polys import ring, ZZ
    >>> R, x = ring("x", ZZ)

    >>> R.dup_decompose(x**4 - 2*x**3 + x**2)
    [x**2, x**2 - x]

    References
    ==========

    .. [1] [Kozen89]_

    N)r   )r5   r7   FrZ   re   r=   r=   r>   dup_decompose  s    $
r   c                 C   s   |j r | }t| |||} |}|js.tdt| |g g   }}}| D ]\}}|jsL|| qLt	ddgt
|dd}	|	D ]H}
t|}t|
|D ]\}}|dkr||  ||< q|t||| qtt||||||jS )a^  
    Convert algebraic coefficients to integers in ``K[X]``.

    Examples
    ========

    >>> from sympy.polys import ring, QQ
    >>> from sympy import I

    >>> K = QQ.algebraic_field(I)
    >>> R, x = ring("x", K)

    >>> f = x**2 + K([QQ(1), QQ(0)])*x + K([QQ(2), QQ(0)])

    >>> R.dmp_lift(f)
    x**8 + 2*x**6 + 9*x**4 - 8*x**2 + 16

    z3computation can be done only in an algebraic domainrN   r.   T)
repetition)is_GaussianFieldas_AlgebraicFieldr   is_Algebraicr*   r   r   	is_groundrO   r+   rd   dictr   r   r   dom)r5   r@   r7   K1r   monomspolysmonomrQ   permspermGsignr=   r=   r>   dmp_liftG  s(    r   c                 C   s8   |j d }}| D ]"}||| r*|d7 }|r|}q|S )z
    Compute the number of sign variations of ``f`` in ``K[x]``.

    Examples
    ========

    >>> from sympy.polys import ring, ZZ
    >>> R, x = ring("x", ZZ)

    >>> R.dup_sign_variations(x**4 - x**2 - x + 1)
    2

    r   r.   )r/   is_negative)r5   r7   prevrR   rQ   r=   r=   r>   dup_sign_variationsw  s    r   NFc                 C   st   |du r|j r| }n|}|j}| D ]}||||}q&||sTt| ||} |s`|| fS |t| ||fS dS )a@  
    Clear denominators, i.e. transform ``K_0`` to ``K_1``.

    Examples
    ========

    >>> from sympy.polys import ring, QQ
    >>> R, x = ring("x", QQ)

    >>> f = QQ(1,2)*x + QQ(1,3)

    >>> R.dup_clear_denoms(f, convert=False)
    (6, 3*x + 2)
    >>> R.dup_clear_denoms(f, convert=True)
    (6, 3*x + 2)

    N)has_assoc_Ringget_ringr   lcmdenomru   r   r   )r5   K0r   rX   commonr:   r=   r=   r>   dup_clear_denoms  s    

r   c              	   C   sT   |j }|s(| D ]}||||}qn(|d }| D ]}||t||||}q4|S )z.Recursive helper for :func:`dmp_clear_denoms`.r.   )r   r   r   _rec_clear_denoms)r8   rA   r   r   r   r:   rG   r=   r=   r>   r     s    r   c                 C   sx   |st | |||dS |du r0|jr,| }n|}t| |||}||sVt| |||} |sb|| fS |t| |||fS dS )aV  
    Clear denominators, i.e. transform ``K_0`` to ``K_1``.

    Examples
    ========

    >>> from sympy.polys import ring, QQ
    >>> R, x,y = ring("x,y", QQ)

    >>> f = QQ(1,2)*x + QQ(1,3)*y + 1

    >>> R.dmp_clear_denoms(f, convert=False)
    (6, 3*x + 2*y + 6)
    >>> R.dmp_clear_denoms(f, convert=True)
    (6, 3*x + 2*y + 6)

    )rX   N)r   r   r   r   ru   r   r   )r5   r@   r   r   rX   r   r=   r=   r>   dmp_clear_denoms  s    

r   c           	      C   s   | t| |g}|j|j|jg}ttt|d}td|d D ]J}t||d|}t	| t
|||}tt|||||}t|t||}qB|S )a  
    Compute ``f**(-1)`` mod ``x**n`` using Newton iteration.

    This function computes first ``2**n`` terms of a polynomial that
    is a result of inversion of a polynomial modulo ``x**n``. This is
    useful to efficiently compute series expansion of ``1/f``.

    Examples
    ========

    >>> from sympy.polys import ring, QQ
    >>> R, x = ring("x", QQ)

    >>> f = -QQ(1,720)*x**6 + QQ(1,24)*x**4 - QQ(1,2)*x**2 + 1

    >>> R.dup_revert(f, 8)
    61/720*x**6 + 5/24*x**4 + 1/2*x**2 + 1

    rk   r.   )revertr!   r   r/   int_ceil_logr2   r   r	   r   r   r   r   r   )	r5   r;   r7   r8   re   Nr9   rY   r   r=   r=   r>   
dup_revert  s    r   c                 C   s   |st | ||S t| |dS )z
    Compute ``f**(-1)`` mod ``x**n`` using Newton iteration.

    Examples
    ========

    >>> from sympy.polys import ring, QQ
    >>> R, x,y = ring("x,y", QQ)

    N)r   r)   )r5   r8   r@   r7   r=   r=   r>   
dmp_revert  s    r   )NF)NF)a__doc__sympy.polys.densearithr   r   r   r   r   r   r   r	   r
   r   r   r   r   r   r   r   r   r   r   r   sympy.polys.densebasicr   r   r   r   r   r   r   r   r   r   r    r!   r"   r#   r$   r%   r&   r'   r(   sympy.polys.polyerrorsr)   r*   sympy.utilitiesr+   mathr,   r   r-   r   r?   rB   rC   rM   rS   rT   rU   rW   r[   r\   r]   r_   ra   rg   rh   rj   ro   rr   rs   rw   rx   r~   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r=   r=   r=   r>   <module>   sd   XT #
+/

 
$*-!$/20
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