# Source code for sympy.polys.densearith

"""Arithmetics for dense recursive polynomials in K[x] or K[X]. """

from sympy.polys.densebasic import (
dup_LC, dmp_LC,
dup_degree, dmp_degree,
dup_normal,
dup_strip, dmp_strip,
dmp_zero_p, dmp_zero,
dmp_one_p, dmp_one,
dmp_ground, dmp_zeros)

from sympy.polys.polyerrors import (
ExactQuotientFailed)

from sympy.utilities import cythonized

@cythonized("i,n,m")
"""
Add c*x**i to f in K[x].

Examples
========

>>> from sympy.polys.domains import ZZ

>>> f = ZZ.map([1, 0, -1])

[2, 0, 1, 0, -1]

"""
if not c:
return f

n = len(f)
m = n-i-1

if i == n-1:
return dup_strip([f[0]+c] + f[1:])
else:
if i >= n:
return [c] + [K.zero]*(i-n) + f
else:
return f[:m] + [f[m]+c] + f[m+1:]

@cythonized("i,u,v,n,m")
[docs]def dmp_add_term(f, c, i, u, K):
"""
Add c(x_2..x_u)*x_0**i to f in K[X].

Examples
========

>>> from sympy.polys.domains import ZZ

>>> f = ZZ.map([[1, 0], [1]])
>>> c = ZZ.map([2])

>>> dmp_add_term(f, c, 2, 1, ZZ)
[[2], [1, 0], [1]]

"""
if not u:

v = u-1

if dmp_zero_p(c, v):
return f

n = len(f)
m = n-i-1

if i == n-1:
return dmp_strip([dmp_add(f[0], c, v, K)] + f[1:], u)
else:
if i >= n:
return [c] + dmp_zeros(i-n, v, K) + f
else:
return f[:m] + [dmp_add(f[m], c, v, K)] + f[m+1:]

@cythonized("i,n,m")
def dup_sub_term(f, c, i, K):
"""
Subtract c*x**i from f in K[x].

Examples
========

>>> from sympy.polys.domains import ZZ
>>> from sympy.polys.densearith import dup_sub_term

>>> f = ZZ.map([2, 0, 1, 0, -1])

>>> dup_sub_term(f, ZZ(2), 4, ZZ)
[1, 0, -1]

"""
if not c:
return f

n = len(f)
m = n-i-1

if i == n-1:
return dup_strip([f[0]-c] + f[1:])
else:
if i >= n:
return [-c] + [K.zero]*(i-n) + f
else:
return f[:m] + [f[m]-c] + f[m+1:]

@cythonized("i,u,v,n,m")
[docs]def dmp_sub_term(f, c, i, u, K):
"""
Subtract c(x_2..x_u)*x_0**i from f in K[X].

Examples
========

>>> from sympy.polys.domains import ZZ
>>> from sympy.polys.densearith import dmp_sub_term

>>> f = ZZ.map([[2], [1, 0], [1]])
>>> c = ZZ.map([2])

>>> dmp_sub_term(f, c, 2, 1, ZZ)
[[1, 0], [1]]

"""
if not u:

v = u-1

if dmp_zero_p(c, v):
return f

n = len(f)
m = n-i-1

if i == n-1:
return dmp_strip([dmp_sub(f[0], c, v, K)] + f[1:], u)
else:
if i >= n:
return [dmp_neg(c, v, K)] + dmp_zeros(i-n, v, K) + f
else:
return f[:m] + [dmp_sub(f[m], c, v, K)] + f[m+1:]

@cythonized("i")
def dup_mul_term(f, c, i, K):
"""
Multiply f by c*x**i in K[x].

Examples
========

>>> from sympy.polys.domains import ZZ
>>> from sympy.polys.densearith import dup_mul_term

>>> f = ZZ.map([1, 0, -1])

>>> dup_mul_term(f, ZZ(3), 2, ZZ)
[3, 0, -3, 0, 0]

"""
if not c or not f:
return []
else:
return [ cf * c for cf in f ] + [K.zero]*i

@cythonized("i,u,v")
[docs]def dmp_mul_term(f, c, i, u, K):
"""
Multiply f by c(x_2..x_u)*x_0**i in K[X].

Examples
========

>>> from sympy.polys.domains import ZZ
>>> from sympy.polys.densearith import dmp_mul_term

>>> f = ZZ.map([[1, 0], [1], []])
>>> c = ZZ.map([3, 0])

>>> dmp_mul_term(f, c, 2, 1, ZZ)
[[3, 0, 0], [3, 0], [], [], []]

"""
if not u:
return dup_mul_term(f, c, i, K)

v = u-1

if dmp_zero_p(f, u):
return f
if dmp_zero_p(c, v):
return dmp_zero(u)
else:
return [ dmp_mul(cf, c, v, K) for cf in f ] + dmp_zeros(i, v, K)

"""
Add an element of the ground domain to f.

Examples
========

>>> from sympy.polys.domains import ZZ

>>> f = ZZ.map([1, 2, 3, 4])

[1, 2, 3, 8]

"""

"""
Add an element of the ground domain to f.

Examples
========

>>> from sympy.polys.domains import ZZ

>>> f = ZZ.map([[1], [2], [3], [4]])

[[1], [2], [3], [8]]

"""
return dmp_add_term(f, dmp_ground(c, u-1), 0, u, K)

def dup_sub_ground(f, c, K):
"""
Subtract an element of the ground domain from f.

Examples
========

>>> from sympy.polys.domains import ZZ
>>> from sympy.polys.densearith import dup_sub_ground

>>> f = ZZ.map([1, 2, 3, 4])

>>> dup_sub_ground(f, ZZ(4), ZZ)
[1, 2, 3, 0]

"""
return dup_sub_term(f, c, 0, K)

[docs]def dmp_sub_ground(f, c, u, K):
"""
Subtract an element of the ground domain from f.

Examples
========

>>> from sympy.polys.domains import ZZ
>>> from sympy.polys.densearith import dmp_sub_ground

>>> f = ZZ.map([[1], [2], [3], [4]])

>>> dmp_sub_ground(f, ZZ(4), 1, ZZ)
[[1], [2], [3], []]

"""
return dmp_sub_term(f, dmp_ground(c, u-1), 0, u, K)

def dup_mul_ground(f, c, K):
"""
Multiply f by a constant value in K[x].

Examples
========

>>> from sympy.polys.domains import ZZ
>>> from sympy.polys.densearith import dup_mul_ground

>>> f = ZZ.map([1, 2, -1])

>>> dup_mul_ground(f, ZZ(3), ZZ)
[3, 6, -3]

"""
if not c or not f:
return []
else:
return [ cf * c for cf in f ]

@cythonized("u,v")
[docs]def dmp_mul_ground(f, c, u, K):
"""
Multiply f by a constant value in K[X].

Examples
========

>>> from sympy.polys.domains import ZZ
>>> from sympy.polys.densearith import dmp_mul_ground

>>> f = ZZ.map([[2], [2, 0]])

>>> dmp_mul_ground(f, ZZ(3), 1, ZZ)
[[6], [6, 0]]

"""
if not u:
return dup_mul_ground(f, c, K)

v = u-1

return [ dmp_mul_ground(cf, c, v, K) for cf in f ]

def dup_quo_ground(f, c, K):
"""
Quotient by a constant in K[x].

Examples
========

>>> from sympy.polys.domains import ZZ, QQ
>>> from sympy.polys.densearith import dup_quo_ground

>>> f = ZZ.map([3, 0, 2])
>>> g = QQ.map([3, 0, 2])

>>> dup_quo_ground(f, ZZ(2), ZZ)
[1, 0, 1]

>>> dup_quo_ground(g, QQ(2), QQ)
[3/2, 0/1, 1/1]

"""
if not c:
raise ZeroDivisionError('polynomial division')
if not f:
return f

if K.has_Field or not K.is_Exact:
return [ K.quo(cf, c) for cf in f ]
else:
return [ cf // c for cf in f ]

@cythonized("u,v")
[docs]def dmp_quo_ground(f, c, u, K):
"""
Quotient by a constant in K[X].

Examples
========

>>> from sympy.polys.domains import ZZ, QQ
>>> from sympy.polys.densearith import dmp_quo_ground

>>> f = ZZ.map([[2, 0], [3], []])
>>> g = QQ.map([[2, 0], [3], []])

>>> dmp_quo_ground(f, ZZ(2), 1, ZZ)
[[1, 0], [1], []]

>>> dmp_quo_ground(g, QQ(2), 1, QQ)
[[1/1, 0/1], [3/2], []]

"""
if not u:
return dup_quo_ground(f, c, K)

v = u-1

return [ dmp_quo_ground(cf, c, v, K) for cf in f ]

def dup_exquo_ground(f, c, K):
"""
Exact quotient by a constant in K[x].

Examples
========

>>> from sympy.polys.domains import ZZ, QQ
>>> from sympy.polys.densearith import dup_exquo_ground

>>> f = QQ.map([1, 0, 2])

>>> dup_exquo_ground(f, QQ(2), QQ)
[1/2, 0/1, 1/1]

"""
if not c:
raise ZeroDivisionError('polynomial division')
if not f:
return f

return [ K.exquo(cf, c) for cf in f ]

@cythonized("u,v")
[docs]def dmp_exquo_ground(f, c, u, K):
"""
Exact quotient by a constant in K[X].

Examples
========

>>> from sympy.polys.domains import ZZ, QQ
>>> from sympy.polys.densearith import dmp_exquo_ground

>>> f = QQ.map([[1, 0], [2], []])

>>> dmp_exquo_ground(f, QQ(2), 1, QQ)
[[1/2, 0/1], [1/1], []]

"""
if not u:
return dup_exquo_ground(f, c, K)

v = u-1

return [ dmp_exquo_ground(cf, c, v, K) for cf in f ]

@cythonized("n")
[docs]def dup_lshift(f, n, K):
"""
Efficiently multiply f by x**n in K[x].

Examples
========

>>> from sympy.polys.domains import ZZ
>>> from sympy.polys.densearith import dup_lshift

>>> f = ZZ.map([1, 0, 1])

>>> dup_lshift(f, 2, ZZ)
[1, 0, 1, 0, 0]

"""
if not f:
return f
else:
return f + [K.zero]*n

@cythonized("n")
[docs]def dup_rshift(f, n, K):
"""
Efficiently divide f by x**n in K[x].

Examples
========

>>> from sympy.polys.domains import ZZ
>>> from sympy.polys.densearith import dup_rshift

>>> f = ZZ.map([1, 0, 1, 0, 0])
>>> g = ZZ.map([1, 0, 1, 0, 2])

>>> dup_rshift(f, 2, ZZ)
[1, 0, 1]

>>> dup_rshift(g, 2, ZZ)
[1, 0, 1]

"""
return f[:-n]

def dup_abs(f, K):
"""
Make all coefficients positive in K[x].

Examples
========

>>> from sympy.polys.domains import ZZ
>>> from sympy.polys.densearith import dup_abs

>>> f = ZZ.map([1, 0, -1])

>>> dup_abs(f, ZZ)
[1, 0, 1]

"""
return [ K.abs(coeff) for coeff in f ]

@cythonized("u,v")
[docs]def dmp_abs(f, u, K):
"""
Make all coefficients positive in K[X].

Examples
========

>>> from sympy.polys.domains import ZZ
>>> from sympy.polys.densearith import dmp_abs

>>> f = ZZ.map([[1, 0], [-1], []])

>>> dmp_abs(f, 1, ZZ)
[[1, 0], [1], []]

"""
if not u:
return dup_abs(f, K)

v = u-1

return [ dmp_abs(cf, v, K) for cf in f ]

def dup_neg(f, K):
"""
Negate a polynomial in K[x].

Examples
========

>>> from sympy.polys.domains import ZZ
>>> from sympy.polys.densearith import dup_neg

>>> f = ZZ.map([1, 0, -1])

>>> dup_neg(f, ZZ)
[-1, 0, 1]

"""
return [ -coeff for coeff in f ]

@cythonized("u,v")
[docs]def dmp_neg(f, u, K):
"""
Negate a polynomial in K[X].

Examples
========

>>> from sympy.polys.domains import ZZ
>>> from sympy.polys.densearith import dmp_neg

>>> f = ZZ.map([[1, 0], [-1], []])

>>> dmp_neg(f, 1, ZZ)
[[-1, 0], [1], []]

"""
if not u:
return dup_neg(f, K)

v = u-1

return [ dmp_neg(cf, v, K) for cf in f ]

@cythonized("df,dg,k")
"""
Add dense polynomials in K[x].

Examples
========

>>> from sympy.polys.domains import ZZ

>>> f = ZZ.map([1, 0, -1])
>>> g = ZZ.map([1, -2])

[1, 1, -3]

"""
if not f:
return g
if not g:
return f

df = dup_degree(f)
dg = dup_degree(g)

if df == dg:
return dup_strip([ a + b for a, b in zip(f, g) ])
else:
k = abs(df - dg)

if df > dg:
h, f = f[:k], f[k:]
else:
h, g = g[:k], g[k:]

return h + [ a + b for a, b in zip(f, g) ]

@cythonized("u,v,df,dg,k")
"""
Add dense polynomials in K[X].

Examples
========

>>> from sympy.polys.domains import ZZ

>>> f = ZZ.map([[1], [], [1, 0]])
>>> g = ZZ.map([[1, 0], [1], []])

[[1, 1], [1], [1, 0]]

"""
if not u:

df = dmp_degree(f, u)

if df < 0:
return g

dg = dmp_degree(g, u)

if dg < 0:
return f

v = u-1

if df == dg:
return dmp_strip([ dmp_add(a, b, v, K) for a, b in zip(f, g) ], u)
else:
k = abs(df - dg)

if df > dg:
h, f = f[:k], f[k:]
else:
h, g = g[:k], g[k:]

return h + [ dmp_add(a, b, v, K) for a, b in zip(f, g) ]

@cythonized("df,dg,k")
def dup_sub(f, g, K):
"""
Subtract dense polynomials in K[x].

Examples
========

>>> from sympy.polys.domains import ZZ
>>> from sympy.polys.densearith import dup_sub

>>> f = ZZ.map([1, 0, -1])
>>> g = ZZ.map([1, -2])

>>> dup_sub(f, g, ZZ)
[1, -1, 1]

"""
if not f:
return dup_neg(g, K)
if not g:
return f

df = dup_degree(f)
dg = dup_degree(g)

if df == dg:
return dup_strip([ a - b for a, b in zip(f, g) ])
else:
k = abs(df - dg)

if df > dg:
h, f = f[:k], f[k:]
else:
h, g = dup_neg(g[:k], K), g[k:]

return h + [ a - b for a, b in zip(f, g) ]

@cythonized("u,v,df,dg,k")
[docs]def dmp_sub(f, g, u, K):
"""
Subtract dense polynomials in K[X].

Examples
========

>>> from sympy.polys.domains import ZZ
>>> from sympy.polys.densearith import dmp_sub

>>> f = ZZ.map([[1], [], [1, 0]])
>>> g = ZZ.map([[1, 0], [1], []])

>>> dmp_sub(f, g, 1, ZZ)
[[-1, 1], [-1], [1, 0]]

"""
if not u:
return dup_sub(f, g, K)

df = dmp_degree(f, u)

if df < 0:
return dmp_neg(g, u, K)

dg = dmp_degree(g, u)

if dg < 0:
return f

v = u-1

if df == dg:
return dmp_strip([ dmp_sub(a, b, v, K) for a, b in zip(f, g) ], u)
else:
k = abs(df - dg)

if df > dg:
h, f = f[:k], f[k:]
else:
h, g = dmp_neg(g[:k], u, K), g[k:]

return h + [ dmp_sub(a, b, v, K) for a, b in zip(f, g) ]

"""
Returns f + g*h where f, g, h are in K[x].

Examples
========

>>> from sympy.polys.domains import ZZ

>>> f = ZZ.map([1, 0, -1])
>>> g = ZZ.map([1, -2])
>>> h = ZZ.map([1, 2])

[2, 0, -5]

"""
return dup_add(f, dup_mul(g, h, K), K)

@cythonized("u")
[docs]def dmp_add_mul(f, g, h, u, K):
"""
Returns f + g*h where f, g, h are in K[X].

Examples
========

>>> from sympy.polys.domains import ZZ

>>> f = ZZ.map([[1], [], [1, 0]])
>>> g = ZZ.map([[1], []])
>>> h = ZZ.map([[1], [2]])

>>> dmp_add_mul(f, g, h, 1, ZZ)
[[2], [2], [1, 0]]

"""
return dmp_add(f, dmp_mul(g, h, u, K), u, K)

def dup_sub_mul(f, g, h, K):
"""
Returns f - g*h where f, g, h are in K[x].

Examples
========

>>> from sympy.polys.domains import ZZ
>>> from sympy.polys.densearith import dup_sub_mul

>>> f = ZZ.map([1, 0, -1])
>>> g = ZZ.map([1, -2])
>>> h = ZZ.map([1, 2])

>>> dup_sub_mul(f, g, h, ZZ)
[3]

"""
return dup_sub(f, dup_mul(g, h, K), K)

@cythonized("u")
[docs]def dmp_sub_mul(f, g, h, u, K):
"""
Returns f - g*h where f, g, h are in K[X].

Examples
========

>>> from sympy.polys.domains import ZZ
>>> from sympy.polys.densearith import dmp_sub_mul

>>> f = ZZ.map([[1], [], [1, 0]])
>>> g = ZZ.map([[1], []])
>>> h = ZZ.map([[1], [2]])

>>> dmp_sub_mul(f, g, h, 1, ZZ)
[[-2], [1, 0]]

"""
return dmp_sub(f, dmp_mul(g, h, u, K), u, K)

@cythonized("df,dg,i,j")
def dup_mul(f, g, K):
"""
Multiply dense polynomials in K[x].

Examples
========

>>> from sympy.polys.domains import ZZ
>>> from sympy.polys.densearith import dup_mul

>>> f = ZZ.map([1, -2])
>>> g = ZZ.map([1, 2])

>>> dup_mul(f, g, ZZ)
[1, 0, -4]

"""
if f == g:
return dup_sqr(f, K)

if not (f and g):
return []

df = dup_degree(f)
dg = dup_degree(g)

h = []

for i in xrange(0, df+dg+1):
coeff = K.zero

for j in xrange(max(0, i-dg), min(df, i)+1):
coeff += f[j]*g[i-j]

h.append(coeff)

return dup_strip(h)

@cythonized("u,v,df,dg,i,j")
[docs]def dmp_mul(f, g, u, K):
"""
Multiply dense polynomials in K[X].

Examples
========

>>> from sympy.polys.domains import ZZ
>>> from sympy.polys.densearith import dmp_mul

>>> f = ZZ.map([[1, 0], [1]])
>>> g = ZZ.map([[1], []])

>>> dmp_mul(f, g, 1, ZZ)
[[1, 0], [1], []]

"""
if not u:
return dup_mul(f, g, K)

if f == g:
return dmp_sqr(f, u, K)

df = dmp_degree(f, u)

if df < 0:
return f

dg = dmp_degree(g, u)

if dg < 0:
return g

h, v = [], u-1

for i in xrange(0, df+dg+1):
coeff = dmp_zero(v)

for j in xrange(max(0, i-dg), min(df, i)+1):
coeff = dmp_add(coeff, dmp_mul(f[j], g[i-j], v, K), v, K)

h.append(coeff)

return dmp_strip(h, u)

@cythonized("df,jmin,jmax,n,i,j")
def dup_sqr(f, K):
"""
Square dense polynomials in K[x].

Examples
========

>>> from sympy.polys.domains import ZZ
>>> from sympy.polys.densearith import dup_sqr

>>> f = ZZ.map([1, 0, 1])

>>> dup_sqr(f, ZZ)
[1, 0, 2, 0, 1]

"""
df, h = dup_degree(f), []

for i in xrange(0, 2*df+1):
c = K.zero

jmin = max(0, i-df)
jmax = min(i, df)

n = jmax - jmin + 1

jmax = jmin + n // 2 - 1

for j in xrange(jmin, jmax+1):
c += f[j]*f[i-j]

c += c

if n & 1:
elem = f[jmax+1]
c += elem**2

h.append(c)

return dup_strip(h)

@cythonized("u,v,df,jmin,jmax,n,i,j")
[docs]def dmp_sqr(f, u, K):
"""
Square dense polynomials in K[X].

Examples
========

>>> from sympy.polys.domains import ZZ
>>> from sympy.polys.densearith import dmp_sqr

>>> f = ZZ.map([[1], [1, 0], [1, 0, 0]])

>>> dmp_sqr(f, 1, ZZ)
[[1], [2, 0], [3, 0, 0], [2, 0, 0, 0], [1, 0, 0, 0, 0]]

"""
if not u:
return dup_sqr(f, K)

df = dmp_degree(f, u)

if df < 0:
return f

h, v = [], u-1

for i in xrange(0, 2*df+1):
c = dmp_zero(v)

jmin = max(0, i-df)
jmax = min(i, df)

n = jmax - jmin + 1

jmax = jmin + n // 2 - 1

for j in xrange(jmin, jmax+1):
c = dmp_add(c, dmp_mul(f[j], f[i-j], v, K), v, K)

c = dmp_mul_ground(c, K(2), v, K)

if n & 1:
elem = dmp_sqr(f[jmax+1], v, K)
c = dmp_add(c, elem, v, K)

h.append(c)

return dmp_strip(h, u)

@cythonized("n,m")
def dup_pow(f, n, K):
"""
Raise f to the n-th power in K[x].

Examples
========

>>> from sympy.polys.domains import ZZ
>>> from sympy.polys.densearith import dup_pow

>>> dup_pow([ZZ(1), -ZZ(2)], 3, ZZ)
[1, -6, 12, -8]

"""
if not n:
return [K.one]
if n < 0:
raise ValueError("can't raise polynomial to a negative power")
if n == 1 or not f or f == [K.one]:
return f

g = [K.one]

while True:
n, m = n//2, n

if m % 2:
g = dup_mul(g, f, K)

if not n:
break

f = dup_sqr(f, K)

return g

@cythonized("u,n,m")
[docs]def dmp_pow(f, n, u, K):
"""
Raise f to the n-th power in K[X].

Examples
========

>>> from sympy.polys.domains import ZZ
>>> from sympy.polys.densearith import dmp_pow

>>> f = ZZ.map([[1, 0], [1]])

>>> dmp_pow(f, 3, 1, ZZ)
[[1, 0, 0, 0], [3, 0, 0], [3, 0], [1]]

"""
if not u:
return dup_pow(f, n, K)

if not n:
return dmp_one(u, K)
if n < 0:
raise ValueError("can't raise polynomial to a negative power")
if n == 1 or dmp_zero_p(f, u) or dmp_one_p(f, u, K):
return f

g = dmp_one(u, K)

while True:
n, m = n//2, n

if m & 1:
g = dmp_mul(g, f, u, K)

if not n:
break

f = dmp_sqr(f, u, K)

return g

@cythonized("df,dg,dr,N,j")
def dup_pdiv(f, g, K):
"""
Polynomial pseudo-division in K[x].

Examples
========

>>> from sympy.polys.domains import ZZ
>>> from sympy.polys.densearith import dup_pdiv

>>> f = ZZ.map([1, 0, 1])
>>> g = ZZ.map([2, -4])

>>> dup_pdiv(f, g, ZZ)
([2, 4], [20])

"""
df = dup_degree(f)
dg = dup_degree(g)

q, r = [], f

if not g:
raise ZeroDivisionError("polynomial division")
elif df < dg:
return q, r

N = df - dg + 1
lc_g = dup_LC(g, K)

while True:
dr = dup_degree(r)

if dr < dg:
break

lc_r = dup_LC(r, K)
j, N = dr-dg, N-1

Q = dup_mul_ground(q, lc_g, K)
q = dup_add_term(Q, lc_r, j, K)

R = dup_mul_ground(r, lc_g, K)
G = dup_mul_term(g, lc_r, j, K)
r = dup_sub(R, G, K)

c = lc_g**N

q = dup_mul_ground(q, c, K)
r = dup_mul_ground(r, c, K)

return q, r

@cythonized("df,dg,dr,N,j")
def dup_prem(f, g, K):
"""
Polynomial pseudo-remainder in K[x].

Examples
========

>>> from sympy.polys.domains import ZZ
>>> from sympy.polys.densearith import dup_prem

>>> f = ZZ.map([1, 0, 1])
>>> g = ZZ.map([2, -4])

>>> dup_prem(f, g, ZZ)
[20]

"""
df = dup_degree(f)
dg = dup_degree(g)

r = f

if not g:
raise ZeroDivisionError("polynomial division")
elif df < dg:
return r

N = df - dg + 1
lc_g = dup_LC(g, K)

while True:
dr = dup_degree(r)

if dr < dg:
break

lc_r = dup_LC(r, K)
j, N = dr-dg, N-1

R = dup_mul_ground(r, lc_g, K)
G = dup_mul_term(g, lc_r, j, K)
r = dup_sub(R, G, K)

return dup_mul_ground(r, lc_g**N, K)

def dup_pquo(f, g, K):
"""
Polynomial exact pseudo-quotient in K[X].

Examples
========

>>> from sympy.polys.domains import ZZ
>>> from sympy.polys.densearith import dup_pquo

>>> f = ZZ.map([1, 0, -1])
>>> g = ZZ.map([2, -2])

>>> dup_pquo(f, g, ZZ)
[2, 2]

>>> f = ZZ.map([1, 0, 1])
>>> g = ZZ.map([2, -4])

>>> dup_pquo(f, g, ZZ)
[2, 4]

"""
return dup_pdiv(f, g, K)[0]

def dup_pexquo(f, g, K):
"""
Polynomial pseudo-quotient in K[x].

Examples
========

>>> from sympy.polys.domains import ZZ
>>> from sympy.polys.densearith import dup_pexquo

>>> f = ZZ.map([1, 0, -1])
>>> g = ZZ.map([2, -2])

>>> dup_pexquo(f, g, ZZ)
[2, 2]

>>> f = ZZ.map([1, 0, 1])
>>> g = ZZ.map([2, -4])

>>> dup_pexquo(f, g, ZZ)
Traceback (most recent call last):
...
ExactQuotientFailed: [2, -4] does not divide [1, 0, 1]

"""
q, r = dup_pdiv(f, g, K)

if not r:
return q
else:
raise ExactQuotientFailed(f, g)

@cythonized("u,df,dg,dr,N,j")
[docs]def dmp_pdiv(f, g, u, K):
"""
Polynomial pseudo-division in K[X].

Examples
========

>>> from sympy.polys.domains import ZZ
>>> from sympy.polys.densearith import dmp_pdiv

>>> f = ZZ.map([[1], [1, 0], []])
>>> g = ZZ.map([[2], [2]])

>>> dmp_pdiv(f, g, 1, ZZ)
([[2], [2, -2]], [[-4, 4]])

"""
if not u:
return dup_pdiv(f, g, K)

df = dmp_degree(f, u)
dg = dmp_degree(g, u)

if dg < 0:
raise ZeroDivisionError("polynomial division")

q, r = dmp_zero(u), f

if df < dg:
return q, r

N = df - dg + 1
lc_g = dmp_LC(g, K)

while True:
dr = dmp_degree(r, u)

if dr < dg:
break

lc_r = dmp_LC(r, K)
j, N = dr-dg, N-1

Q = dmp_mul_term(q, lc_g, 0, u, K)
q = dmp_add_term(Q, lc_r, j, u, K)

R = dmp_mul_term(r, lc_g, 0, u, K)
G = dmp_mul_term(g, lc_r, j, u, K)
r = dmp_sub(R, G, u, K)

c = dmp_pow(lc_g, N, u-1, K)

q = dmp_mul_term(q, c, 0, u, K)
r = dmp_mul_term(r, c, 0, u, K)

return q, r

@cythonized("u,df,dg,dr,N,j")
[docs]def dmp_prem(f, g, u, K):
"""
Polynomial pseudo-remainder in K[X].

Examples
========

>>> from sympy.polys.domains import ZZ
>>> from sympy.polys.densearith import dmp_prem

>>> f = ZZ.map([[1], [1, 0], []])
>>> g = ZZ.map([[2], [2]])

>>> dmp_prem(f, g, 1, ZZ)
[[-4, 4]]

"""
if not u:
return dup_prem(f, g, K)

df = dmp_degree(f, u)
dg = dmp_degree(g, u)

if dg < 0:
raise ZeroDivisionError("polynomial division")

r = f

if df < dg:
return r

N = df - dg + 1
lc_g = dmp_LC(g, K)

while True:
dr = dmp_degree(r, u)

if dr < dg:
break

lc_r = dmp_LC(r, K)
j, N = dr-dg, N-1

R = dmp_mul_term(r, lc_g, 0, u, K)
G = dmp_mul_term(g, lc_r, j, u, K)
r = dmp_sub(R, G, u, K)

c = dmp_pow(lc_g, N, u-1, K)

return dmp_mul_term(r, c, 0, u, K)

[docs]def dmp_pquo(f, g, u, K):
"""
Polynomial exact pseudo-quotient in K[X].

Examples
========

>>> from sympy.polys.domains import ZZ
>>> from sympy.polys.densearith import dmp_pquo

>>> f = ZZ.map([[1], [1, 0], []])
>>> g = ZZ.map([[2], [2, 0]])
>>> h = ZZ.map([[2], [2]])

>>> dmp_pquo(f, g, 1, ZZ)
[[2], []]

>>> dmp_pquo(f, h, 1, ZZ)
[[2], [2, -2]]

"""
return dmp_pdiv(f, g, u, K)[0]

[docs]def dmp_pexquo(f, g, u, K):
"""
Polynomial pseudo-quotient in K[X].

Examples
========

>>> from sympy.polys.domains import ZZ
>>> from sympy.polys.densearith import dmp_pexquo

>>> f = ZZ.map([[1], [1, 0], []])
>>> g = ZZ.map([[2], [2, 0]])
>>> h = ZZ.map([[2], [2]])

>>> dmp_pexquo(f, g, 1, ZZ)
[[2], []]

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

"""
q, r = dmp_pdiv(f, g, u, K)

if dmp_zero_p(r, u):
return q
else:
raise ExactQuotientFailed(f, g)

@cythonized("df,dg,dr,j")
def dup_rr_div(f, g, K):
"""
Univariate division with remainder over a ring.

Examples
========

>>> from sympy.polys.domains import ZZ
>>> from sympy.polys.densearith import dup_rr_div

>>> f = ZZ.map([1, 0, 1])
>>> g = ZZ.map([2, -4])

>>> dup_rr_div(f, g, ZZ)
([], [1, 0, 1])

"""
df = dup_degree(f)
dg = dup_degree(g)

q, r = [], f

if not g:
raise ZeroDivisionError("polynomial division")
elif df < dg:
return q, r

lc_g = dup_LC(g, K)

while True:
dr = dup_degree(r)

if dr < dg:
break

lc_r = dup_LC(r, K)

if lc_r % lc_g:
break

c = K.exquo(lc_r, lc_g)
j = dr - dg

q = dup_add_term(q, c, j, K)
h = dup_mul_term(g, c, j, K)

r = dup_sub(r, h, K)

return q, r

@cythonized("u,df,dg,dr,j")
[docs]def dmp_rr_div(f, g, u, K):
"""
Multivariate division with remainder over a ring.

Examples
========

>>> from sympy.polys.domains import ZZ
>>> from sympy.polys.densearith import dmp_rr_div

>>> f = ZZ.map([[1], [1, 0], []])
>>> g = ZZ.map([[2], [2]])

>>> dmp_rr_div(f, g, 1, ZZ)
([[]], [[1], [1, 0], []])

"""
if not u:
return dup_rr_div(f, g, K)

df = dmp_degree(f, u)
dg = dmp_degree(g, u)

if dg < 0:
raise ZeroDivisionError("polynomial division")

q, r = dmp_zero(u), f

if df < dg:
return q, r

lc_g, v = dmp_LC(g, K), u-1

while True:
dr = dmp_degree(r, u)

if dr < dg:
break

lc_r = dmp_LC(r, K)

c, R = dmp_rr_div(lc_r, lc_g, v, K)

if not dmp_zero_p(R, v):
break

j = dr - dg

q = dmp_add_term(q, c, j, u, K)
h = dmp_mul_term(g, c, j, u, K)

r = dmp_sub(r, h, u, K)

return q, r

@cythonized("df,dg,dr,j")
def dup_ff_div(f, g, K):
"""
Polynomial division with remainder over a field.

Examples
========

>>> from sympy.polys.domains import QQ
>>> from sympy.polys.densearith import dup_ff_div

>>> f = QQ.map([1, 0, 1])
>>> g = QQ.map([2, -4])

>>> dup_ff_div(f, g, QQ)
([1/2, 1/1], [5/1])

"""
df = dup_degree(f)
dg = dup_degree(g)

q, r = [], f

if not g:
raise ZeroDivisionError("polynomial division")
elif df < dg:
return q, r

lc_g = dup_LC(g, K)

while True:
dr = dup_degree(r)

if dr < dg:
break

lc_r = dup_LC(r, K)

c = K.exquo(lc_r, lc_g)
j = dr - dg

q = dup_add_term(q, c, j, K)
h = dup_mul_term(g, c, j, K)

r = dup_sub(r, h, K)

if not K.is_Exact:
r = dup_normal(r, K)

return q, r

@cythonized("u,df,dg,dr,j")
[docs]def dmp_ff_div(f, g, u, K):
"""
Polynomial division with remainder over a field.

Examples
========

>>> from sympy.polys.domains import QQ
>>> from sympy.polys.densearith import dmp_ff_div

>>> f = QQ.map([[1], [1, 0], []])
>>> g = QQ.map([[2], [2]])

>>> dmp_ff_div(f, g, 1, QQ)
([[1/2], [1/2, -1/2]], [[-1/1, 1/1]])

"""
if not u:
return dup_ff_div(f, g, K)

df = dmp_degree(f, u)
dg = dmp_degree(g, u)

if dg < 0:
raise ZeroDivisionError("polynomial division")

q, r = dmp_zero(u), f

if df < dg:
return q, r

lc_g, v = dmp_LC(g, K), u-1

while True:
dr = dmp_degree(r, u)

if dr < dg:
break

lc_r = dmp_LC(r, K)

c, R = dmp_ff_div(lc_r, lc_g, v, K)

if not dmp_zero_p(R, v):
break

j = dr - dg

q = dmp_add_term(q, c, j, u, K)
h = dmp_mul_term(g, c, j, u, K)

r = dmp_sub(r, h, u, K)

return q, r

def dup_div(f, g, K):
"""
Polynomial division with remainder in K[x].

Examples
========

>>> from sympy.polys.domains import ZZ, QQ
>>> from sympy.polys.densearith import dup_div

>>> f = ZZ.map([1, 0, 1])
>>> g = ZZ.map([2, -4])

>>> dup_div(f, g, ZZ)
([], [1, 0, 1])

>>> f = QQ.map([1, 0, 1])
>>> g = QQ.map([2, -4])

>>> dup_div(f, g, QQ)
([1/2, 1/1], [5/1])

"""
if K.has_Field or not K.is_Exact:
return dup_ff_div(f, g, K)
else:
return dup_rr_div(f, g, K)

def dup_rem(f, g, K):
"""
Returns polynomial remainder in K[x].

Examples
========

>>> from sympy.polys.domains import ZZ, QQ
>>> from sympy.polys.densearith import dup_rem

>>> f = ZZ.map([1, 0, 1])
>>> g = ZZ.map([2, -4])

>>> dup_rem(f, g, ZZ)
[1, 0, 1]

>>> f = QQ.map([1, 0, 1])
>>> g = QQ.map([2, -4])

>>> dup_rem(f, g, QQ)
[5/1]

"""
return dup_div(f, g, K)[1]

def dup_quo(f, g, K):
"""
Returns exact polynomial quotient in K[x].

Examples
========

>>> from sympy.polys.domains import ZZ, QQ
>>> from sympy.polys.densearith import dup_quo

>>> f = ZZ.map([1, 0, 1])
>>> g = ZZ.map([2, -4])

>>> dup_quo(f, g, ZZ)
[]

>>> f = QQ.map([1, 0, 1])
>>> g = QQ.map([2, -4])

>>> dup_quo(f, g, QQ)
[1/2, 1/1]

"""
return dup_div(f, g, K)[0]

def dup_exquo(f, g, K):
"""
Returns polynomial quotient in K[x].

Examples
========

>>> from sympy.polys.domains import ZZ
>>> from sympy.polys.densearith import dup_exquo

>>> f = ZZ.map([1, 0, -1])
>>> g = ZZ.map([1, -1])

>>> dup_exquo(f, g, ZZ)
[1, 1]

>>> f = ZZ.map([1, 0, 1])
>>> g = ZZ.map([2, -4])

>>> dup_exquo(f, g, ZZ)
Traceback (most recent call last):
...
ExactQuotientFailed: [2, -4] does not divide [1, 0, 1]

"""
q, r = dup_div(f, g, K)

if not r:
return q
else:
raise ExactQuotientFailed(f, g)

@cythonized("u")
[docs]def dmp_div(f, g, u, K):
"""
Polynomial division with remainder in K[X].

Examples
========

>>> from sympy.polys.domains import ZZ, QQ
>>> from sympy.polys.densearith import dmp_div

>>> f = ZZ.map([[1], [1, 0], []])
>>> g = ZZ.map([[2], [2]])

>>> dmp_div(f, g, 1, ZZ)
([[]], [[1], [1, 0], []])

>>> f = QQ.map([[1], [1, 0], []])
>>> g = QQ.map([[2], [2]])

>>> dmp_div(f, g, 1, QQ)
([[1/2], [1/2, -1/2]], [[-1/1, 1/1]])

"""
if K.has_Field or not K.is_Exact:
return dmp_ff_div(f, g, u, K)
else:
return dmp_rr_div(f, g, u, K)

@cythonized("u")
[docs]def dmp_rem(f, g, u, K):
"""
Returns polynomial remainder in K[X].

Examples
========

>>> from sympy.polys.domains import ZZ, QQ
>>> from sympy.polys.densearith import dmp_rem

>>> f = ZZ.map([[1], [1, 0], []])
>>> g = ZZ.map([[2], [2]])

>>> dmp_rem(f, g, 1, ZZ)
[[1], [1, 0], []]

>>> f = QQ.map([[1], [1, 0], []])
>>> g = QQ.map([[2], [2]])

>>> dmp_rem(f, g, 1, QQ)
[[-1/1, 1/1]]

"""
return dmp_div(f, g, u, K)[1]

@cythonized("u")
[docs]def dmp_quo(f, g, u, K):
"""
Returns exact polynomial quotient in K[X].

Examples
========

>>> from sympy.polys.domains import ZZ, QQ
>>> from sympy.polys.densearith import dmp_quo

>>> f = ZZ.map([[1], [1, 0], []])
>>> g = ZZ.map([[2], [2]])

>>> dmp_quo(f, g, 1, ZZ)
[[]]

>>> f = QQ.map([[1], [1, 0], []])
>>> g = QQ.map([[2], [2]])

>>> dmp_quo(f, g, 1, QQ)
[[1/2], [1/2, -1/2]]

"""
return dmp_div(f, g, u, K)[0]

@cythonized("u")
[docs]def dmp_exquo(f, g, u, K):
"""
Returns polynomial quotient in K[X].

Examples
========

>>> from sympy.polys.domains import ZZ
>>> from sympy.polys.densearith import dmp_exquo

>>> f = ZZ.map([[1], [1, 0], []])
>>> g = ZZ.map([[1], [1, 0]])
>>> h = ZZ.map([[2], [2]])

>>> dmp_exquo(f, g, 1, ZZ)
[[1], []]

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

"""
q, r = dmp_div(f, g, u, K)

if dmp_zero_p(r, u):
return q
else:
raise ExactQuotientFailed(f, g)

def dup_max_norm(f, K):
"""
Returns maximum norm of a polynomial in K[x].

Examples
========

>>> from sympy.polys.domains import ZZ
>>> from sympy.polys.densearith import dup_max_norm

>>> f = ZZ.map([-1, 2, -3])

>>> dup_max_norm(f, ZZ)
3

"""
if not f:
return K.zero
else:
return max(dup_abs(f, K))

@cythonized("u,v")
[docs]def dmp_max_norm(f, u, K):
"""
Returns maximum norm of a polynomial in K[X].

Examples
========

>>> from sympy.polys.domains import ZZ
>>> from sympy.polys.densearith import dmp_max_norm

>>> f = ZZ.map([[2, -1], [-3]])

>>> dmp_max_norm(f, 1, ZZ)
3

"""
if not u:
return dup_max_norm(f, K)

v = u-1

return max([ dmp_max_norm(c, v, K) for c in f ])

def dup_l1_norm(f, K):
"""
Returns l1 norm of a polynomial in K[x].

Examples
========

>>> from sympy.polys.domains import ZZ
>>> from sympy.polys.densearith import dup_l1_norm

>>> f = ZZ.map([2, -3, 0, 1])

>>> dup_l1_norm(f, ZZ)
6

"""
if not f:
return K.zero
else:
return sum(dup_abs(f, K))

@cythonized("u,v")
[docs]def dmp_l1_norm(f, u, K):
"""
Returns l1 norm of a polynomial in K[X].

Examples
========

>>> from sympy.polys.domains import ZZ
>>> from sympy.polys.densearith import dmp_l1_norm

>>> f = ZZ.map([[2, -1], [-3]])

>>> dmp_l1_norm(f, 1, ZZ)
6

"""
if not u:
return dup_l1_norm(f, K)

v = u-1

return sum([ dmp_l1_norm(c, v, K) for c in f ])

def dup_expand(polys, K):
"""
Multiply together several polynomials in K[x].

Examples
========

>>> from sympy.polys.domains import ZZ
>>> from sympy.polys.densearith import dup_expand

>>> f = ZZ.map([1, 0, -1])
>>> g = ZZ.map([1, 0])
>>> h = ZZ.map([2])

>>> dup_expand([f, g, h], ZZ)
[2, 0, -2, 0]

"""
if not polys:
return [K.one]

f = polys[0]

for g in polys[1:]:
f = dup_mul(f, g, K)

return f

@cythonized("u")
[docs]def dmp_expand(polys, u, K):
"""
Multiply together several polynomials in K[X].

Examples
========

>>> from sympy.polys.domains import ZZ
>>> from sympy.polys.densearith import dmp_expand

>>> f = ZZ.map([[1], [], [1, 0, 0]])
>>> g = ZZ.map([[1], [1]])

>>> dmp_expand([f, g], 1, ZZ)
[[1], [1], [1, 0, 0], [1, 0, 0]]

"""
if not polys:
return dmp_one(u, K)

f = polys[0]

for g in polys[1:]:
f = dmp_mul(f, g, u, K)

return f