def field_isomorphism_pslq(a, b):
"""Construct field isomorphism using PSLQ algorithm. """
if not a.root.is_real or not b.root.is_real:
raise NotImplementedError("PSLQ doesn't support complex coefficients")
f = a.minpoly
g = b.minpoly.replace(f.gen)
n, m, prev = 100, b.minpoly.degree(), None
for i in range(1, 5):
A = a.root.evalf(n)
B = b.root.evalf(n)
basis = [1, B] + [ B**i for i in xrange(2, m) ] + [A]
dps, mp.dps = mp.dps, n
coeffs = pslq(basis, maxcoeff=int(1e10), maxsteps=1000)
mp.dps = dps
if coeffs is None:
break
if coeffs != prev:
prev = coeffs
else:
break
coeffs = [S(c)/coeffs[-1] for c in coeffs[:-1]]
while not coeffs[-1]:
coeffs.pop()
coeffs = list(reversed(coeffs))
h = Poly(coeffs, f.gen, domain='QQ')
if f.compose(h).rem(g).is_zero:
d, approx = len(coeffs) - 1, 0
for i, coeff in enumerate(coeffs):
approx += coeff*B**(d - i)
if A*approx < 0:
return [ -c for c in coeffs ]
else:
return coeffs
elif f.compose(-h).rem(g).is_zero:
return [ -c for c in coeffs ]
else:
n *= 2
return None
def field_isomorphism_pslq(a, b):
"""Construct field isomorphism using PSLQ algorithm. """
if not a.root.is_real or not b.root.is_real:
raise NotImplementedError("PSLQ doesn't support complex coefficients")
f = a.minpoly
g = b.minpoly.replace(f.gen)
n, m, prev = 100, b.minpoly.degree(), None
for i in range(1, 5):
A = a.root.evalf(n)
B = b.root.evalf(n)
basis = [1, B] + [B**i for i in xrange(2, m)] + [A]
dps, mp.dps = mp.dps, n
coeffs = pslq(basis, maxcoeff=int(1e10), maxsteps=1000)
mp.dps = dps
if coeffs is None:
break
if coeffs != prev:
prev = coeffs
else:
break
coeffs = [S(c) / coeffs[-1] for c in coeffs[:-1]]
while not coeffs[-1]:
coeffs.pop()
coeffs = list(reversed(coeffs))
h = Poly(coeffs, f.gen, domain='QQ')
if f.compose(h).rem(g).is_zero:
d, approx = len(coeffs) - 1, 0
for i, coeff in enumerate(coeffs):
approx += coeff * B**(d - i)
if A * approx < 0:
return [-c for c in coeffs]
else:
return coeffs
elif f.compose(-h).rem(g).is_zero:
return [-c for c in coeffs]
else:
n *= 2
return None
##Consider r = 1.2599210498948732. Apply the PSLQ ##algorithm to find the algebraic number that is closest to r. from sympy.mpmath import pslq r = 1.2599210498948732 print pslq([r**n for n in xrange(5)]) # [0, 2, 0, 0, -1] # This shows that r is a root of 0+2*x+0*x**2+0*x**3-1x**4 # =2*x-x**4=x(2-x^3) # The algebraic number corresponding to r is 2**(1/3.0) print 2**(1/3.0)
# L-1 MCS 507 Mon 27 Aug 2012 : hex4pi.py # Discovery of the formula to compute hexadecimal digits of pi # with the PSLQ algorithm using the PSLQ algorithm of sympy. from sympy.mpmath import pslq S = [sum([1.0/(16**k*(8*k+j)) \ for k in xrange(8)]) \ for j in xrange(1,8)] print S import math S.append(math.pi) P = pslq(S) print P
##Consider r = 1.2599210498948732. Apply the PSLQ ##algorithm to find the algebraic number that is closest to r. from sympy.mpmath import pslq r = 1.2599210498948732 print pslq([r**n for n in xrange(5)]) # [0, 2, 0, 0, -1] # This shows that r is a root of 0+2*x+0*x**2+0*x**3-1x**4 # =2*x-x**4=x(2-x^3) # The algebraic number corresponding to r is 2**(1/3.0) print 2**(1 / 3.0)
