def rs_integrate(self, x):
"""
integrate ``p`` with respect to ``x``
Examples
========
>>> from diofant.polys.domains import QQ
>>> from diofant.polys.rings import ring
>>> R, x, y = ring('x, y', QQ)
>>> p = x + x**2*y**3
>>> rs_integrate(p, x)
1/3*x**3*y**3 + 1/2*x**2
"""
ring = self.ring
p1 = ring.zero
n = ring.gens.index(x)
mn = [0] * ring.ngens
mn[n] = 1
mn = tuple(mn)
for expv in self:
e = monomial_mul(expv, mn)
p1[e] = self[expv] / (expv[n] + 1)
return p1
def representing_matrix(m):
M = [[domain.zero] * len(basis) for _ in range(len(basis))]
for i, v in enumerate(basis):
r = ring.term_new(monomial_mul(m, v), domain.one).rem(G)
for monom, coeff in r.terms():
j = basis.index(monom)
M[j][i] = coeff
return M
def rs_square(p1, x, prec):
"""
square modulo ``O(x**prec)``
Examples
========
>>> from diofant.polys.domains import QQ
>>> from diofant.polys.rings import ring
>>> R, x = ring('x', QQ)
>>> p = x**2 + 2*x + 1
>>> rs_square(p, x, 3)
6*x**2 + 4*x + 1
"""
ring = p1.ring
p = ring.zero
iv = ring.gens.index(x)
get = p.get
items = list(p1.items())
items.sort(key=lambda e: e[0][iv])
monomial_mul = ring.monomial_mul
for i in range(len(items)):
exp1, v1 = items[i]
for j in range(i):
exp2, v2 = items[j]
if exp1[iv] + exp2[iv] < prec:
exp = monomial_mul(exp1, exp2)
p[exp] = get(exp, 0) + v1 * v2
else:
break
p = p.imul_num(2)
get = p.get
for expv, v in p1.items():
if 2 * expv[iv] < prec:
e2 = monomial_mul(expv, expv)
p[e2] = get(e2, 0) + v**2
p.strip_zero()
return p
def rs_mul(p1, p2, x, prec):
"""
product of series modulo ``O(x**prec)``
``x`` is the series variable or its position in the generators.
Examples
========
>>> from diofant.polys.domains import QQ
>>> from diofant.polys.rings import ring
>>> R, x = ring('x', QQ)
>>> p1 = x**2 + 2*x + 1
>>> p2 = x + 1
>>> rs_mul(p1, p2, x, 3)
3*x**2 + 3*x + 1
"""
ring = p1.ring
p = ring.zero
if ring.__class__ != p2.ring.__class__ or ring != p2.ring:
raise ValueError('p1 and p2 must have the same ring')
iv = ring.gens.index(x)
if not isinstance(p2, PolyElement):
raise ValueError('p1 and p2 must have the same ring')
if ring == p2.ring:
get = p.get
items2 = list(p2.items())
items2.sort(key=lambda e: e[0][iv])
if ring.ngens == 1:
for exp1, v1 in p1.items():
for exp2, v2 in items2:
exp = exp1[0] + exp2[0]
if exp < prec:
exp = (exp, )
p[exp] = get(exp, 0) + v1 * v2
else:
break
else:
monomial_mul = ring.monomial_mul
for exp1, v1 in p1.items():
for exp2, v2 in items2:
if exp1[iv] + exp2[iv] < prec:
exp = monomial_mul(exp1, exp2)
p[exp] = get(exp, 0) + v1 * v2
else:
break
p.strip_zero()
return p
def sdm_monomial_mul(M, X):
"""
Multiply tuple ``X`` representing a monomial of `K[X]` into the tuple
``M`` representing a monomial of `F`.
Examples
========
Multiplying `xy^3` into `x f_1` yields `x^2 y^3 f_1`:
>>> sdm_monomial_mul((1, 1, 0), (1, 3))
(1, 2, 3)
"""
return (M[0],) + monomial_mul(X, M[1:])
def test_monomial_mul():
assert monomial_mul((3, 4, 1), (1, 2, 0)) == (4, 6, 1)