def poly_factorize(poly):
"""Factorize multivariate polynomials into a sum of products of monomials.
This function can be used to decompose polynomials into a form which
minimizes the number of additions and multiplications, and which thus
can be evaluated efficently."""
max_deg = {}
if 'horner' in dir(sympy):
return sympy.horner(poly)
if not isinstance(poly, Poly):
poly = Poly(sympy.expand(poly), *poly.atoms(Symbol))
denom, poly = poly.as_integer()
# Determine the order of factorization. We proceed through the
# symbols, starting with the one present in the highest order
# in the polynomial.
for i, sym in enumerate(poly.symbols):
max_deg[i] = 0
for monom in poly.monoms:
for i, symvar in enumerate(monom):
max_deg[i] = max(max_deg[i], symvar)
ret_poly = 0
monoms = list(poly.monoms)
for isym, maxdeg in sorted(max_deg.items(),
key=itemgetter(1),
reverse=True):
drop_idx = []
new_poly = []
for i, monom in enumerate(monoms):
if monom[isym] > 0:
drop_idx.append(i)
new_poly.append((poly.coeff(*monom), monom))
if not new_poly:
continue
ret_poly += sympy.factor(Poly(new_poly, *poly.symbols))
for idx in reversed(drop_idx):
del monoms[idx]
# Add any remaining O(1) terms.
new_poly = []
for i, monom in enumerate(monoms):
new_poly.append((poly.coeff(*monom), monom))
if new_poly:
ret_poly += Poly(new_poly, *poly.symbols)
return ret_poly / denom
def poly_factorize(poly):
"""Factorize multivariate polynomials into a sum of products of monomials.
This function can be used to decompose polynomials into a form which
minimizes the number of additions and multiplications, and which thus
can be evaluated efficently."""
max_deg = {}
if 'horner' in dir(sympy):
return sympy.horner(poly)
if not isinstance(poly, Poly):
poly = Poly(sympy.expand(poly), *poly.atoms(Symbol))
denom, poly = poly.as_integer()
# Determine the order of factorization. We proceed through the
# symbols, starting with the one present in the highest order
# in the polynomial.
for i, sym in enumerate(poly.symbols):
max_deg[i] = 0
for monom in poly.monoms:
for i, symvar in enumerate(monom):
max_deg[i] = max(max_deg[i], symvar)
ret_poly = 0
monoms = list(poly.monoms)
for isym, maxdeg in sorted(max_deg.items(), key=itemgetter(1), reverse=True):
drop_idx = []
new_poly = []
for i, monom in enumerate(monoms):
if monom[isym] > 0:
drop_idx.append(i)
new_poly.append((poly.coeff(*monom), monom))
if not new_poly:
continue
ret_poly += sympy.factor(Poly(new_poly, *poly.symbols))
for idx in reversed(drop_idx):
del monoms[idx]
# Add any remaining O(1) terms.
new_poly = []
for i, monom in enumerate(monoms):
new_poly.append((poly.coeff(*monom), monom))
if new_poly:
ret_poly += Poly(new_poly, *poly.symbols)
return ret_poly / denom
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 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
