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 Symbol('t', dummy=True)
a, b = g, f - g.diff().mul_term(t)
res, R = poly_subresultants(a, b)
Q = poly_sqf(Poly(res, t))
R_map, H, i = {}, [], 1
for r in R:
R_map[r.degree] = r
for q in Q:
if q.degree > 0:
_, q = q.as_primitive()
if g.degree == i:
H.append((g, q))
else:
h = R_map[i]
A = poly_sqf(h.LC, t)
for j in xrange(0, len(A)):
T = poly_gcd(A[j], q)**(j+1)
h = poly_div(h, Poly(T, x))[0]
# NOTE: h.LC is always invertible in K[t]
inv, coeffs = Poly(h.LC, t).invert(q), [S(1)]
for coeff in h.coeffs[1:]:
T = poly_div(inv*coeff, q)[1]
coeffs.append(T.as_basic())
h = Poly(zip(coeffs, h.monoms), x)
H.append((h, q))
i += 1
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 Symbol('t', dummy=True)
a, b = g, f - g.diff().mul_term(t)
res, R = poly_subresultants(a, b)
Q = poly_sqf(Poly(res, t))
R_map, H, i = {}, [], 1
for r in R:
R_map[r.degree] = r
for q in Q:
if q.degree > 0:
_, q = q.as_primitive()
if g.degree == i:
H.append((g, q))
else:
h = R_map[i]
A = poly_sqf(h.LC, t)
for j in xrange(0, len(A)):
T = poly_gcd(A[j], q)**(j+1)
h = poly_div(h, Poly(T, x))[0]
# NOTE: h.LC is always invertible in K[t]
inv, coeffs = Poly(h.LC, t).invert(q), [S(1)]
for coeff in h.coeffs[1:]:
T = poly_div(inv*coeff, q)[1]
coeffs.append(T.as_basic())
h = Poly(zip(coeffs, h.monoms), x)
H.append((h, q))
i += 1
return H
def ratint_ratpart(f, g, x):
"""Horowitz-Ostrogradsky algorithm.
Given a field K and polynomials f and g in K[x], such that f and g
are coprime and deg(f) < deg(g), returns fractions A and B in K(x),
such that f/g = A' + B and B has square-free denominator.
"""
f, g = Poly(f, x), Poly(g, x)
u = poly_gcd(g, g.diff())
v = poly_div(g, u)[0]
n = u.degree - 1
m = v.degree - 1
d = g.degree
A_coeff = [ Symbol('a' + str(n-i), dummy=True) for i in xrange(0, n+1) ]
B_coeff = [ Symbol('b' + str(m-i), dummy=True) for i in xrange(0, m+1) ]
symbols = A_coeff + B_coeff
A = Poly(zip(A_coeff, xrange(n, -1, -1)), x)
B = Poly(zip(B_coeff, xrange(m, -1, -1)), x)
H = f - A.diff()*v + A*poly_div(u.diff()*v, u)[0] - B*u
result = solve(H.coeffs, symbols)
A = A.subs(result)
B = B.subs(result)
rat_part = Poly.cancel((A, u), x)
log_part = Poly.cancel((B, v), x)
return rat_part, log_part
def ratint_ratpart(f, g, x):
"""Horowitz-Ostrogradsky algorithm.
Given a field K and polynomials f and g in K[x], such that f and g
are coprime and deg(f) < deg(g), returns fractions A and B in K(x),
such that f/g = A' + B and B has square-free denominator.
"""
f, g = Poly(f, x), Poly(g, x)
u = poly_gcd(g, g.diff())
v = poly_div(g, u)[0]
n = u.degree - 1
m = v.degree - 1
d = g.degree
A_coeff = [ Symbol('a' + str(n-i), dummy=True) for i in xrange(0, n+1) ]
B_coeff = [ Symbol('b' + str(m-i), dummy=True) for i in xrange(0, m+1) ]
symbols = A_coeff + B_coeff
A = Poly(zip(A_coeff, xrange(n, -1, -1)), x)
B = Poly(zip(B_coeff, xrange(m, -1, -1)), x)
H = f - A.diff()*v + A*poly_div(u.diff()*v, u)[0] - B*u
result = solve(H.coeffs, symbols)
A = A.subs(result)
B = B.subs(result)
rat_part = Poly.cancel((A, u), x)
log_part = Poly.cancel((B, v), x)
return rat_part, log_part
def ratint(f, x, **flags):
"""Performs indefinite integration of rational functions.
Given a field K and a rational function f = p/q, where p and q
are polynomials in K[x], returns a function g such that f = g'.
>>> from sympy.integrals.rationaltools import ratint
>>> from sympy.abc import x
>>> ratint(36/(x**5 - 2*x**4 - 2*x**3 + 4*x**2 + x - 2), x)
-4*log(1 + x) + 4*log(-2 + x) - (6 + 12*x)/(1 - x**2)
References
==========
.. [Bro05] M. Bronstein, Symbolic Integration I: Transcendental
Functions, Second Edition, Springer-Verlag, 2005, pp. 35-70
"""
if type(f) is not tuple:
p, q = f.as_numer_denom()
else:
p, q = f
p, q = Poly(p, x), Poly(q, x)
g = poly_gcd(p, q)
p = poly_div(p, g)[0]
q = poly_div(q, g)[0]
result, p = poly_div(p, q)
result = result.integrate(x).as_basic()
if p.is_zero:
return result
g, h = ratint_ratpart(p, q, x)
P, Q = h.as_numer_denom()
q, r = poly_div(P, Q, x)
result += g + q.integrate(x).as_basic()
if not r.is_zero:
symbol = flags.get('symbol', 't')
if not isinstance(symbol, Symbol):
t = Symbol(symbol, dummy=True)
else:
t = symbol
L = ratint_logpart(r, Q, x, t)
real = flags.get('real')
if real is None:
if type(f) is not tuple:
atoms = f.atoms()
else:
p, q = f
atoms = p.atoms() \
| q.atoms()
for elt in atoms - set([x]):
if not elt.is_real:
real = False
break
else:
real = True
eps = S(0)
if not real:
for h, q in L:
eps += RootSum(Lambda(t, t*log(h.as_basic())), q)
else:
for h, q in L:
R = log_to_real(h, q, x, t)
if R is not None:
eps += R
else:
eps += RootSum(Lambda(t, t*log(h.as_basic())), q)
result += eps
return result
def test_poly_gcd():
assert poly_gcd(0, 0, x) == Poly(0, x)
assert poly_gcd(0, 1, x) == Poly(1, x)
assert poly_gcd(1, 0, x) == Poly(1, x)
assert poly_gcd(1, 1, x) == Poly(1, x)
assert poly_gcd(x-1, 0, x) == Poly(x-1, x)
assert poly_gcd(0, x-1, x) == Poly(x-1, x)
assert poly_gcd(-x-1, 0, x) == Poly(x+1, x)
assert poly_gcd(0, -x-1, x) == Poly(x+1, x)
assert poly_gcd(2, 6, x) == Poly(2, x)
assert poly_gcd(2, 6, x, y) == Poly(2, x, y)
assert poly_gcd(x, y, x, y) == Poly(1, x, y)
assert poly_gcd(2*x**3, 6*x, x) == Poly(2*x, x)
assert poly_gcd(2*x**3, 3*x, x) == Poly(x, x)
assert poly_gcd(2*x**3*y, 6*x*y**2, x, y) == Poly(2*x*y, x, y)
assert poly_gcd(2*x**3*y, 3*x*y**2, x, y) == Poly(x*y, x, y)
assert poly_gcd(x**2+2*x+1, x+1, x) == Poly(x+1, x)
assert poly_gcd(x**2+2*x+2, x+1, x) == Poly(1, x)
assert poly_gcd(x**2+2*x+1, 2+2*x, x) == Poly(x+1, x)
assert poly_gcd(x**2+2*x+2, 2+2*x, x) == Poly(1, x)
assert poly_gcd(sin(z)*(x+y), x**2+2*x*y+y**2,
x, y) == Poly(x+y, x, y)
f = x**8+x**6-3*x**4-3*x**3+8*x**2+2*x-5
g = 3*x**6+5*x**4-4*x**2-9*x+21
assert poly_gcd(f, g, x) == Poly(1, x)
def test_poly_gcd():
assert poly_gcd(0, 0, x) == Poly(0, x)
assert poly_gcd(0, 1, x) == Poly(1, x)
assert poly_gcd(1, 0, x) == Poly(1, x)
assert poly_gcd(1, 1, x) == Poly(1, x)
assert poly_gcd(x-1, 0, x) == Poly(x-1, x)
assert poly_gcd(0, x-1, x) == Poly(x-1, x)
assert poly_gcd(-x-1, 0, x) == Poly(x+1, x)
assert poly_gcd(0, -x-1, x) == Poly(x+1, x)
assert poly_gcd(2, 6, x) == Poly(2, x)
assert poly_gcd(2, 6, x, y) == Poly(2, x, y)
assert poly_gcd(x, y, x, y) == Poly(1, x, y)
assert poly_gcd(2*x**3, 6*x, x) == Poly(2*x, x)
assert poly_gcd(2*x**3, 3*x, x) == Poly(x, x)
assert poly_gcd(2*x**3*y, 6*x*y**2, x, y) == Poly(2*x*y, x, y)
assert poly_gcd(2*x**3*y, 3*x*y**2, x, y) == Poly(x*y, x, y)
assert poly_gcd(x**2+2*x+1, x+1, x) == Poly(x+1, x)
assert poly_gcd(x**2+2*x+2, x+1, x) == Poly(1, x)
assert poly_gcd(x**2+2*x+1, 2+2*x, x) == Poly(x+1, x)
assert poly_gcd(x**2+2*x+2, 2+2*x, x) == Poly(1, x)
assert poly_gcd(sin(z)*(x+y), x**2+2*x*y+y**2,
x, y) == Poly(x+y, x, y)
f = x**8+x**6-3*x**4-3*x**3+8*x**2+2*x-5
g = 3*x**6+5*x**4-4*x**2-9*x+21
assert poly_gcd(f, g, x) == Poly(1, x)
def apart(f, z, **flags):
"""Compute partial fraction decomposition of a rational function.
Given a rational function 'f', performing only gcd operations
over the algebraic closue of the initial field of definition,
compute full partial fraction decomposition with fractions
having linear denominators.
For all other kinds of expressions the input is returned in an
unchanged form. Note however, that 'apart' function can thread
over sums and relational operators.
Note that no factorization of the initial denominator of 'f' is
needed. The final decomposition is formed in terms of a sum of
RootSum instances. By default RootSum tries to compute all its
roots to simplify itself. This behaviour can be however avoided
by seting the keyword flag evaluate=False, which will make this
function return a formal decomposition.
>>> from sympy import *
>>> x,y = symbols('xy')
>>> apart(y/(x+2)/(x+1), x)
y/(1 + x) - y/(2 + x)
>>> apart(1/(1+x**5), x, evaluate=False)
RootSum(Lambda(_a, -1/5/(x - _a)*_a), x**5 + 1, x)
For more information on the implemented algorithm refer to:
[1] M. Bronstein, B. Salvy, Full partial fraction decomposition
of rational functions, in: M. Bronstein, ed., Proceedings
ISSAC '93, ACM Press, Kiev, Ukraine, 1993, pp. 157-160.
"""
if not f.has(z):
return f
f = Poly.cancel(f, z)
P, Q = f.as_numer_denom()
if not Q.has(z):
return f
partial, r = div(P, Q, z)
f, q, U = r / Q, Q, []
u = Function('u')(z)
a = Symbol('a', dummy=True)
for k, d in enumerate(poly_sqf(q, z)):
n, b = k + 1, d.as_basic()
U += [u.diff(z, k)]
h = together(Poly.cancel(f * b**n, z) / u**n)
H, subs = [h], []
for j in range(1, n):
H += [H[-1].diff(z) / j]
for j in range(1, n + 1):
subs += [(U[j - 1], b.diff(z, j) / j)]
for j in range(0, n):
P, Q = together(H[j]).as_numer_denom()
for i in range(0, j + 1):
P = P.subs(*subs[j - i])
Q = Q.subs(*subs[0])
P, Q = Poly(P, z), Poly(Q, z)
G = poly_gcd(P, d)
D = poly_quo(d, G)
B, g = poly_half_gcdex(Q, D)
b = poly_rem(P * poly_quo(B, g), D)
numer = b.as_basic()
denom = (z - a)**(n - j)
expr = numer.subs(z, a) / denom
partial += RootSum(Lambda(a, expr), D, **flags)
return partial
def apart(f, z, **flags):
"""Compute partial fraction decomposition of a rational function.
Given a rational function 'f', performing only gcd operations
over the algebraic closue of the initial field of definition,
compute full partial fraction decomposition with fractions
having linear denominators.
For all other kinds of expressions the input is returned in an
unchanged form. Note however, that 'apart' function can thread
over sums and relational operators.
Note that no factorization of the initial denominator of 'f' is
needed. The final decomposition is formed in terms of a sum of
RootSum instances. By default RootSum tries to compute all its
roots to simplify itself. This behaviour can be however avoided
by seting the keyword flag evaluate=False, which will make this
function return a formal decomposition.
>>> from sympy import *
>>> x,y = symbols('xy')
>>> apart(y/(x+2)/(x+1), x)
y/(1 + x) - y/(2 + x)
>>> apart(1/(1+x**5), x, evaluate=False)
RootSum(Lambda(_a, -1/5/(x - _a)*_a), x**5 + 1, x)
For more information on the implemented algorithm refer to:
[1] M. Bronstein, B. Salvy, Full partial fraction decomposition
of rational functions, in: M. Bronstein, ed., Proceedings
ISSAC '93, ACM Press, Kiev, Ukraine, 1993, pp. 157-160.
"""
if not f.has(z):
return f
f = Poly.cancel(f, z)
P, Q = f.as_numer_denom()
if not Q.has(z):
return f
partial, r = div(P, Q, z)
f, q, U = r / Q, Q, []
u = Function('u')(z)
a = Symbol('a', dummy=True)
for k, d in enumerate(poly_sqf(q, z)):
n, b = k + 1, d.as_basic()
U += [ u.diff(z, k) ]
h = together(Poly.cancel(f*b**n, z) / u**n)
H, subs = [h], []
for j in range(1, n):
H += [ H[-1].diff(z) / j ]
for j in range(1, n+1):
subs += [ (U[j-1], b.diff(z, j) / j) ]
for j in range(0, n):
P, Q = together(H[j]).as_numer_denom()
for i in range(0, j+1):
P = P.subs(*subs[j-i])
Q = Q.subs(*subs[0])
P, Q = Poly(P, z), Poly(Q, z)
G = poly_gcd(P, d)
D = poly_quo(d, G)
B, g = poly_half_gcdex(Q, D)
b = poly_rem(P * poly_quo(B, g), D)
numer = b.as_basic()
denom = (z-a)**(n-j)
expr = numer.subs(z, a) / denom
partial += RootSum(Lambda(a, expr), D, **flags)
return partial
def ratint(f, x, **flags):
"""Performs indefinite integration of rational functions.
Given a field K and a rational function f = p/q, where p and q
are polynomials in K[x], returns a function g such that f = g'.
>>> from sympy import *
>>> x = Symbol('x')
>>> ratint(36/(x**5 - 2*x**4 - 2*x**3 + 4*x**2 + x - 2), x)
-4*log(1 + x) + 4*log(-2 + x) - (6 + 12*x)/(1 - x**2)
References
==========
.. [Bro05] M. Bronstein, Symbolic Integration I: Transcendental
Functions, Second Edition, Springer-Verlang, 2005, pp. 35-70
"""
if type(f) is not tuple:
p, q = f.as_numer_denom()
else:
p, q = f
p, q = Poly(p, x), Poly(q, x)
g = poly_gcd(p, q)
p = poly_div(p, g)[0]
q = poly_div(q, g)[0]
result, p = poly_div(p, q)
result = result.integrate(x).as_basic()
if p.is_zero:
return result
g, h = ratint_ratpart(p, q, x)
P, Q = h.as_numer_denom()
q, r = poly_div(P, Q, x)
result += g + q.integrate(x).as_basic()
if not r.is_zero:
symbol = flags.get('symbol', 't')
if not isinstance(symbol, Symbol):
t = Symbol(symbol, dummy=True)
else:
t = symbol
L = ratint_logpart(r, Q, x, t)
real = flags.get('real')
if real is None:
if type(f) is not tuple:
atoms = f.atoms()
else:
p, q = f
atoms = p.atoms() \
| q.atoms()
for elt in atoms - set([x]):
if not elt.is_real:
real = False
break
else:
real = True
eps = S(0)
if not real:
for h, q in L:
eps += RootSum(Lambda(t, t*log(h.as_basic())), q)
else:
for h, q in L:
R = log_to_real(h, q, x, t)
if R is not None:
eps += R
else:
eps += RootSum(Lambda(t, t*log(h.as_basic())), q)
result += eps
return result