def ratint_logpart(f, g, x, t=None):
"""Lazard-Rioboo-Trager algorithm.
Given a field K and polynomials f and g in K[x], such that f and g
are coprime, deg(f) < deg(g) and g is square-free, returns a list
of tuples (s_i, q_i) of polynomials, for i = 1..n, such that s_i
in K[t, x] and q_i in K[t], and:
___ ___
d f d \ ` \ `
-- - = -- ) ) a log(s_i(a, x))
dx g dx /__, /__,
i=1..n a | q_i(a) = 0
"""
f, g = Poly(f, x), Poly(g, x)
t = t or Dummy('t')
a, b = g, f - g.diff()*Poly(t, x)
R = subresultants(a, b)
res = Poly(resultant(a, b), t)
R_map, H = {}, []
for r in R:
R_map[r.degree()] = r
def _include_sign(c, sqf):
if c < 0:
h, k = sqf[0]
sqf[0] = h*c, k
C, res_sqf = res.sqf_list()
_include_sign(C, res_sqf)
for q, i in res_sqf:
_, q = q.primitive()
if g.degree() == i:
H.append((g, q))
else:
h = R_map[i]
h_lc = Poly(h.LC(), t, field=True)
c, h_lc_sqf = h_lc.sqf_list(all=True)
_include_sign(c, h_lc_sqf)
for a, j in h_lc_sqf:
h = h.exquo(Poly(a.gcd(q)**j, x))
inv, coeffs = h_lc.invert(q), [S(1)]
for coeff in h.coeffs()[1:]:
T = (inv*coeff).rem(q)
coeffs.append(T.as_basic())
h = Poly(dict(zip(h.monoms(), coeffs)), x)
H.append((h, q))
return H
def ratint_logpart(f, g, x, t=None):
"""
Lazard-Rioboo-Trager algorithm.
Given a field K and polynomials f and g in K[x], such that f and g
are coprime, deg(f) < deg(g) and g is square-free, returns a list
of tuples (s_i, q_i) of polynomials, for i = 1..n, such that s_i
in K[t, x] and q_i in K[t], and:
___ ___
d f d \ ` \ `
-- - = -- ) ) a log(s_i(a, x))
dx g dx /__, /__,
i=1..n a | q_i(a) = 0
Examples
========
>>> from sympy.integrals.rationaltools import ratint_logpart
>>> from sympy.abc import x
>>> from sympy import Poly
>>> ratint_logpart(Poly(1, x, domain='ZZ'),
... Poly(x**2 + x + 1, x, domain='ZZ'), x)
[(Poly(x + 3*_t/2 + 1/2, x, domain='QQ[_t]'),
...Poly(3*_t**2 + 1, _t, domain='ZZ'))]
>>> ratint_logpart(Poly(12, x, domain='ZZ'),
... Poly(x**2 - x - 2, x, domain='ZZ'), x)
[(Poly(x - 3*_t/8 - 1/2, x, domain='QQ[_t]'),
...Poly(-_t**2 + 16, _t, domain='ZZ'))]
"""
f, g = Poly(f, x), Poly(g, x)
t = t or Dummy('t')
a, b = g, f - g.diff()*Poly(t, x)
R = subresultants(a, b)
res = Poly(resultant(a, b), t, composite=False)
R_map, H = {}, []
for r in R:
R_map[r.degree()] = r
def _include_sign(c, sqf):
if c < 0:
h, k = sqf[0]
sqf[0] = h*c, k
C, res_sqf = res.sqf_list()
_include_sign(C, res_sqf)
for q, i in res_sqf:
_, q = q.primitive()
if g.degree() == i:
H.append((g, q))
else:
h = R_map[i]
h_lc = Poly(h.LC(), t, field=True)
c, h_lc_sqf = h_lc.sqf_list(all=True)
_include_sign(c, h_lc_sqf)
for a, j in h_lc_sqf:
h = h.quo(Poly(a.gcd(q)**j, x))
inv, coeffs = h_lc.invert(q), [S(1)]
for coeff in h.coeffs()[1:]:
T = (inv*coeff).rem(q)
coeffs.append(T.as_expr())
h = Poly(dict(zip(h.monoms(), coeffs)), x)
H.append((h, q))
return H
def ratint_logpart(f, g, x, t=None):
"""Lazard-Rioboo-Trager algorithm.
Given a field K and polynomials f and g in K[x], such that f and g
are coprime, deg(f) < deg(g) and g is square-free, returns a list
of tuples (s_i, q_i) of polynomials, for i = 1..n, such that s_i
in K[t, x] and q_i in K[t], and:
___ ___
d f d \ ` \ `
-- - = -- ) ) a log(s_i(a, x))
dx g dx /__, /__,
i=1..n a | q_i(a) = 0
"""
f, g = Poly(f, x), Poly(g, x)
t = t or Dummy('t')
a, b = g, f - g.diff() * Poly(t, x)
R = subresultants(a, b)
res = Poly(resultant(a, b), t)
R_map, H = {}, []
for r in R:
R_map[r.degree()] = r
def _include_sign(c, sqf):
if c < 0:
h, k = sqf[0]
sqf[0] = h * c, k
C, res_sqf = res.sqf_list()
_include_sign(C, res_sqf)
for q, i in res_sqf:
_, q = q.primitive()
if g.degree() == i:
H.append((g, q))
else:
h = R_map[i]
h_lc = Poly(h.LC(), t, field=True)
c, h_lc_sqf = h_lc.sqf_list(all=True)
_include_sign(c, h_lc_sqf)
for a, j in h_lc_sqf:
h = h.exquo(Poly(a.gcd(q)**j, x))
inv, coeffs = h_lc.invert(q), [S(1)]
for coeff in h.coeffs()[1:]:
T = (inv * coeff).rem(q)
coeffs.append(T.as_basic())
h = Poly(dict(zip(h.monoms(), coeffs)), x)
H.append((h, q))
return H
def ratint_logpart(f, g, x, t=None):
"""
Lazard-Rioboo-Trager algorithm.
Given a field K and polynomials f and g in K[x], such that f and g
are coprime, deg(f) < deg(g) and g is square-free, returns a list
of tuples (s_i, q_i) of polynomials, for i = 1..n, such that s_i
in K[t, x] and q_i in K[t], and:
___ ___
d f d \ ` \ `
-- - = -- ) ) a log(s_i(a, x))
dx g dx /__, /__,
i=1..n a | q_i(a) = 0
Examples
========
>>> from sympy.integrals.rationaltools import ratint_logpart
>>> from sympy.abc import x
>>> from sympy import Poly
>>> ratint_logpart(Poly(1, x, domain='ZZ'),
... Poly(x**2 + x + 1, x, domain='ZZ'), x)
[(Poly(x + 3*_t/2 + 1/2, x, domain='QQ[_t]'),
...Poly(3*_t**2 + 1, _t, domain='ZZ'))]
>>> ratint_logpart(Poly(12, x, domain='ZZ'),
... Poly(x**2 - x - 2, x, domain='ZZ'), x)
[(Poly(x - 3*_t/8 - 1/2, x, domain='QQ[_t]'),
...Poly(-_t**2 + 16, _t, domain='ZZ'))]
"""
f, g = Poly(f, x), Poly(g, x)
t = t or Dummy('t')
a, b = g, f - g.diff() * Poly(t, x)
R = subresultants(a, b)
res = Poly(resultant(a, b), t, composite=False)
R_map, H = {}, []
for r in R:
R_map[r.degree()] = r
def _include_sign(c, sqf):
if c < 0:
h, k = sqf[0]
sqf[0] = h * c, k
C, res_sqf = res.sqf_list()
_include_sign(C, res_sqf)
for q, i in res_sqf:
_, q = q.primitive()
if g.degree() == i:
H.append((g, q))
else:
h = R_map[i]
h_lc = Poly(h.LC(), t, field=True)
c, h_lc_sqf = h_lc.sqf_list(all=True)
_include_sign(c, h_lc_sqf)
for a, j in h_lc_sqf:
h = h.quo(Poly(a.gcd(q)**j, x))
inv, coeffs = h_lc.invert(q), [S(1)]
for coeff in h.coeffs()[1:]:
T = (inv * coeff).rem(q)
coeffs.append(T.as_expr())
h = Poly(dict(zip(h.monoms(), coeffs)), x)
H.append((h, q))
return H