def closed_form(polynomial, roots, symbolic=False):
'''
A simple heuristic to check if an approximate root is close to some algebraically
expressible number.
:param symbolic: Return an exact symbolic representation (via sympy), for LaTeX
'''
x = []
for root in roots:
rl = mpmath.identify(root.real, tol=1e-4, maxcoeff=30)
im = mpmath.identify(root.imag, tol=1e-4, maxcoeff=30)
if not rl:
rl = root.real if not symbolic else sympify(rl).evalf(5)
if not im:
im = root.imag if not symbolic else sympify(im).evalf(5)
new = float(sympify(rl)) + float(sympify(im))*1j
# Arbitrary roundoff
if abs(polynomial(root)) > abs(polynomial(new)) or abs(polynomial(new)) < 1e-10:
if symbolic:
x.append(sympify(rl) + sympify(im) * 1j)
else:
x.append(new)
else:
if symbolic:
x.append(sympify(root))
else:
x.append(root)
return array(x)
def inverse_symbolic(n, threshold=1e-5):
rl = mpmath.identify(n.real, tol=1e-3, maxcoeff=30)
im = mpmath.identify(n.imag, tol=1e-3, maxcoeff=30)
if not rl or abs(n.real - (nsimplify(rl).evalf())) > threshold:
rl = nsimplify(n.real).evalf(5)
if not im or abs(n.imag - (nsimplify(im).evalf())) > threshold:
im = nsimplify(n.imag).evalf(5)
return (sympify(rl) + sympify(im) * 1j)
def handleIdentify( result ):
formula = identify( result )
if formula is None:
base = [ 'pi', 'e', 'euler' ]
formula = identify( result, base )
if formula is None:
print( ' = [formula cannot be found]' )
else:
print( ' = ' + formula )
def handleIdentify( result, file=sys.stdout ):
'''Calls the mpmath identify function to try to identify a constant.'''
formula = identify( result )
if formula is None:
base = [ 'pi', 'e', 'euler' ]
formula = identify( result, base )
if formula is None:
print( ' = [formula cannot be found]', file=file )
else:
print( ' = ' + formula, file=file )
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.findpoly(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 xv not 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 handleIdentify(result, file=sys.stdout):
'''Calls the mpmath identify function to try to identify a constant.'''
if isinstance(result, (list, RPNGenerator)):
print('foo!', result)
for i in list(result):
print('i', i)
handleIdentify(i)
return
formula = identify(result)
if formula is None:
base = ['pi', 'e', 'euler', 'sqrt(pi)', 'phi']
formula = identify(result, base)
if formula is None:
print(' = [formula cannot be found]', file=file)
else:
print(' = ' + formula, file=file)
def identify(a, abs_tol=1.0e-15):
sols = mpmath.identify(a, ["pi"], tol=abs_tol, full=True)
return list(set([sympify(sol) for sol in sols]))