def field_isomorphism_pslq(a, b):
"""Construct field isomorphism using PSLQ algorithm."""
if not all(_.domain.is_RationalField and _.ext.is_real for _ in (a, b)):
raise NotImplementedError("PSLQ doesn't support complex coefficients")
f = a.minpoly
x = f.gen
g = b.minpoly.replace(x)
m = g.degree()
a, b = a.ext, b.ext
for n in mpmath.libmp.libintmath.giant_steps(32, 256): # pragma: no branch
with mpmath.workdps(n):
A, B = lambdify((), [a, b], 'mpmath')()
basis = [B**i for i in range(m)] + [A]
coeffs = mpmath.pslq(basis, maxcoeff=10**10, maxsteps=10**3)
if coeffs:
assert coeffs[-1] # basis[:-1] elements are linearly independent
h = -Poly(coeffs[:-1], x, field=True).quo_ground(coeffs[-1])
if f.compose(h).rem(g).is_zero:
return h.rep.all_coeffs()
else:
break
def nsimplify_real(x):
orig = mpmath.mp.dps
xv = x._to_mpmath(bprec)
try:
# We'll be happy with low precision if a simple fraction
if not (tolerance or full):
mpmath.mp.dps = 15
rat = mpmath.pslq([xv, 1])
if rat is not None:
return Rational(-int(rat[1]), int(rat[0]))
mpmath.mp.dps = prec
newexpr = mpmath.identify(xv, constants=constants_dict,
tol=tolerance, full=full)
if not newexpr:
raise ValueError
if full:
newexpr = newexpr[0]
expr = sympify(newexpr)
if x and not expr: # don't let x become 0
raise ValueError
if expr.is_finite is False and not xv in [mpmath.inf, mpmath.ninf]:
raise ValueError
return expr
finally:
# even though there are returns above, this is executed
# before leaving
mpmath.mp.dps = orig
def nsimplify_real(x):
orig = mpmath.mp.dps
xv = x._to_mpmath(bprec)
try:
# We'll be happy with low precision if a simple fraction
if not (tolerance or full):
mpmath.mp.dps = 15
rat = mpmath.pslq([xv, 1])
if rat is not None:
return Rational(-int(rat[1]), int(rat[0]))
mpmath.mp.dps = prec
newexpr = mpmath.identify(xv, constants=constants_dict,
tol=tolerance, full=full)
if not newexpr:
raise ValueError
if full:
newexpr = newexpr[0]
expr = sympify(newexpr)
if x and not expr: # don't let x become 0
raise ValueError
if expr.is_finite is False and not xv in [mpmath.inf, mpmath.ninf]:
raise ValueError
return expr
finally:
# even though there are returns above, this is executed
# before leaving
mpmath.mp.dps = orig
def field_isomorphism_pslq(a, b):
"""Construct field isomorphism using PSLQ algorithm. """
if not all(_.domain.is_RationalField and _.ext.is_real for _ in (a, b)):
raise NotImplementedError("PSLQ doesn't support complex coefficients")
f = a.minpoly
g = b.minpoly.replace(f.gen)
m = b.minpoly.degree()
for n in mpmath.libmp.libintmath.giant_steps(32, 256): # pragma: no branch
with mpmath.workdps(n):
A = lambdify((), a.ext, "mpmath")()
B = lambdify((), b.ext, "mpmath")()
basis = [B**i for i in range(m)] + [A]
coeffs = mpmath.pslq(basis, maxcoeff=int(1e10), maxsteps=1000)
if coeffs is None:
break
coeffs = [QQ(c, coeffs[-1]) for c in coeffs[:-1]]
while not coeffs[-1]:
coeffs.pop()
coeffs.reverse()
h = Poly(coeffs, f.gen, domain='QQ')
if f.compose(h).rem(g).is_zero or f.compose(-h).rem(g).is_zero:
return [-c for c in coeffs]
def _improve_results_precision(self, intermediate_results, verbose=True):
"""
Calculates GCFs to a higher depth using RelativeGCFEnumerator's implementation.
We then feed those results and the constant given to a PSLQ, that tries to find a suitable LHS.
Notice-
The second part of this function (PSLQ), logically belongs to the next step of the algorithm - the
result refinement part. It is implemented here, because the next function is not parallelized over
different processes or clients, and we want the PSLQ to be parallelized as well.
"""
precise_intermediate_results = super()._improve_results_precision(intermediate_results, verbose)
for i in precise_intermediate_results:
print(i)
pslq_results = []
const = self.constants_generator[0]() # using only one constant for now.
for match, val, precision in precise_intermediate_results:
mpf_val = mpmath.mpf(val)
print(match)
try:
pslq_res = mpmath.pslq(
[1, const, const**2, -mpf_val, -const * mpf_val, -(const**2) * mpf_val],
tol=10 ** (1 - precision))
except Exception as e:
import ipdb
ipdb.set_trace()
if pslq_res:
# Sometimes, PSLQ can find several results for the same value (e.g. z(3)/(z(3)^2) = 1/z(3))
# we'll reduce fraction found to get uniform results
reduced_num, reduced_denom = get_reduced_fraction(pslq_res[:3], pslq_res[3:], 2)
print(reduced_num, reduced_denom)
pslq_results.append(RefinedMatch(*match, val, reduced_num, reduced_denom, precision))
else:
pslq_results.append(RefinedMatch(*match, val, None, None, precision))
return pslq_results
def real_to_simple_algebraic_maybe(x, **param):
''' Tries to turn a float into a rational combination of roots.
If it succeeds, returns a sympy number
If it fails, returns directly the float
We return a number, a certificate, and a latex string
It would be nice to also take into account other constants like pi?
NB : this is very expensive. Can it be improved?
'''
list_roots = nonsquare_int(param['root_max'])
L = [np.sqrt(n) for n in list_roots]
L.append(x)
coeff = pslq(L, tol=param['tol']) # TOO EXPENSIVE
#coeff = [1,2,3,5,6,7,-1]
if coeff is None: # we found no algebraic combination
return x, False, ''
denominator = -coeff[-1]
if denominator == 0: # this shouldn't happen, but who knows..
return x, False, ''
fracs = [Fraction(c, denominator) for c in coeff[:-1]]
for frac in fracs:
if frac.denominator > param['denominator_max']:
return x, False, ''
approx = 0
latex = ''
for k in range(len(fracs)):
if fracs[k] != 0:
approx = approx + fracs[k] * sympy.sqrt(list_roots[
k]) # it is expensive to use sympy.sqrt. Maybe hard code it?
latex = latex + fraction_x_root_to_latex(
(0, fracs[k], list_roots[k]), **param) + '+'
return approx, True, latex[:-1]
def field_isomorphism_pslq(a, b):
"""Construct field isomorphism using PSLQ algorithm."""
if not all(_.domain.is_RationalField and _.ext.is_real for _ in (a, b)):
raise NotImplementedError("PSLQ doesn't support complex coefficients")
f = a.minpoly
g = b.minpoly.replace(f.gen)
m = b.minpoly.degree()
for n in mpmath.libmp.libintmath.giant_steps(32, 256): # pragma: no branch
with mpmath.workdps(n):
A = lambdify((), a.ext, "mpmath")()
B = lambdify((), b.ext, "mpmath")()
basis = [B**i for i in range(m)] + [A]
coeffs = mpmath.pslq(basis, maxcoeff=int(1e10), maxsteps=1000)
if coeffs is None:
break
coeffs = [QQ(c, coeffs[-1]) for c in coeffs[:-1]]
while not coeffs[-1]:
coeffs.pop()
coeffs.reverse()
h = Poly(coeffs, f.gen, domain='QQ')
if f.compose(h).rem(g).is_zero or f.compose(-h).rem(g).is_zero:
return [-c for c in coeffs]
def find_simple_recurrence_vector(v, maxcoeff=1024):
"""
This function is used internally by other functions from the
sympy.concrete.guess module. While most users may want to rather use the
function find_simple_recurrence when looking for recurrence relations
among rational numbers, the current function may still be useful when
some post-processing has to be done.
The function returns a vector of length n when a recurrence relation of
order n is detected in the sequence of rational numbers v.
If the returned vector has a length 1, then the returned value is always
the list [0], which means that no relation has been found.
While the functions is intended to be used with rational numbers, it should
work for other kinds of real numbers except for some cases involving
quadratic numbers; for that reason it should be used with some caution when
the argument is not a list of rational numbers.
Examples
========
>>> from sympy.concrete.guess import find_simple_recurrence_vector
>>> from sympy import fibonacci
>>> find_simple_recurrence_vector([fibonacci(k) for k in range(12)])
[1, -1, -1]
See also
========
See the function sympy.concrete.guess.find_simple_recurrence which is more
user-friendly.
"""
l = len(v) >> 1
previous = mp.prec # save current precision
mp.prec = 128
b = [
sum(sqrt((l >> 1)**2 + k) * v[-1 - k - i] for k in range(l))
for i in range(l)
]
p = pslq(b, maxcoeff=maxcoeff, maxsteps=128 + 4 * l)
mp.prec = previous # restore current precision
if p == None: return [0]
first, last = 0, l - 1
while p[first] == 0:
first += 1
while p[last] == 0:
last -= 1
if first == last: return [0] # TODO: probably never occuring
return p[first:last + 1]
def find_simple_recurrence_vector(v, maxcoeff=1024):
"""
This function is used internally by other functions from the
sympy.concrete.guess module. While most users may want to rather use the
function find_simple_recurrence when looking for recurrence relations
among rational numbers, the current function may still be useful when
some post-processing has to be done.
The function returns a vector of length n when a recurrence relation of
order n is detected in the sequence of rational numbers v.
If the returned vector has a length 1, then the returned value is always
the list [0], which means that no relation has been found.
While the functions is intended to be used with rational numbers, it should
work for other kinds of real numbers except for some cases involving
quadratic numbers; for that reason it should be used with some caution when
the argument is not a list of rational numbers.
Examples
========
>>> from sympy.concrete.guess import find_simple_recurrence_vector
>>> from sympy import fibonacci
>>> find_simple_recurrence_vector([fibonacci(k) for k in range(12)])
[1, -1, -1]
See also
========
See the function sympy.concrete.guess.find_simple_recurrence which is more
user-friendly.
"""
l = len(v)>>1
previous = mp.prec # save current precision
mp.prec = 128
b = [sum(sqrt((l>>1)**2 + k)*v[-1-k-i] for k in range(l))
for i in range(l)]
p = pslq(b,
maxcoeff = maxcoeff,
maxsteps = 128 + 4*l)
mp.prec = previous # restore current precision
if p == None: return [0]
first, last = 0, l-1
while p[first]==0: first += 1
while p[last]==0:
last -= 1
if first == last: return [0] # TODO: probably never occuring
return p[first:last+1]
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 range(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 range(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
