Source code for diofant.simplify.ratsimp

from itertools import combinations_with_replacement

from ..core import Add, Dummy, Rational, symbols
from ..polys import (ComputationFailed, Poly, cancel, parallel_poly_from_expr,
reduced)
from ..polys.monomials import Monomial
from ..utilities.misc import debug

[docs]def ratsimp(expr):
"""
Put an expression over a common denominator, cancel and reduce.

Examples
========

>>> ratsimp(1/x + 1/y)
(x + y)/(x*y)

"""

f, g = cancel(expr).as_numer_denom()
try:
Q, r = reduced(f, [g], field=True, expand=False)
except ComputationFailed:
return f/g

def ratsimpmodprime(expr, G, *gens, **args):
"""
Simplifies a rational expression expr modulo the prime ideal
generated by G.  G should be a Gröbner basis of the
ideal.

>>> eq = (x + y**5 + y)/(x - y)
>>> ratsimpmodprime(eq, [x*y**5 - x - y], x, y, order='lex')
(x**2 + x*y + x + y)/(x**2 - x*y)

If polynomial is False, the algorithm computes a rational
simplification which minimizes the sum of the total degrees of
the numerator and the denominator.

If polynomial is True, this function just brings numerator and
denominator into a canonical form. This is much faster, but has
potentially worse results.

References
==========

M. Monagan, R. Pearce, Rational Simplification Modulo a Polynomial
Ideal,
http://citeseer.ist.psu.edu/viewdoc/summary?doi=10.1.1.163.6984
(specifically, the second algorithm)

"""
from ..matrices import Matrix, zeros
from ..solvers.solvers import minsolve_linear_system

quick = args.pop('quick', True)
polynomial = args.pop('polynomial', False)
debug('ratsimpmodprime', expr)

# usual preparation of polynomials:

num, denom = cancel(expr).as_numer_denom()

polys, opt = parallel_poly_from_expr([num, denom] + G, *gens, **args)
domain = opt.domain
opt.domain = domain.field

# compute only once
leading_monomials = [g.LM(opt.order) for g in polys[2:]]
tested = set()

def staircase(n):
"""
Compute all monomials with degree less than n that are
not divisible by any element of leading_monomials.

"""
if n == 0:
return 
S = []
for mi in combinations_with_replacement(range(len(opt.gens)), n):
m = *len(opt.gens)
for i in mi:
m[i] += 1
if all(not lmg.divides(m) for lmg in leading_monomials):
S.append(m)

return [Monomial(s).as_expr(*opt.gens) for s in S] + staircase(n - 1)

def _ratsimpmodprime(a, b, allsol, N=0, D=0):
r"""
Computes a rational simplification of a/b which minimizes
the sum of the total degrees of the numerator and the denominator.

The algorithm proceeds by looking at a * d - b * c modulo
the ideal generated by G for some c and d with degree
less than a and b respectively.
The coefficients of c and d are indeterminates and thus
the coefficients of the normalform of a * d - b * c are
linear polynomials in these indeterminates.
If these linear polynomials, considered as system of
equations, have a nontrivial solution, then \frac{a}{b}
\equiv \frac{c}{d} modulo the ideal generated by G. So,
by construction, the degree of c and d is less than
the degree of a and b, so a simpler representation
has been found.
After a simpler representation has been found, the algorithm
tries to reduce the degree of the numerator and denominator
and returns the result afterwards.

As an extension, if quick=False, we look at all possible degrees such
that the total degree is less than *or equal to* the best current
solution. We retain a list of all solutions of minimal degree, and try
to find the best one at the end.

"""
c, d = a, b
steps = 0

maxdeg = a.total_degree() + b.total_degree()
if quick:
bound = maxdeg - 1
else:
bound = maxdeg
while N + D <= bound:
if (N, D) in tested:
break

M1 = staircase(N)
M2 = staircase(D)
debug('%s / %s: %s, %s' % (N, D, M1, M2))

Cs = symbols("c:%d" % len(M1), cls=Dummy)
Ds = symbols("d:%d" % len(M2), cls=Dummy)
ng = Cs + Ds

c_hat = Poly(
sum(Cs[i] * M1[i] for i in range(len(M1))), opt.gens + ng)
d_hat = Poly(
sum(Ds[i] * M2[i] for i in range(len(M2))), opt.gens + ng)

r = reduced(a * d_hat - b * c_hat, G, opt.gens + ng,
order=opt.order, polys=True)

S = Poly(r, gens=opt.gens).coeffs()
Sv = Cs + Ds
Sm = zeros(len(S), len(Sv) + 1)
Sm[:, :-1] = Matrix([[a.diff(b) for b in Sv] for a in S])
sol = minsolve_linear_system(Sm, *Sv, quick=True)

if sol and not all(s == 0 for s in sol.values()):
c = c_hat.subs(sol)
d = d_hat.subs(sol)

# The "free" variables occuring before as parameters
# might still be in the substituted c, d, so set them
# to the value chosen before:
c = c.subs(dict(zip(Cs + Ds,  * (len(Cs) + len(Ds)))))
d = d.subs(dict(zip(Cs + Ds,  * (len(Cs) + len(Ds)))))

c = Poly(c, opt.gens)
d = Poly(d, opt.gens)
if d == 0:
raise ValueError('Ideal not prime?')

allsol.append((c_hat, d_hat, S, Cs + Ds))
if N + D != maxdeg:
allsol = [allsol[-1]]

break

steps += 1
N += 1
D += 1

if steps > 0:
c, d, allsol = _ratsimpmodprime(c, d, allsol, N, D - steps)
c, d, allsol = _ratsimpmodprime(c, d, allsol, N - steps, D)

return c, d, allsol

# preprocessing. this improves performance a bit when deg(num)
# and deg(denom) are large:
num = reduced(num, G, opt.gens, order=opt.order)
denom = reduced(denom, G, opt.gens, order=opt.order)

if polynomial:
return (num/denom).cancel()

c, d, allsol = _ratsimpmodprime(
Poly(num, opt.gens, domain=opt.domain), Poly(denom, opt.gens, domain=opt.domain), [])
if not quick and allsol:
debug('Looking for best minimal solution. Got: %s' % len(allsol))
newsol = []
for c_hat, d_hat, S, ng in allsol:
Sm = zeros(len(S), len(ng) + 1)
Sm[:, :-1] = Matrix([[a.diff(b) for b in ng] for a in S])
sol = minsolve_linear_system(Sm, *ng)
newsol.append((c_hat.subs(sol), d_hat.subs(sol)))
c, d = min(newsol, key=lambda x: len(x.terms()) + len(x.terms()))

if not domain.is_Field:
cn, c = c.clear_denoms(convert=True)
dn, d = d.clear_denoms(convert=True)
else:
cn, dn = 1, 1

cf, c, d = cancel((c, d), opt.gens, order=opt.order)  # canonicalize signs
r = cf*Rational(cn, dn)

return (c*r.denominator)/(d*r.numerator)