def dmp_fateman_poly_F_1(n, K):
"""Fateman's GCD benchmark: trivial GCD """
u = [K(1), K(0)]
for i in range(0, n):
u = [dmp_one(i, K), u]
v = [K(1), K(0), K(0)]
for i in range(0, n):
v = [dmp_one(i, K), dmp_zero(i), v]
m = n - 1
U = dmp_add_term(u, dmp_ground(K(1), m), 0, n, K)
V = dmp_add_term(u, dmp_ground(K(2), m), 0, n, K)
f = [[-K(3), K(0)], [], [K(1), K(0), -K(1)]]
W = dmp_add_term(v, dmp_ground(K(1), m), 0, n, K)
Y = dmp_raise(f, m, 1, K)
F = dmp_mul(U, V, n, K)
G = dmp_mul(W, Y, n, K)
H = dmp_one(n, K)
return F, G, H
def dmp_fateman_poly_F_1(n, K):
"""Fateman's GCD benchmark: trivial GCD """
u = [K(1), K(0)]
for i in xrange(0, n):
u = [dmp_one(i, K), u]
v = [K(1), K(0), K(0)]
for i in xrange(0, n):
v = [dmp_one(i, K), dmp_zero(i), v]
m = n - 1
U = dmp_add_term(u, dmp_ground(K(1), m), 0, n, K)
V = dmp_add_term(u, dmp_ground(K(2), m), 0, n, K)
f = [[-K(3), K(0)], [], [K(1), K(0), -K(1)]]
W = dmp_add_term(v, dmp_ground(K(1), m), 0, n, K)
Y = dmp_raise(f, m, 1, K)
F = dmp_mul(U, V, n, K)
G = dmp_mul(W, Y, n, K)
H = dmp_one(n, K)
return F, G, H
def dmp_compose(f, g, u, K):
"""
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
"""
if not u:
return dup_compose(f, g, K)
if dmp_zero_p(f, u):
return f
h = [f[0]]
for c in f[1:]:
h = dmp_mul(h, g, u, K)
h = dmp_add_term(h, c, 0, u, K)
return h
def dmp_compose(f, g, u, K):
"""
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
"""
if not u:
return dup_compose(f, g, K)
if dmp_zero_p(f, u):
return f
h = [f[0]]
for c in f[1:]:
h = dmp_mul(h, g, u, K)
h = dmp_add_term(h, c, 0, u, K)
return h
def dmp_compose(f, g, u, K):
"""
Evaluate functional composition ``f(g)`` in ``K[X]``.
Examples
========
>>> from sympy.polys.domains import ZZ
>>> from sympy.polys.densetools import dmp_compose
>>> f = ZZ.map([[1, 2], [1, 0]])
>>> g = ZZ.map([[1, 0]])
>>> dmp_compose(f, g, 1, ZZ)
[[1, 3, 0]]
"""
if not u:
return dup_compose(f, g, K)
if dmp_zero_p(f, u):
return f
h = [f[0]]
for c in f[1:]:
h = dmp_mul(h, g, u, K)
h = dmp_add_term(h, c, 0, u, K)
return h
def dmp_compose(f, g, u, K):
"""
Evaluate functional composition ``f(g)`` in ``K[X]``.
Examples
========
>>> from sympy.polys.domains import ZZ
>>> from sympy.polys.densetools import dmp_compose
>>> f = ZZ.map([[1, 2], [1, 0]])
>>> g = ZZ.map([[1, 0]])
>>> dmp_compose(f, g, 1, ZZ)
[[1, 3, 0]]
"""
if not u:
return dup_compose(f, g, K)
if dmp_zero_p(f, u):
return f
h = [f[0]]
for c in f[1:]:
h = dmp_mul(h, g, u, K)
h = dmp_add_term(h, c, 0, u, K)
return h
def dmp_fateman_poly_F_2(n, K):
"""Fateman's GCD benchmark: linearly dense quartic inputs """
u = [K(1), K(0)]
for i in range(0, n - 1):
u = [dmp_one(i, K), u]
m = n - 1
v = dmp_add_term(u, dmp_ground(K(2), m - 1), 0, n, K)
f = dmp_sqr([dmp_one(m, K), dmp_neg(v, m, K)], n, K)
g = dmp_sqr([dmp_one(m, K), v], n, K)
v = dmp_add_term(u, dmp_one(m - 1, K), 0, n, K)
h = dmp_sqr([dmp_one(m, K), v], n, K)
return dmp_mul(f, h, n, K), dmp_mul(g, h, n, K), h
def dmp_fateman_poly_F_2(n, K):
"""Fateman's GCD benchmark: linearly dense quartic inputs """
u = [K(1), K(0)]
for i in xrange(0, n - 1):
u = [dmp_one(i, K), u]
m = n - 1
v = dmp_add_term(u, dmp_ground(K(2), m - 1), 0, n, K)
f = dmp_sqr([dmp_one(m, K), dmp_neg(v, m, K)], n, K)
g = dmp_sqr([dmp_one(m, K), v], n, K)
v = dmp_add_term(u, dmp_one(m - 1, K), 0, n, K)
h = dmp_sqr([dmp_one(m, K), v], n, K)
return dmp_mul(f, h, n, K), dmp_mul(g, h, n, K), h
def dup_real_imag(f, K):
"""
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)
"""
if not K.is_ZZ and not K.is_QQ:
raise DomainError(
"computing real and imaginary parts is not supported over %s" % K)
f1 = dmp_zero(1)
f2 = dmp_zero(1)
if not f:
return f1, f2
g = [[[K.one, K.zero]], [[K.one], []]]
h = dmp_ground(f[0], 2)
for c in f[1:]:
h = dmp_mul(h, g, 2, K)
h = dmp_add_term(h, dmp_ground(c, 1), 0, 2, K)
H = dup_to_raw_dict(h)
for k, h in H.items():
m = k % 4
if not m:
f1 = dmp_add(f1, h, 1, K)
elif m == 1:
f2 = dmp_add(f2, h, 1, K)
elif m == 2:
f1 = dmp_sub(f1, h, 1, K)
else:
f2 = dmp_sub(f2, h, 1, K)
return f1, f2
def dup_real_imag(f, K):
"""
Return bivariate polynomials ``f1`` and ``f2``, such that ``f = f1 + f2*I``.
Examples
========
>>> from sympy.polys.domains import ZZ
>>> from sympy.polys.densetools import dup_real_imag
>>> dup_real_imag([ZZ(1), ZZ(1), ZZ(1), ZZ(1)], ZZ)
([[1], [1], [-3, 0, 1], [-1, 0, 1]], [[3, 0], [2, 0], [-1, 0, 1, 0]])
"""
if not K.is_ZZ and not K.is_QQ:
raise DomainError(
"computing real and imaginary parts is not supported over %s" % K)
f1 = dmp_zero(1)
f2 = dmp_zero(1)
if not f:
return f1, f2
g = [[[K.one, K.zero]], [[K.one], []]]
h = dmp_ground(f[0], 2)
for c in f[1:]:
h = dmp_mul(h, g, 2, K)
h = dmp_add_term(h, dmp_ground(c, 1), 0, 2, K)
H = dup_to_raw_dict(h)
for k, h in H.iteritems():
m = k % 4
if not m:
f1 = dmp_add(f1, h, 1, K)
elif m == 1:
f2 = dmp_add(f2, h, 1, K)
elif m == 2:
f1 = dmp_sub(f1, h, 1, K)
else:
f2 = dmp_sub(f2, h, 1, K)
return f1, f2
def dup_real_imag(f, K):
"""
Return bivariate polynomials ``f1`` and ``f2``, such that ``f = f1 + f2*I``.
Examples
========
>>> from sympy.polys.domains import ZZ
>>> from sympy.polys.densetools import dup_real_imag
>>> dup_real_imag([ZZ(1), ZZ(1), ZZ(1), ZZ(1)], ZZ)
([[1], [1], [-3, 0, 1], [-1, 0, 1]], [[3, 0], [2, 0], [-1, 0, 1, 0]])
"""
if not K.is_ZZ and not K.is_QQ:
raise DomainError(
"computing real and imaginary parts is not supported over %s" % K)
f1 = dmp_zero(1)
f2 = dmp_zero(1)
if not f:
return f1, f2
g = [[[K.one, K.zero]], [[K.one], []]]
h = dmp_ground(f[0], 2)
for c in f[1:]:
h = dmp_mul(h, g, 2, K)
h = dmp_add_term(h, dmp_ground(c, 1), 0, 2, K)
H = dup_to_raw_dict(h)
for k, h in H.iteritems():
m = k % 4
if not m:
f1 = dmp_add(f1, h, 1, K)
elif m == 1:
f2 = dmp_add(f2, h, 1, K)
elif m == 2:
f1 = dmp_sub(f1, h, 1, K)
else:
f2 = dmp_sub(f2, h, 1, K)
return f1, f2
def dup_real_imag(f, K):
"""
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)
"""
if not K.is_ZZ and not K.is_QQ:
raise DomainError("computing real and imaginary parts is not supported over %s" % K)
f1 = dmp_zero(1)
f2 = dmp_zero(1)
if not f:
return f1, f2
g = [[[K.one, K.zero]], [[K.one], []]]
h = dmp_ground(f[0], 2)
for c in f[1:]:
h = dmp_mul(h, g, 2, K)
h = dmp_add_term(h, dmp_ground(c, 1), 0, 2, K)
H = dup_to_raw_dict(h)
for k, h in H.items():
m = k % 4
if not m:
f1 = dmp_add(f1, h, 1, K)
elif m == 1:
f2 = dmp_add(f2, h, 1, K)
elif m == 2:
f1 = dmp_sub(f1, h, 1, K)
else:
f2 = dmp_sub(f2, h, 1, K)
return f1, f2
def dmp_fateman_poly_F_3(n, K):
"""Fateman's GCD benchmark: sparse inputs (deg f ~ vars f) """
u = dup_from_raw_dict({n+1: K.one}, K)
for i in xrange(0, n-1):
u = dmp_add_term([u], dmp_one(i, K), n+1, i+1, K)
v = dmp_add_term(u, dmp_ground(K(2), n-2), 0, n, K)
f = dmp_sqr(dmp_add_term([dmp_neg(v, n-1, K)], dmp_one(n-1, K), n+1, n, K), n, K)
g = dmp_sqr(dmp_add_term([v], dmp_one(n-1, K), n+1, n, K), n, K)
v = dmp_add_term(u, dmp_one(n-2, K), 0, n-1, K)
h = dmp_sqr(dmp_add_term([v], dmp_one(n-1, K), n+1, n, K), n, K)
return dmp_mul(f, h, n, K), dmp_mul(g, h, n, K), h
def dmp_fateman_poly_F_3(n, K):
"""Fateman's GCD benchmark: sparse inputs (deg f ~ vars f) """
u = dup_from_raw_dict({n+1: K.one}, K)
for i in xrange(0, n-1):
u = dmp_add_term([u], dmp_one(i, K), n+1, i+1, K)
v = dmp_add_term(u, dmp_ground(K(2), n-2), 0, n, K)
f = dmp_sqr(dmp_add_term([dmp_neg(v, n-1, K)], dmp_one(n-1, K), n+1, n, K), n, K)
g = dmp_sqr(dmp_add_term([v], dmp_one(n-1, K), n+1, n, K), n, K)
v = dmp_add_term(u, dmp_one(n-2, K), 0, n-1, K)
h = dmp_sqr(dmp_add_term([v], dmp_one(n-1, K), n+1, n, K), n, K)
return dmp_mul(f, h, n, K), dmp_mul(g, h, n, K), h
def test_dmp_add_term():
assert dmp_add_term([ZZ(1),ZZ(1),ZZ(1)], ZZ(1), 2, 0, ZZ) == \
dup_add_term([ZZ(1),ZZ(1),ZZ(1)], ZZ(1), 2, ZZ)
assert dmp_add_term(f_0, [[]], 3, 2, ZZ) == f_0
assert dmp_add_term(F_0, [[]], 3, 2, QQ) == F_0