"""Arithmetics for dense recursive polynomials in ``K[x]`` or ``K[X]``. """
from __future__ import print_function, division
from sympy.core.compatibility import range
from sympy.polys.densebasic import (
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)
from sympy.polys.polyerrors import (ExactQuotientFailed, PolynomialDivisionFailed)
def dup_add_term(f, c, i, K):
"""
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
"""
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:]
[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 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
"""
if not u:
return dup_add_term(f, c, i, K)
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:]
def dup_sub_term(f, c, i, K):
"""
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
"""
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:]
[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 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
"""
if not u:
return dup_add_term(f, -c, i, K)
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:]
def dup_mul_term(f, c, i, K):
"""
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
"""
if not c or not f:
return []
else:
return [ cf * c for cf in f ] + [K.zero]*i
[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 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
"""
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)
def dup_add_ground(f, c, K):
"""
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
"""
return dup_add_term(f, c, 0, K)
[docs]def dmp_add_ground(f, c, u, K):
"""
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
"""
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 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
"""
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 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
"""
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 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
"""
if not c or not f:
return []
else:
return [ cf * c for cf in f ]
[docs]def dmp_mul_ground(f, c, u, K):
"""
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
"""
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 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
"""
if not c:
raise ZeroDivisionError('polynomial division')
if not f:
return f
if K.is_Field:
return [ K.quo(cf, c) for cf in f ]
else:
return [ cf // c for cf in f ]
[docs]def dmp_quo_ground(f, c, u, K):
"""
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
"""
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 import ring, QQ
>>> R, x = ring("x", QQ)
>>> R.dup_exquo_ground(x**2 + 2, QQ(2))
1/2*x**2 + 1
"""
if not c:
raise ZeroDivisionError('polynomial division')
if not f:
return f
return [ K.exquo(cf, c) for cf in f ]
[docs]def dmp_exquo_ground(f, c, u, K):
"""
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
"""
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 ]
[docs]def dup_lshift(f, n, K):
"""
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
"""
if not f:
return f
else:
return f + [K.zero]*n
[docs]def dup_rshift(f, n, K):
"""
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
"""
return f[:-n]
def dup_abs(f, K):
"""
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
"""
return [ K.abs(coeff) for coeff in f ]
[docs]def dmp_abs(f, u, K):
"""
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
"""
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 import ring, ZZ
>>> R, x = ring("x", ZZ)
>>> R.dup_neg(x**2 - 1)
-x**2 + 1
"""
return [ -coeff for coeff in f ]
[docs]def dmp_neg(f, u, K):
"""
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
"""
if not u:
return dup_neg(f, K)
v = u - 1
return [ dmp_neg(cf, v, K) for cf in f ]
def dup_add(f, g, K):
"""
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
"""
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) ]
[docs]def dmp_add(f, g, u, K):
"""
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
"""
if not u:
return dup_add(f, g, K)
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) ]
def dup_sub(f, g, K):
"""
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
"""
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) ]
[docs]def dmp_sub(f, g, u, K):
"""
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
"""
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) ]
def dup_add_mul(f, g, h, K):
"""
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
"""
return dup_add(f, dup_mul(g, h, K), K)
[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 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
"""
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 import ring, ZZ
>>> R, x = ring("x", ZZ)
>>> R.dup_sub_mul(x**2 - 1, x - 2, x + 2)
3
"""
return dup_sub(f, dup_mul(g, h, K), K)
[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 import ring, ZZ
>>> R, x,y = ring("x,y", ZZ)
>>> R.dmp_sub_mul(x**2 + y, x, x + 2)
-2*x + y
"""
return dmp_sub(f, dmp_mul(g, h, u, K), u, K)
def dup_mul(f, g, K):
"""
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
"""
if f == g:
return dup_sqr(f, K)
if not (f and g):
return []
df = dup_degree(f)
dg = dup_degree(g)
n = max(df, dg) + 1
if n < 100:
h = []
for i in range(0, df + dg + 1):
coeff = K.zero
for j in range(max(0, i - dg), min(df, i) + 1):
coeff += f[j]*g[i - j]
h.append(coeff)
return dup_strip(h)
else:
# Use Karatsuba's algorithm (divide and conquer), see e.g.:
# Joris van der Hoeven, Relax But Don't Be Too Lazy,
# J. Symbolic Computation, 11 (2002), section 3.1.1.
n2 = n//2
fl, gl = dup_slice(f, 0, n2, K), dup_slice(g, 0, n2, K)
fh = dup_rshift(dup_slice(f, n2, n, K), n2, K)
gh = dup_rshift(dup_slice(g, n2, n, K), n2, K)
lo, hi = dup_mul(fl, gl, K), dup_mul(fh, gh, K)
mid = dup_mul(dup_add(fl, fh, K), dup_add(gl, gh, K), K)
mid = dup_sub(mid, dup_add(lo, hi, K), K)
return dup_add(dup_add(lo, dup_lshift(mid, n2, K), K),
dup_lshift(hi, 2*n2, K), K)
[docs]def dmp_mul(f, g, u, K):
"""
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
"""
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 range(0, df + dg + 1):
coeff = dmp_zero(v)
for j in range(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)
def dup_sqr(f, K):
"""
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
"""
df, h = len(f) - 1, []
for i in range(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 range(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)
[docs]def dmp_sqr(f, u, K):
"""
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
"""
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 range(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 range(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)
def dup_pow(f, n, K):
"""
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
"""
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
[docs]def dmp_pow(f, n, u, K):
"""
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
"""
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
def dup_pdiv(f, g, K):
"""
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)
"""
df = dup_degree(f)
dg = dup_degree(g)
q, r, dr = [], f, df
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:
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)
_dr, dr = dr, dup_degree(r)
if dr < dg:
break
elif not (dr < _dr):
raise PolynomialDivisionFailed(f, g, K)
c = lc_g**N
q = dup_mul_ground(q, c, K)
r = dup_mul_ground(r, c, K)
return q, r
def dup_prem(f, g, K):
"""
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
"""
df = dup_degree(f)
dg = dup_degree(g)
r, dr = f, df
if not g:
raise ZeroDivisionError("polynomial division")
elif df < dg:
return r
N = df - dg + 1
lc_g = dup_LC(g, K)
while True:
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)
_dr, dr = dr, dup_degree(r)
if dr < dg:
break
elif not (dr < _dr):
raise PolynomialDivisionFailed(f, 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 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
"""
return dup_pdiv(f, g, K)[0]
def dup_pexquo(f, g, K):
"""
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]
"""
q, r = dup_pdiv(f, g, K)
if not r:
return q
else:
raise ExactQuotientFailed(f, g)
[docs]def dmp_pdiv(f, g, u, K):
"""
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)
"""
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, dr = dmp_zero(u), f, df
if df < dg:
return q, r
N = df - dg + 1
lc_g = dmp_LC(g, K)
while True:
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)
_dr, dr = dr, dmp_degree(r, u)
if dr < dg:
break
elif not (dr < _dr):
raise PolynomialDivisionFailed(f, g, 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
[docs]def dmp_prem(f, g, u, K):
"""
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
"""
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, dr = f, df
if df < dg:
return r
N = df - dg + 1
lc_g = dmp_LC(g, K)
while True:
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)
_dr, dr = dr, dmp_degree(r, u)
if dr < dg:
break
elif not (dr < _dr):
raise PolynomialDivisionFailed(f, g, 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 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
"""
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 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], []]
"""
q, r = dmp_pdiv(f, g, u, K)
if dmp_zero_p(r, u):
return q
else:
raise ExactQuotientFailed(f, g)
def dup_rr_div(f, g, K):
"""
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)
"""
df = dup_degree(f)
dg = dup_degree(g)
q, r, dr = [], f, df
if not g:
raise ZeroDivisionError("polynomial division")
elif df < dg:
return q, r
lc_g = dup_LC(g, K)
while True:
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)
_dr, dr = dr, dup_degree(r)
if dr < dg:
break
elif not (dr < _dr):
raise PolynomialDivisionFailed(f, g, K)
return q, r
[docs]def dmp_rr_div(f, g, u, K):
"""
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)
"""
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, dr = dmp_zero(u), f, df
if df < dg:
return q, r
lc_g, v = dmp_LC(g, K), u - 1
while True:
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)
_dr, dr = dr, dmp_degree(r, u)
if dr < dg:
break
elif not (dr < _dr):
raise PolynomialDivisionFailed(f, g, K)
return q, r
def dup_ff_div(f, g, K):
"""
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)
"""
df = dup_degree(f)
dg = dup_degree(g)
q, r, dr = [], f, df
if not g:
raise ZeroDivisionError("polynomial division")
elif df < dg:
return q, r
lc_g = dup_LC(g, K)
while True:
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)
_dr, dr = dr, dup_degree(r)
if dr < dg:
break
elif not (dr < _dr):
raise PolynomialDivisionFailed(f, g, K)
return q, r
[docs]def dmp_ff_div(f, g, u, K):
"""
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)
"""
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, dr = dmp_zero(u), f, df
if df < dg:
return q, r
lc_g, v = dmp_LC(g, K), u - 1
while True:
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)
_dr, dr = dr, dmp_degree(r, u)
if dr < dg:
break
elif not (dr < _dr):
raise PolynomialDivisionFailed(f, g, K)
return q, r
def dup_div(f, g, K):
"""
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)
"""
if K.is_Field:
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 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
"""
return dup_div(f, g, K)[1]
def dup_quo(f, g, K):
"""
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
"""
return dup_div(f, g, K)[0]
def dup_exquo(f, g, K):
"""
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]
"""
q, r = dup_div(f, g, K)
if not r:
return q
else:
raise ExactQuotientFailed(f, g)
[docs]def dmp_div(f, g, u, K):
"""
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)
"""
if K.is_Field:
return dmp_ff_div(f, g, u, K)
else:
return dmp_rr_div(f, g, u, K)
[docs]def dmp_rem(f, g, u, K):
"""
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
"""
return dmp_div(f, g, u, K)[1]
[docs]def dmp_quo(f, g, u, K):
"""
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
"""
return dmp_div(f, g, u, K)[0]
[docs]def dmp_exquo(f, g, u, K):
"""
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], []]
"""
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 import ring, ZZ
>>> R, x = ring("x", ZZ)
>>> R.dup_max_norm(-x**2 + 2*x - 3)
3
"""
if not f:
return K.zero
else:
return max(dup_abs(f, K))
[docs]def dmp_max_norm(f, u, K):
"""
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
"""
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 import ring, ZZ
>>> R, x = ring("x", ZZ)
>>> R.dup_l1_norm(2*x**3 - 3*x**2 + 1)
6
"""
if not f:
return K.zero
else:
return sum(dup_abs(f, K))
[docs]def dmp_l1_norm(f, u, K):
"""
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
"""
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 import ring, ZZ
>>> R, x = ring("x", ZZ)
>>> R.dup_expand([x**2 - 1, x, 2])
2*x**3 - 2*x
"""
if not polys:
return [K.one]
f = polys[0]
for g in polys[1:]:
f = dup_mul(f, g, K)
return f
[docs]def dmp_expand(polys, u, K):
"""
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
"""
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
