a
    RG5d<                     @   sT  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 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d Zdd Zdd Zdd Zdd Zdd Zdd Zd d! Z d"d# Z!d$d% Z"d&d' Z#d(d) Z$d*d+ Z%d,d- Z&d.d/ Z'd0d1 Z(d2d3 Z)d4d5 Z*d6d7 Z+d8d9 Z,d:d; Z-d<d= Z.d>d? Z/d@dA Z0dBdC Z1dDdE Z2dFdG Z3dHdI Z4dJdK Z5dLdM Z6dNdO Z7dPdQ Z8dRdS Z9dTdU Z:dVdW Z;dXdY Z<dZd[ Z=d\d] Z>d^d_ Z?d`da Z@dbdc ZAddde ZBdfdg ZCdhdi ZDdjdk ZEdldm ZFdndo ZGdpdq ZHdrds ZIdtdu ZJdvdw ZKdxdy ZLdzd{ ZMd|d} ZNd~d ZOdd ZPdd ZQdS )zEArithmetics for dense recursive polynomials in ``K[x]`` or ``K[X]``.     )	dup_slicedup_LCdmp_LC
dup_degree
dmp_degree	dup_strip	dmp_strip
dmp_zero_pdmp_zero	dmp_one_pdmp_one
dmp_ground	dmp_zeros)ExactQuotientFailedPolynomialDivisionFailedc                 C   s   |s| S t | }|| d }||d krFt| d | g| dd  S ||krh|g|jg||   |  S | d| | | | g | |d d  S dS )z
    Add ``c*x**i`` to ``f`` in ``K[x]``.

    Examples
    ========

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

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

       r   Nlenr   zerofciKnm r   R/var/www/html/django/DPS/env/lib/python3.9/site-packages/sympy/polys/densearith.pydup_add_term   s    r   c                 C   s   |st | |||S |d }t||r(| S t| }|| d }||d krntt| d |||g| dd  |S ||kr|gt|| || |  S | d| t| | |||g | |d d  S dS )z
    Add ``c(x_2..x_u)*x_0**i`` to ``f`` in ``K[X]``.

    Examples
    ========

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

    >>> R.dmp_add_term(x*y + 1, 2, 2)
    2*x**2 + x*y + 1

    r   r   N)r   r	   r   r   dmp_addr   r   r   r   ur   vr   r   r   r   r   dmp_add_term+   s    
&r#   c                 C   s   |s| S t | }|| d }||d krFt| d | g| dd  S ||krj| g|jg||   |  S | d| | | | g | |d d  S dS )z
    Subtract ``c*x**i`` from ``f`` in ``K[x]``.

    Examples
    ========

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

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

    r   r   Nr   r   r   r   r   dup_sub_termM   s    r$   c                 C   s   |st | | ||S |d }t||r*| S t| }|| d }||d krptt| d |||g| dd  |S ||krt|||gt|| || |  S | d| t| | |||g | |d d  S dS )z
    Subtract ``c(x_2..x_u)*x_0**i`` from ``f`` in ``K[X]``.

    Examples
    ========

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

    >>> R.dmp_sub_term(2*x**2 + x*y + 1, 2, 2)
    x*y + 1

    r   r   N)r   r	   r   r   dmp_subdmp_negr   r    r   r   r   dmp_sub_termj   s    
&"r'   c                    s.    r| sg S  fdd| D |j g|  S dS )z
    Multiply ``f`` by ``c*x**i`` in ``K[x]``.

    Examples
    ========

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

    >>> R.dup_mul_term(x**2 - 1, ZZ(3), 2)
    3*x**4 - 3*x**2

    c                    s   g | ]}|  qS r   r   .0cfr   r   r   
<listcomp>       z dup_mul_term.<locals>.<listcomp>Nr   )r   r   r   r   r   r+   r   dup_mul_term   s    r/   c                    s`   |st | | S |d t| |r(| S tr:t|S  fdd| D t|  S dS )z
    Multiply ``f`` by ``c(x_2..x_u)*x_0**i`` in ``K[X]``.

    Examples
    ========

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

    >>> R.dmp_mul_term(x**2*y + x, 3*y, 2)
    3*x**4*y**2 + 3*x**3*y

    r   c                    s   g | ]}t | qS r   )dmp_mulr(   r   r   r"   r   r   r,      r-   z dmp_mul_term.<locals>.<listcomp>N)r/   r	   r
   r   )r   r   r   r!   r   r   r1   r   dmp_mul_term   s    

r2   c                 C   s   t | |d|S )z
    Add an element of the ground domain to ``f``.

    Examples
    ========

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

    >>> R.dup_add_ground(x**3 + 2*x**2 + 3*x + 4, ZZ(4))
    x**3 + 2*x**2 + 3*x + 8

    r   )r   r   r   r   r   r   r   dup_add_ground   s    r4   c                 C   s   t | t||d d||S )z
    Add an element of the ground domain to ``f``.

    Examples
    ========

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

    >>> R.dmp_add_ground(x**3 + 2*x**2 + 3*x + 4, ZZ(4))
    x**3 + 2*x**2 + 3*x + 8

    r   r   )r#   r   r   r   r!   r   r   r   r   dmp_add_ground   s    r6   c                 C   s   t | |d|S )z
    Subtract an element of the ground domain from ``f``.

    Examples
    ========

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

    >>> R.dup_sub_ground(x**3 + 2*x**2 + 3*x + 4, ZZ(4))
    x**3 + 2*x**2 + 3*x

    r   )r$   r3   r   r   r   dup_sub_ground   s    r7   c                 C   s   t | t||d d||S )z
    Subtract an element of the ground domain from ``f``.

    Examples
    ========

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

    >>> R.dmp_sub_ground(x**3 + 2*x**2 + 3*x + 4, ZZ(4))
    x**3 + 2*x**2 + 3*x

    r   r   )r'   r   r5   r   r   r   dmp_sub_ground   s    r8   c                    s"    r| sg S  fdd| D S dS )z
    Multiply ``f`` by a constant value in ``K[x]``.

    Examples
    ========

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

    >>> R.dup_mul_ground(x**2 + 2*x - 1, ZZ(3))
    3*x**2 + 6*x - 3

    c                    s   g | ]}|  qS r   r   r(   r+   r   r   r,     r-   z"dup_mul_ground.<locals>.<listcomp>Nr   r3   r   r+   r   dup_mul_ground   s    r9   c                    s.   |st |  S |d  fdd| D S )z
    Multiply ``f`` by a constant value in ``K[X]``.

    Examples
    ========

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

    >>> R.dmp_mul_ground(2*x + 2*y, ZZ(3))
    6*x + 6*y

    r   c                    s   g | ]}t | qS r   )dmp_mul_groundr(   r1   r   r   r,   &  r-   z"dmp_mul_ground.<locals>.<listcomp>)r9   r5   r   r1   r   r:     s    r:   c                    sD   st d| s| S  jr. fdd| D S fdd| D S dS )a)  
    Quotient by a constant in ``K[x]``.

    Examples
    ========

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

    >>> R, x = ring("x", ZZ)
    >>> R.dup_quo_ground(3*x**2 + 2, ZZ(2))
    x**2 + 1

    >>> R, x = ring("x", QQ)
    >>> R.dup_quo_ground(3*x**2 + 2, QQ(2))
    3/2*x**2 + 1

    polynomial divisionc                    s   g | ]}  |qS r   )quor(   r   r   r   r   r,   A  r-   z"dup_quo_ground.<locals>.<listcomp>c                    s   g | ]}|  qS r   r   r(   r+   r   r   r,   C  r-   N)ZeroDivisionErroris_Fieldr3   r   r=   r   dup_quo_ground)  s    r@   c                    s.   |st |  S |d  fdd| D S )a=  
    Quotient by a constant in ``K[X]``.

    Examples
    ========

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

    >>> R, x,y = ring("x,y", ZZ)
    >>> R.dmp_quo_ground(2*x**2*y + 3*x, ZZ(2))
    x**2*y + x

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

    r   c                    s   g | ]}t | qS r   )dmp_quo_groundr(   r1   r   r   r,   ]  r-   z"dmp_quo_ground.<locals>.<listcomp>)r@   r5   r   r1   r   rA   F  s    rA   c                    s(   st d| s| S  fdd| D S )z
    Exact quotient by a constant in ``K[x]``.

    Examples
    ========

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

    >>> R.dup_exquo_ground(x**2 + 2, QQ(2))
    1/2*x**2 + 1

    r;   c                    s   g | ]}  |qS r   )exquor(   r=   r   r   r,   s  r-   z$dup_exquo_ground.<locals>.<listcomp>)r>   r3   r   r=   r   dup_exquo_ground`  s
    rC   c                    s.   |st |  S |d  fdd| D S )z
    Exact quotient by a constant in ``K[X]``.

    Examples
    ========

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

    >>> R.dmp_exquo_ground(x**2*y + 2*x, QQ(2))
    1/2*x**2*y + x

    r   c                    s   g | ]}t | qS r   )dmp_exquo_groundr(   r1   r   r   r,     r-   z$dmp_exquo_ground.<locals>.<listcomp>)rC   r5   r   r1   r   rD   v  s    rD   c                 C   s   | s| S | |j g|  S dS )z
    Efficiently multiply ``f`` by ``x**n`` in ``K[x]``.

    Examples
    ========

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

    >>> R.dup_lshift(x**2 + 1, 2)
    x**4 + x**2

    Nr.   r   r   r   r   r   r   
dup_lshift  s    rF   c                 C   s   | d|  S )a  
    Efficiently divide ``f`` by ``x**n`` in ``K[x]``.

    Examples
    ========

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

    >>> R.dup_rshift(x**4 + x**2, 2)
    x**2 + 1
    >>> R.dup_rshift(x**4 + x**2 + 2, 2)
    x**2 + 1

    Nr   rE   r   r   r   
dup_rshift  s    rG   c                    s    fdd| D S )z
    Make all coefficients positive in ``K[x]``.

    Examples
    ========

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

    >>> R.dup_abs(x**2 - 1)
    x**2 + 1

    c                    s   g | ]}  |qS r   )absr)   coeffr   r   r   r,     r-   zdup_abs.<locals>.<listcomp>r   r   r   r   rK   r   dup_abs  s    rM   c                    s*   |st |  S |d  fdd| D S )z
    Make all coefficients positive in ``K[X]``.

    Examples
    ========

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

    >>> R.dmp_abs(x**2*y - x)
    x**2*y + x

    r   c                    s   g | ]}t | qS r   )dmp_absr(   r   r"   r   r   r,     r-   zdmp_abs.<locals>.<listcomp>)rM   r   r!   r   r   rO   r   rN     s    
rN   c                 C   s   dd | D S )z
    Negate a polynomial in ``K[x]``.

    Examples
    ========

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

    >>> R.dup_neg(x**2 - 1)
    -x**2 + 1

    c                 S   s   g | ]
}| qS r   r   rI   r   r   r   r,     r-   zdup_neg.<locals>.<listcomp>r   rL   r   r   r   dup_neg  s    rQ   c                    s*   |st |  S |d  fdd| D S )z
    Negate a polynomial in ``K[X]``.

    Examples
    ========

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

    >>> R.dmp_neg(x**2*y - x)
    -x**2*y + x

    r   c                    s   g | ]}t | qS r   )r&   r(   rO   r   r   r,     r-   zdmp_neg.<locals>.<listcomp>)rQ   rP   r   rO   r   r&     s    
r&   c                 C   s   | s|S |s| S t | }t |}||kr@tdd t| |D S t|| }||krp| d| | |d  }} n|d| ||d  }}|dd t| |D  S dS )z
    Add dense polynomials in ``K[x]``.

    Examples
    ========

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

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

    c                 S   s   g | ]\}}|| qS r   r   r)   abr   r   r   r,     r-   zdup_add.<locals>.<listcomp>Nc                 S   s   g | ]\}}|| qS r   r   rR   r   r   r   r,   !  r-   )r   r   ziprH   r   gr   dfdgkhr   r   r   dup_add  s    r\   c                    s   |st | | S t| |}|dk r&|S t||}|dk r<| S |d ||krlt fddt| |D |S t|| }||kr| d| | |d  }} n|d| ||d  }}| fddt| |D  S dS )z
    Add dense polynomials in ``K[X]``.

    Examples
    ========

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

    >>> R.dmp_add(x**2 + y, x**2*y + x)
    x**2*y + x**2 + x + y

    r   r   c                    s   g | ]\}}t || qS r   r   rR   rO   r   r   r,   B  r-   zdmp_add.<locals>.<listcomp>Nc                    s   g | ]\}}t || qS r   r]   rR   rO   r   r   r,   K  r-   )r\   r   r   rU   rH   r   rW   r!   r   rX   rY   rZ   r[   r   rO   r   r   $  s     

 r   c                 C   s   | st ||S |s| S t| }t|}||krFtdd t| |D S t|| }||krv| d| | |d  }} n t |d| |||d  }}|dd t| |D  S dS )z
    Subtract dense polynomials in ``K[x]``.

    Examples
    ========

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

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

    c                 S   s   g | ]\}}|| qS r   r   rR   r   r   r   r,   e  r-   zdup_sub.<locals>.<listcomp>Nc                 S   s   g | ]\}}|| qS r   r   rR   r   r   r   r,   n  r-   )rQ   r   r   rU   rH   rV   r   r   r   dup_subN  s    
 r_   c                    s   |st | | S t| |}|dk r.t|| S t||}|dk rD| S |d ||krtt fddt| |D |S t|| }||kr| d| | |d  }} n"t|d| | ||d  }}| fddt| |D  S dS )z
    Subtract dense polynomials in ``K[X]``.

    Examples
    ========

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

    >>> R.dmp_sub(x**2 + y, x**2*y + x)
    -x**2*y + x**2 - x + y

    r   r   c                    s   g | ]\}}t || qS r   r%   rR   rO   r   r   r,     r-   zdmp_sub.<locals>.<listcomp>Nc                    s   g | ]\}}t || qS r   r`   rR   rO   r   r   r,     r-   )r_   r   r&   r   rU   rH   r^   r   rO   r   r%   q  s     

 "r%   c                 C   s   t | t||||S )z
    Returns ``f + g*h`` where ``f, g, h`` are in ``K[x]``.

    Examples
    ========

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

    >>> R.dup_add_mul(x**2 - 1, x - 2, x + 2)
    2*x**2 - 5

    )r\   dup_mulr   rW   r[   r   r   r   r   dup_add_mul  s    rc   c                 C   s   t | t||||||S )z
    Returns ``f + g*h`` where ``f, g, h`` are in ``K[X]``.

    Examples
    ========

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

    >>> R.dmp_add_mul(x**2 + y, x, x + 2)
    2*x**2 + 2*x + y

    )r   r0   r   rW   r[   r!   r   r   r   r   dmp_add_mul  s    re   c                 C   s   t | t||||S )z
    Returns ``f - g*h`` where ``f, g, h`` are in ``K[x]``.

    Examples
    ========

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

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

    )r_   ra   rb   r   r   r   dup_sub_mul  s    rf   c                 C   s   t | t||||||S )z
    Returns ``f - g*h`` where ``f, g, h`` are in ``K[X]``.

    Examples
    ========

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

    >>> R.dmp_sub_mul(x**2 + y, x, x + 2)
    -2*x + y

    )r%   r0   rd   r   r   r   dmp_sub_mul  s    rg   c                 C   s|  | |krt | |S | r|sg S t| }t|}t||d }|dk rg }td|| d D ]P}|j}ttd|| t||d D ]}	|| |	 |||	   7 }q|| qZt|S |d }
t| d|
|t|d|
| }}t	t| |
|||
|}t	t||
|||
|}t
|||t
||| }}t
t|||t||||}t|t||||}tt|t||
||t|d|
 ||S dS )z
    Multiply dense polynomials in ``K[x]``.

    Examples
    ========

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

    >>> R.dup_mul(x - 2, x + 2)
    x**2 - 4

    r   d   r      N)dup_sqrr   maxranger   minappendr   r   rG   ra   r\   r_   rF   )r   rW   r   rX   rY   r   r[   r   rJ   jn2flglfhghlohimidr   r   r   ra     s2    
"ra   c              	   C   s   |st | ||S | |kr$t| ||S t| |}|dk r:| S t||}|dk rP|S g |d  }}td|| d D ]^}t|}	ttd|| t||d D ](}
t|	t| |
 |||
  ||||}	q|	|	 qpt
||S )z
    Multiply dense polynomials in ``K[X]``.

    Examples
    ========

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

    >>> R.dmp_mul(x*y + 1, x)
    x**2*y + x

    r   r   )ra   dmp_sqrr   rl   r
   rk   rm   r   r0   rn   r   )r   rW   r!   r   rX   rY   r[   r"   r   rJ   ro   r   r   r   r0     s"    

"&r0   c                 C   s   t | d g  }}tdd| d D ]}|j}td|| }t||}|| d }||d  d }t||d D ]}	|| |	 | ||	   7 }qp||7 }|d@ r| |d  }
||
d 7 }|| q$t|S )z
    Square dense polynomials in ``K[x]``.

    Examples
    ========

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

    >>> R.dup_sqr(x**2 + 1)
    x**4 + 2*x**2 + 1

    r   r   ri   )r   rl   r   rk   rm   rn   r   )r   r   rX   r[   r   r   jminjmaxr   ro   elemr   r   r   rj   C  s    
rj   c              	   C   s  |st | |S t| |}|dk r$| S g |d  }}tdd| d D ]}t|}td|| }t||}	|	| d }
||
d  d }	t||	d D ](}t|t| | | ||  ||||}qt||d||}|
d@ rt	| |	d  ||}t||||}|
| qDt||S )z
    Square dense polynomials in ``K[X]``.

    Examples
    ========

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

    >>> R.dmp_sqr(x**2 + x*y + y**2)
    x**4 + 2*x**3*y + 3*x**2*y**2 + 2*x*y**3 + y**4

    r   r   ri   )rj   r   rl   r
   rk   rm   r   r0   r:   rx   rn   r   )r   r!   r   rX   r[   r"   r   r   ry   rz   r   ro   r{   r   r   r   rx   k  s(    


&rx   c                 C   sx   |s|j gS |dk rtd|dks4| r4| |j gkr8| S |j g}|d | }}|d rht|| |}|shqtt| |} q@|S )z
    Raise ``f`` to the ``n``-th power in ``K[x]``.

    Examples
    ========

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

    >>> R.dup_pow(x - 2, 3)
    x**3 - 6*x**2 + 12*x - 8

    r   +Cannot raise polynomial to a negative powerr   ri   )one
ValueErrorra   rj   )r   r   r   rW   r   r   r   r   dup_pow  s    r   c                 C   s   |st | ||S |st||S |dk r.td|dksLt| |sLt| ||rP| S t||}|d | }}|d@ rt|| ||}|sqt| ||} qZ|S )z
    Raise ``f`` to the ``n``-th power in ``K[X]``.

    Examples
    ========

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

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

    r   r|   r   ri   )r   r   r~   r	   r   r0   rx   )r   r   r!   r   rW   r   r   r   r   dmp_pow  s     

r   c                 C   s  t | }t |}g | |  }}}|s.tdn||k r>||fS || d }t||}	t||}
|| |d  }}t||	|}t||
||}t||	|}t||
||}t|||}|t | }}||k rqqT||k sTt| ||qT|	| }t|||}t|||}||fS )z
    Polynomial pseudo-division in ``K[x]``.

    Examples
    ========

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

    >>> R.dup_pdiv(x**2 + 1, 2*x - 4)
    (2*x + 4, 20)

    r;   r   )r   r>   r   r9   r   r/   r_   r   )r   rW   r   rX   rY   qrdrNlc_glc_rro   QRG_drr   r   r   r   dup_pdiv  s2    


r   c                 C   s   t | }t |}| | }}|s(tdn||k r4|S || d }t||}t||}	|| |d  }
}t|||}t||	|
|}t|||}|t | }}||k rqqJ||k sJt| ||qJt||| |S )z
    Polynomial pseudo-remainder in ``K[x]``.

    Examples
    ========

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

    >>> R.dup_prem(x**2 + 1, 2*x - 4)
    20

    r;   r   )r   r>   r   r9   r/   r_   r   )r   rW   r   rX   rY   r   r   r   r   r   ro   r   r   r   r   r   r   dup_prem  s(    



r   c                 C   s   t | ||d S )a   
    Polynomial exact pseudo-quotient in ``K[X]``.

    Examples
    ========

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

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

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

    r   )r   r   rW   r   r   r   r   dup_pquoJ  s    r   c                 C   s&   t | ||\}}|s|S t| |dS )a\  
    Polynomial pseudo-quotient in ``K[x]``.

    Examples
    ========

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

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

    >>> R.dup_pexquo(x**2 + 1, 2*x - 4)
    Traceback (most recent call last):
    ...
    ExactQuotientFailed: [2, -4] does not divide [1, 0, 1]

    N)r   r   r   rW   r   r   r   r   r   r   
dup_pexquo^  s    r   c                 C   sF  |st | ||S t| |}t||}|dk r4tdt|| |  }}}||k rX||fS || d }	t||}
t||}|| |	d  }}	t||
d||}t|||||}t||
d||}t|||||}t||||}|t|| }}||k rqqn||k snt| ||qnt	|
|	|d |}t||d||}t||d||}||fS )z
    Polynomial pseudo-division in ``K[X]``.

    Examples
    ========

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

    >>> R.dmp_pdiv(x**2 + x*y, 2*x + 2)
    (2*x + 2*y - 2, -4*y + 4)

    r   r;   r   )
r   r   r>   r
   r   r2   r#   r%   r   r   )r   rW   r!   r   rX   rY   r   r   r   r   r   r   ro   r   r   r   r   r   r   r   r   dmp_pdivy  s6    



r   c                 C   s   |st | ||S t| |}t||}|dk r4td| | }}||k rJ|S || d }t||}	t||}
|| |d  }}t||	d||}t||
|||}t||||}|t|| }}||k rqq`||k s`t| ||q`t|	||d |}t||d||S )z
    Polynomial pseudo-remainder in ``K[X]``.

    Examples
    ========

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

    >>> R.dmp_prem(x**2 + x*y, 2*x + 2)
    -4*y + 4

    r   r;   r   )r   r   r>   r   r2   r%   r   r   )r   rW   r!   r   rX   rY   r   r   r   r   r   ro   r   r   r   r   r   r   r   dmp_prem  s.    




r   c                 C   s   t | |||d S )a.  
    Polynomial exact pseudo-quotient in ``K[X]``.

    Examples
    ========

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

    >>> f = x**2 + x*y
    >>> g = 2*x + 2*y
    >>> h = 2*x + 2

    >>> R.dmp_pquo(f, g)
    2*x

    >>> R.dmp_pquo(f, h)
    2*x + 2*y - 2

    r   )r   r   rW   r!   r   r   r   r   dmp_pquo  s    r   c                 C   s.   t | |||\}}t||r |S t| |dS )a  
    Polynomial pseudo-quotient in ``K[X]``.

    Examples
    ========

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

    >>> f = x**2 + x*y
    >>> g = 2*x + 2*y
    >>> h = 2*x + 2

    >>> R.dmp_pexquo(f, g)
    2*x

    >>> R.dmp_pexquo(f, h)
    Traceback (most recent call last):
    ...
    ExactQuotientFailed: [[2], [2]] does not divide [[1], [1, 0], []]

    N)r   r	   r   r   rW   r!   r   r   r   r   r   r   
dmp_pexquo  s    
r   c                 C   s   t | }t |}g | |  }}}|s.tdn||k r>||fS t||}t||}	|	| r\q||	|}
|| }t||
||}t||
||}t|||}|t | }}||k rqqH||k sHt| ||qH||fS )z
    Univariate division with remainder over a ring.

    Examples
    ========

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

    >>> R.dup_rr_div(x**2 + 1, 2*x - 4)
    (0, x**2 + 1)

    r;   )r   r>   r   rB   r   r/   r_   r   r   rW   r   rX   rY   r   r   r   r   r   r   ro   r[   r   r   r   r   
dup_rr_div  s,    


r   c                 C   s  |st | ||S t| |}t||}|dk r4tdt|| |  }}}||k rX||fS t|||d  }	}
t||}t||	|
|\}}t||
sq || }t|||||}t|||||}t	||||}|t|| }}||k rq ql||k slt
| ||ql||fS )z
    Multivariate division with remainder over a ring.

    Examples
    ========

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

    >>> R.dmp_rr_div(x**2 + x*y, 2*x + 2)
    (0, x**2 + x*y)

    r   r;   r   )r   r   r>   r
   r   
dmp_rr_divr	   r#   r2   r%   r   r   rW   r!   r   rX   rY   r   r   r   r   r"   r   r   r   ro   r[   r   r   r   r   r   M  s0    



r   c                 C   s   t | }t |}g | |  }}}|s.tdn||k r>||fS t||}t||}	||	|}
|| }t||
||}t||
||}t|||}|t | }}||k rqqH||kr|jst|dd }t |}||k rqqH||k sHt	| ||qH||fS )z
    Polynomial division with remainder over a field.

    Examples
    ========

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

    >>> R.dup_ff_div(x**2 + 1, 2*x - 4)
    (1/2*x + 1, 5)

    r;   r   N)
r   r>   r   rB   r   r/   r_   is_Exactr   r   r   r   r   r   
dup_ff_div  s2    


r   c                 C   s  |st | ||S t| |}t||}|dk r4tdt|| |  }}}||k rX||fS t|||d  }	}
t||}t||	|
|\}}t||
sq || }t|||||}t|||||}t	||||}|t|| }}||k rq ql||k slt
| ||ql||fS )z
    Polynomial division with remainder over a field.

    Examples
    ========

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

    >>> R.dmp_ff_div(x**2 + x*y, 2*x + 2)
    (1/2*x + 1/2*y - 1/2, -y + 1)

    r   r;   r   )r   r   r>   r
   r   
dmp_ff_divr	   r#   r2   r%   r   r   r   r   r   r     s0    



r   c                 C   s"   |j rt| ||S t| ||S dS )a.  
    Polynomial division with remainder in ``K[x]``.

    Examples
    ========

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

    >>> R, x = ring("x", ZZ)
    >>> R.dup_div(x**2 + 1, 2*x - 4)
    (0, x**2 + 1)

    >>> R, x = ring("x", QQ)
    >>> R.dup_div(x**2 + 1, 2*x - 4)
    (1/2*x + 1, 5)

    N)r?   r   r   r   r   r   r   dup_div  s    r   c                 C   s   t | ||d S )a  
    Returns polynomial remainder in ``K[x]``.

    Examples
    ========

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

    >>> R, x = ring("x", ZZ)
    >>> R.dup_rem(x**2 + 1, 2*x - 4)
    x**2 + 1

    >>> R, x = ring("x", QQ)
    >>> R.dup_rem(x**2 + 1, 2*x - 4)
    5

    r   r   r   r   r   r   dup_rem  s    r   c                 C   s   t | ||d S )a  
    Returns exact polynomial quotient in ``K[x]``.

    Examples
    ========

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

    >>> R, x = ring("x", ZZ)
    >>> R.dup_quo(x**2 + 1, 2*x - 4)
    0

    >>> R, x = ring("x", QQ)
    >>> R.dup_quo(x**2 + 1, 2*x - 4)
    1/2*x + 1

    r   r   r   r   r   r   dup_quo  s    r   c                 C   s&   t | ||\}}|s|S t| |dS )aW  
    Returns polynomial quotient in ``K[x]``.

    Examples
    ========

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

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

    >>> R.dup_exquo(x**2 + 1, 2*x - 4)
    Traceback (most recent call last):
    ...
    ExactQuotientFailed: [2, -4] does not divide [1, 0, 1]

    N)r   r   r   r   r   r   	dup_exquo-  s    r   c                 C   s&   |j rt| |||S t| |||S dS )aK  
    Polynomial division with remainder in ``K[X]``.

    Examples
    ========

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

    >>> R, x,y = ring("x,y", ZZ)
    >>> R.dmp_div(x**2 + x*y, 2*x + 2)
    (0, x**2 + x*y)

    >>> R, x,y = ring("x,y", QQ)
    >>> R.dmp_div(x**2 + x*y, 2*x + 2)
    (1/2*x + 1/2*y - 1/2, -y + 1)

    N)r?   r   r   r   r   r   r   dmp_divH  s    r   c                 C   s   t | |||d S )a)  
    Returns polynomial remainder in ``K[X]``.

    Examples
    ========

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

    >>> R, x,y = ring("x,y", ZZ)
    >>> R.dmp_rem(x**2 + x*y, 2*x + 2)
    x**2 + x*y

    >>> R, x,y = ring("x,y", QQ)
    >>> R.dmp_rem(x**2 + x*y, 2*x + 2)
    -y + 1

    r   r   r   r   r   r   dmp_rem`  s    r   c                 C   s   t | |||d S )a2  
    Returns exact polynomial quotient in ``K[X]``.

    Examples
    ========

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

    >>> R, x,y = ring("x,y", ZZ)
    >>> R.dmp_quo(x**2 + x*y, 2*x + 2)
    0

    >>> R, x,y = ring("x,y", QQ)
    >>> R.dmp_quo(x**2 + x*y, 2*x + 2)
    1/2*x + 1/2*y - 1/2

    r   r   r   r   r   r   dmp_quou  s    r   c                 C   s.   t | |||\}}t||r |S t| |dS )a  
    Returns polynomial quotient in ``K[X]``.

    Examples
    ========

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

    >>> f = x**2 + x*y
    >>> g = x + y
    >>> h = 2*x + 2

    >>> R.dmp_exquo(f, g)
    x

    >>> R.dmp_exquo(f, h)
    Traceback (most recent call last):
    ...
    ExactQuotientFailed: [[2], [2]] does not divide [[1], [1, 0], []]

    N)r   r	   r   r   r   r   r   	dmp_exquo  s    
r   c                 C   s   | s
|j S tt| |S dS )z
    Returns maximum norm of a polynomial in ``K[x]``.

    Examples
    ========

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

    >>> R.dup_max_norm(-x**2 + 2*x - 3)
    3

    N)r   rk   rM   rL   r   r   r   dup_max_norm  s    r   c                    s.   |st |  S |d t fdd| D S )z
    Returns maximum norm of a polynomial in ``K[X]``.

    Examples
    ========

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

    >>> R.dmp_max_norm(2*x*y - x - 3)
    3

    r   c                    s   g | ]}t | qS r   )dmp_max_normr)   r   rO   r   r   r,     r-   z dmp_max_norm.<locals>.<listcomp>)r   rk   rP   r   rO   r   r     s    
r   c                 C   s   | s
|j S tt| |S dS )z
    Returns l1 norm of a polynomial in ``K[x]``.

    Examples
    ========

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

    >>> R.dup_l1_norm(2*x**3 - 3*x**2 + 1)
    6

    N)r   sumrM   rL   r   r   r   dup_l1_norm  s    r   c                    s.   |st |  S |d t fdd| D S )z
    Returns l1 norm of a polynomial in ``K[X]``.

    Examples
    ========

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

    >>> R.dmp_l1_norm(2*x*y - x - 3)
    6

    r   c                    s   g | ]}t | qS r   )dmp_l1_normr   rO   r   r   r,     r-   zdmp_l1_norm.<locals>.<listcomp>)r   r   rP   r   rO   r   r     s    
r   c                 C   s   t dd | D |jS )z
    Returns squared l2 norm of a polynomial in ``K[x]``.

    Examples
    ========

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

    >>> R.dup_l2_norm_squared(2*x**3 - 3*x**2 + 1)
    14

    c                 S   s   g | ]}|d  qS )ri   r   rI   r   r   r   r,     r-   z'dup_l2_norm_squared.<locals>.<listcomp>)r   r   rL   r   r   r   dup_l2_norm_squared  s    r   c                    s.   |st |  S |d t fdd| D S )z
    Returns squared l2 norm of a polynomial in ``K[X]``.

    Examples
    ========

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

    >>> R.dmp_l2_norm_squared(2*x*y - x - 3)
    14

    r   c                    s   g | ]}t | qS r   )dmp_l2_norm_squaredr   rO   r   r   r,   !  r-   z'dmp_l2_norm_squared.<locals>.<listcomp>)r   r   rP   r   rO   r   r     s    
r   c                 C   s6   | s|j gS | d }| dd D ]}t|||}q |S )z
    Multiply together several polynomials in ``K[x]``.

    Examples
    ========

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

    >>> R.dup_expand([x**2 - 1, x, 2])
    2*x**3 - 2*x

    r   r   N)r}   ra   )polysr   r   rW   r   r   r   
dup_expand$  s    r   c                 C   s:   | st ||S | d }| dd D ]}t||||}q"|S )z
    Multiply together several polynomials in ``K[X]``.

    Examples
    ========

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

    >>> R.dmp_expand([x**2 + y**2, x + 1])
    x**3 + x**2 + x*y**2 + y**2

    r   r   N)r   r0   )r   r!   r   r   rW   r   r   r   
dmp_expand=  s    
r   N)R__doc__sympy.polys.densebasicr   r   r   r   r   r   r   r	   r
   r   r   r   r   sympy.polys.polyerrorsr   r   r   r#   r$   r'   r/   r2   r4   r6   r7   r8   r9   r:   r@   rA   rC   rD   rF   rG   rM   rN   rQ   r&   r\   r   r_   r%   rc   re   rf   rg   ra   r0   rj   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   r   r   r   r   r   r   <module>   s   <""#*#*9+(0%(5-931545