def daubechis(N):
# make polynomial
q_y = [sm.binomial(N-1+k,k) for k in reversed(range(N))]
# get polynomial roots y[k]
y = sm.mp.polyroots(q_y, maxsteps=200, extraprec=64)
z = []
for yk in y:
# subustitute y = -1/4z + 1/2 - 1/4/z to factor f(y) = y - y[k]
f = [sm.mpf('-1/4'), sm.mpf('1/2') - yk, sm.mpf('-1/4')]
# get polynomial roots z[k]
z += sm.mp.polyroots(f)
# make polynomial using the roots within unit circle
h0z = sm.sqrt('2')
for zk in z:
if sm.fabs(zk) < 1:
h0z *= sympy.sympify('(z-zk)/(1-zk)').subs('zk',zk)
# adapt vanising moments
hz = (sympy.sympify('(1+z)/2')**N*h0z).expand()
# get scaling coefficients
return [sympy.re(hz.coeff('z',k)) for k in reversed(range(N*2))]
def test_mpmath_lambda():
mpmath.mp.dps = 50
sin02 = mpmath.mpf("0.19866933079506121545941262711838975037020672954020")
f = lambdify(x, sin(x), "mpmath")
prec = 1e-49 # mpmath precision is around 50 decimal places
assert -prec < f(mpmath.mpf("0.2")) - sin02 < prec
raises(TypeError, lambda: f(x))
def test_mpmath_lambda():
mpmath.mp.dps = 50
sin02 = mpmath.mpf("0.19866933079506121545941262711838975037020672954020")
f = lambdify(x, sin(x), "mpmath")
prec = 1e-49 # mpmath precision is around 50 decimal places
assert -prec < f(mpmath.mpf("0.2")) - sin02 < prec
raises(TypeError, lambda: f(x))
def _optimize_single(opt_params):
'''mapped function for optimization of a single function in a single domain
opt_params is a tuple (expression, subs, bound, use_basinhopping, use_corners)
returns the interval
'''
(e, subs, bound, use_basinhopping, use_corners) = opt_params
assert not (use_basinhopping and use_corners)
rv = None
e = _simplify_eq(e)
if bound is None:
rv = _eval_eq_direct(e, subs)
else:
under_approx = None
if use_corners:
under_approx = _eval_at_corners(e, subs)
elif use_basinhopping:
under_approx = _basinhopping(e, subs)
else:
under_approx = _eval_at_middle(e, subs)
# under_approx param should be an array of size 1, since it gets updated
rv = _eval_eq_bounded(e, subs, bound, [under_approx])
# convert rv from interval to tuple (for pickling)
return [float(mpf(rv.a)), float(mpf(rv.b))]
def uniform_nodes(n):
"""Return uniform nodes."""
r = set()
for i in range(n):
r.add(mpmath.mpf('-1.0') + i * mpmath.mpf('2.0')/(n-1))
return map_to_zero_one(r)
def mp_real(x):
if type(x) == mpc:
if x.imag == mpf(0):
return mpf(x.real)
else:
raise ValueError
else:
return mpf(x)
def gauss_lobatto(n):
"""Return Gauss-Lobatto nodes.
Gauss-Lobatto nodes are roots of P'_{n-1}(x) and -1, 1.
"""
x = sympy.var('x')
p = legendre_poly(n-1).diff(x)
r = find_roots(p)
r = [mpmath.mpf('-1.0'), mpmath.mpf('1.0')] + r
return sorted(r)
def gauss_lobatto(n):
"""Return Gauss-Lobatto nodes.
Gauss-Lobatto nodes are roots of P'_{n-1}(x) and -1, 1.
"""
x = sympy.var('x')
p = legendre_poly(n - 1).diff(x)
r = find_roots(p)
r = [mpmath.mpf('-1.0'), mpmath.mpf('1.0')] + r
return sorted(r)
def test_mod():
assert mpf(234) % 1 == 0
assert mpf(-3) % 256 == 253
assert mpf(0.25) % 23490.5 == 0.25
assert mpf(0.25) % -23490.5 == -23490.25
assert mpf(-0.25) % 23490.5 == 23490.25
assert mpf(-0.25) % -23490.5 == -0.25
# Check that these cases are handled efficiently
assert mpf('1e10000000000') % 1 == 0
assert mpf('1.23e-1000000000') % 1 == mpf('1.23e-1000000000')
# test __rmod__
assert 3 % mpf('1.75') == 1.25
def as_mpmath(x, prec, options):
x = sympify(x)
if isinstance(x, C.Zero):
return mpf(0)
if isinstance(x, C.Infinity):
return mpf("inf")
if isinstance(x, C.NegativeInfinity):
return mpf("-inf")
# XXX
re, im, _, _ = evalf(x, prec, options)
if im:
return mpc(re or fzero, im)
return mpf(re)
def gauss_lobatto_nodes(n):
"""Return Gauss-Lobatto nodes.
Before mapping to [0,1], the nodes are: roots of :math:`P'_{n-1}(x)`
and -1, 1.
"""
x = sympy.var('x')
p = legendre_poly(n-1).diff(x)
r = find_roots(p)
r = [mpmath.mpf('-1.0'), mpmath.mpf('1.0')] + r
return map_to_zero_one(r)
def as_mpmath(x, prec, options):
x = sympify(x)
if isinstance(x, C.Zero):
return mpf(0)
if isinstance(x, C.Infinity):
return mpf('inf')
if isinstance(x, C.NegativeInfinity):
return mpf('-inf')
# XXX
re, im, _, _ = evalf(x, prec, options)
if im:
return mpc(re or fzero, im)
return mpf(re)
def mc_compute_stationary(P, precision=None, tol=None):
n = P.shape[0]
if precision is None:
# Compute eigenvalues and eigenvectors
eigvals, eigvecs = la.eig(P, left=True, right=False)
# Find the index for unit eigenvalues
index = np.where(abs(eigvals - 1.) < 1e-12)[0]
# Pull out the eigenvectors that correspond to unit eig-vals
uniteigvecs = eigvecs[:, index]
stationary_dists = uniteigvecs / np.sum(uniteigvecs, axis=0)
else:
# Create a list to store eigvals
stationary_dists_list = []
if tol is None:
# If tolerance isn't specified then use 2*precision
tol = mp.mpf(2 * 10**(-precision + 1))
with mp.workdps(precision):
eigvals, eigvecs = mp.eig(mp.matrix(P), left=True, right=False)
for ind, el in enumerate(eigvals):
if el >= (mp.mpf(1) - mp.mpf(tol)) and el <= (mp.mpf(1) +
mp.mpf(tol)):
stationary_dists_list.append(eigvecs[ind, :])
stationary_dists = np.asarray(stationary_dists_list).T
stationary_dists = (stationary_dists /
sum(stationary_dists)).astype(np.float)
# Check to make sure all of the elements of invar_dist are positive
if np.any(stationary_dists < -1e-16):
warn("Elements of your invariant distribution were negative; " +
"Re-trying with additional precision")
if precision is None:
stationary_dists = mc_compute_stationary(P, precision=18, tol=tol)
elif precision is not None:
raise ValueError("Elements of your stationary distribution were" +
"negative. Try computing with higher precision")
# Since we will be accessing the columns of this matrix, we
# might consider adding .astype(np.float, order='F') to make it
# column major at beginning
return stationary_dists.squeeze()
def mc_compute_stationary(P, precision=None, tol=None):
n = P.shape[0]
if precision is None:
# Compute eigenvalues and eigenvectors
eigvals, eigvecs = la.eig(P, left=True, right=False)
# Find the index for unit eigenvalues
index = np.where(abs(eigvals - 1.) < 1e-12)[0]
# Pull out the eigenvectors that correspond to unit eig-vals
uniteigvecs = eigvecs[:, index]
stationary_dists = uniteigvecs/np.sum(uniteigvecs, axis=0)
else:
# Create a list to store eigvals
stationary_dists_list = []
if tol is None:
# If tolerance isn't specified then use 2*precision
tol = mp.mpf(2 * 10**(-precision + 1))
with mp.workdps(precision):
eigvals, eigvecs = mp.eig(mp.matrix(P), left=True, right=False)
for ind, el in enumerate(eigvals):
if el>=(mp.mpf(1)-mp.mpf(tol)) and el<=(mp.mpf(1)+mp.mpf(tol)):
stationary_dists_list.append(eigvecs[ind, :])
stationary_dists = np.asarray(stationary_dists_list).T
stationary_dists = (stationary_dists/sum(stationary_dists)).astype(np.float)
# Check to make sure all of the elements of invar_dist are positive
if np.any(stationary_dists < -1e-16):
warn("Elements of your invariant distribution were negative; " +
"Re-trying with additional precision")
if precision is None:
stationary_dists = mc_compute_stationary(P, precision=18, tol=tol)
elif precision is not None:
raise ValueError("Elements of your stationary distribution were" +
"negative. Try computing with higher precision")
# Since we will be accessing the columns of this matrix, we
# might consider adding .astype(np.float, order='F') to make it
# column major at beginning
return stationary_dists.squeeze()
def test_mpmath_lambda():
dps = mpmath.mp.dps
mpmath.mp.dps = 50
try:
sin02 = mpmath.mpf("0.19866933079506121545941262711838975037020672954020")
f = lambdify(x, sin(x), "mpmath")
prec = 1e-49 # mpmath precision is around 50 decimal places
assert -prec < f(mpmath.mpf("0.2")) - sin02 < prec
try:
f(x) # if this succeeds, it can't be a mpmath function
assert False
except TypeError:
pass
finally:
mpmath.mp.dps = dps
def genScaledVoltage(self):
if self.isZeroT:
self.Vq = self.Q*ELEC*self.V / HBAR
self.scaledVolt = -1j*self.Vq
else:
self.Vq = self.Q*ELEC*self.V/(2*pi*BLTZMN*self.T)
self.scaledVolt = fmpc(self.gtot /mpf(2), - self.Vq )
def f(i, j):
gamDict = self.__genGammaDict(j, gDict)
gamDict[-1] = mpf('0')
def g(n):
return gamDict[n]
_g = vectorize(g)
return _g(i)
def test_3250():
from sympy.mpmath import mpf
assert str(Float(mpf((1, 22, 2, 22)), '')) == '-88.000'
assert Float('23.e3', '')._prec == 10
assert Float('23e3', '')._prec == 20
assert Float('23000', '')._prec == 20
assert Float('-23000', '')._prec == 20
def log(self):
"""
Logaritmo de un intervalo: 'self.log()'
NOTA: Si el intervalo contiene al 0, pero no es estrictamente negativo,
se calcula el logaritmo de la intersecci\'on del intervalo con el dominio
natural del logaritmo, i.e., [0,+inf].
"""
if 0 in self:
domainNatural = Intervalo( 0, mpf('inf') )
intervalRestricted = self.intersection( domainNatural )
txt_warning = "\nWARNING: Interval {} contains 0 or negative numbers.\n".format(self)
print txt_warning
txt_warning = "Restricting to the intersection "\
"with the natural domain of log(x), i.e. {}\n".format(intervalRestricted)
print txt_warning
return Intervalo( mp.log(intervalRestricted.lo), mp.log(intervalRestricted.hi) )
elif 0 > self.hi:
txt_error = 'Interval {} < 0\nlog(x) cannot be computed '\
'for negative numbers.'.format(self)
raise ValueError( txt_error )
else:
return Intervalo( mp.log(self.lo), mp.log(self.hi) )
def genPrefactor(self):
if self.isZeroT:
self.prefac = fmfy(np.ones_like(self.parameters[:, 0]))
else:
fac = 2 * pi * BLTZMN * self.T / HBAR
self.prefac = fexp(
np.sum(self.g * self.parameters, axis=1) * fac / mpf(2))
def log(self):
"""
Logaritmo de un intervalo: 'self.log()'
NOTA: Si el intervalo contiene al 0, pero no es estrictamente negativo,
se calcula el logaritmo de la intersecci\'on del intervalo con el dominio
natural del logaritmo, i.e., [0,+inf].
"""
if 0 in self:
domainNatural = Intervalo(0, mpf('inf'))
intervalRestricted = self.intersection(domainNatural)
txt_warning = "\nWARNING: Interval {} contains 0 or negative numbers.\n".format(
self)
print txt_warning
txt_warning = "Restricting to the intersection "\
"with the natural domain of log(x), i.e. {}\n".format(intervalRestricted)
print txt_warning
return Intervalo(mp.log(intervalRestricted.lo),
mp.log(intervalRestricted.hi))
elif 0 > self.hi:
txt_error = 'Interval {} < 0\nlog(x) cannot be computed '\
'for negative numbers.'.format(self)
raise ValueError(txt_error)
else:
return Intervalo(mp.log(self.lo), mp.log(self.hi))
def test_math_lambda():
mpmath.mp.dps = 50
sin02 = mpmath.mpf("0.19866933079506121545941262711838975037020672954020")
f = lambdify(x, sin(x), "math")
prec = 1e-15
assert -prec < f(0.2) - sin02 < prec
raises(ValueError,"f(x)") # if this succeeds, it can't be a python math function
def test_3250():
from sympy.mpmath import mpf
assert str(Float(mpf((1,22,2,22)), '')) == '-88.000'
assert Float('23.e3', '')._prec == 10
assert Float('23e3', '')._prec == 20
assert Float('23000', '')._prec == 20
assert Float('-23000', '')._prec == 20
def test_mpmath_lambda():
dps = mpmath.mp.dps
mpmath.mp.dps = 50
try:
sin02 = mpmath.mpf(
"0.19866933079506121545941262711838975037020672954020")
f = lambdify(x, sin(x), "mpmath")
prec = 1e-49 # mpmath precision is around 50 decimal places
assert -prec < f(mpmath.mpf("0.2")) - sin02 < prec
try:
f(x) # if this succeeds, it can't be a mpmath function
assert False
except TypeError:
pass
finally:
mpmath.mp.dps = dps
def test_math_lambda():
mpmath.mp.dps = 50
sin02 = mpmath.mpf("0.19866933079506121545941262711838975037020672954020")
f = lambdify(x, sin(x), "math")
prec = 1e-15
assert -prec < f(0.2) - sin02 < prec
raises(ValueError, lambda: f(x))
def __init__(self, parameters, maxParameter,
g, V, prefac, gtot,
scaledVolt, T, Vq):
self.Vq = Vq
self.parameters = parameters
self.V = V
self.T = T
self.isZeroT = mp.almosteq(self.T, mpf(0))
self.scaledVolt = scaledVolt
self.g = g
self.maxParameter = maxParameter
self.prefac = prefac
self.gtot = gtot
### input check ###
if len(self.maxParameter.shape) == 1:
self.maxParameter = self.maxParameter[newaxis,:]
if len(self.prefac.shape) == 1:
self.prefac = self.prefac[newaxis,:]
if self.g.shape != self.parameters.shape or \
self.V.shape != self.scaledVolt.shape or \
self.maxParameter.shape[1] != self.parameters.shape[0]:
raise ValueError
if len(self.g.shape) == 1:
self.g = self.g[:, newaxis]
self.parameters = self.parameters[:, newaxis]
self.prefac = self.prefac * np.power(self.maxParameter,
-self.scaledVolt[...,newaxis])
def _f(i, j):
ldaDICT = dict((k, self.lda[k][j]) for k in self.wijnTerms)
ldaDICT[-1] = mpf(0)
def g(n):
return ldaDICT[n]
_g = np.vectorize(g)
return _g(i)
def _eval_evalf(self, prec):
"""Evaluate this complex root to the given precision. """
_prec, mp.prec = mp.prec, prec
try:
func = lambdify(self.poly.gen, self.expr)
interval = self._get_interval()
refined = False
while True:
if self.is_real:
x0 = mpf(str(interval.center))
else:
x0 = mpc(*map(str, interval.center))
try:
root = findroot(func, x0)
except ValueError:
interval = interval.refine()
refined = True
continue
else:
if refined:
self._set_interval(interval)
break
finally:
mp.prec = _prec
return Float._new(root.real._mpf_, prec) + I*Float._new(root.imag._mpf_, prec)
def _eval_evalf(self, prec):
"""Evaluate this complex root to the given precision. """
_prec, mp.prec = mp.prec, prec
try:
func = lambdify(self.poly.gen, self.expr)
interval = self._get_interval()
refined = False
while True:
if self.is_real:
x0 = mpf(str(interval.center))
else:
x0 = mpc(*map(str, interval.center))
try:
root = findroot(func, x0)
except ValueError:
interval = interval.refine()
refined = True
continue
else:
if refined:
self._set_interval(interval)
break
finally:
mp.prec = _prec
return Float._new(root.real._mpf_, prec) + I*Float._new(root.imag._mpf_, prec)
def __genGammaDict(self, j, gDict):
if self.isZeroT:
return dict((k, self.scaledVolt[j]**k /gDict[k]) for k in self.wijnTerms)
else:
fg = fgamma(self.scaledVolt[j])
return dict((k, fgamma(self.scaledVolt[j] + mpf(str(k))) / \
(fg*gDict[k])) for k in self.wijnTerms)
def test_math_lambda():
mpmath.mp.dps = 50
sin02 = mpmath.mpf("0.19866933079506121545941262711838975037020672954020")
f = lambdify(x, sin(x), "math")
prec = 1e-15
assert -prec < f(0.2) - sin02 < prec
raises(TypeError, lambda: f(x))
def genScaledVoltage(self):
if self.isZeroT:
self.Vq = self.Q * ELEC * self.V / HBAR
self.scaledVolt = -1j * self.Vq
else:
self.Vq = self.Q * ELEC * self.V / (2 * pi * BLTZMN * self.T)
self.scaledVolt = fmpc(self.gtot / mpf(2), -self.Vq)
def clenshaw_curtis_nodes(n):
"""Return Clenshaw-Curtis nodes (actually returns n+1 nodes)."""
r = set()
for k in range(n+1):
r.add(mpmath.cos(k*mpmath.pi/mpmath.mpf(n)))
return sorted(r)
def clenshaw_curtis_nodes(n):
"""Return Clenshaw-Curtis nodes."""
r = set()
for i in range(n):
r.add(mpmath.cos(i*mpmath.pi/mpmath.mpf(n-1)))
return map_to_zero_one(r)
def test_3250():
from sympy.mpmath import mpf
assert str(Float(mpf((1, 22, 2, 22)), "")) == "-88.000"
assert Float("23.e3", "")._prec == 10
assert Float("23e3", "")._prec == 20
assert Float("23000", "")._prec == 20
assert Float("-23000", "")._prec == 20
def get_real(y):
if type(y) == mpc:
if y.imag == mpf(0):
return y.real
else:
raise ValueError
else:
return mp.mpf(y)
def __genGammaDict(self, j, gDict):
if self.isZeroT:
return dict(
(k, self.scaledVolt[j]**k / gDict[k]) for k in self.wijnTerms)
else:
fg = fgamma(self.scaledVolt[j])
return dict((k, fgamma(self.scaledVolt[j] + mpf(str(k))) / \
(fg*gDict[k])) for k in self.wijnTerms)
def f(i, j):
gamDict = self.__genGammaDict(j, gDict)
gamDict[-1] = mpf('0')
def g(n):
return gamDict[n]
_g = vectorize(g)
return _g(i)
def test_mpmath_lambda():
f = lambdify(x, sin(x), "mpmath")
prec = 1e-49 # mpmath precision is around 50 decimal places
assert -prec < f(mpmath.mpf("0.2")) - sin02 < prec
try:
f(x) # if this succeeds, it can't be a mpmath function
raise Exception
except TypeError:
pass
def test_sympy_lambda():
mpmath.mp.dps = 50
sin02 = mpmath.mpf("0.19866933079506121545941262711838975037020672954020")
f = lambdify(x, sin(x), "sympy")
assert f(x) == sin(x)
prec = 1e-15
assert -prec < f(Rational(1, 5)).evalf() - Float(str(sin02)) < prec
# arctan is in numpy module and should not be available
raises(NameError, lambda: lambdify(x, arctan(x), "sympy"))
def test_sympy_lambda():
mpmath.mp.dps = 50
sin02 = mpmath.mpf("0.19866933079506121545941262711838975037020672954020")
f = lambdify(x, sin(x), "sympy")
assert f(x) == sin(x)
prec = 1e-15
assert -prec < f(Rational(1, 5)).evalf() - Float(str(sin02)) < prec
# arctan is in numpy module and should not be available
raises(NameError, lambda: lambdify(x, arctan(x), "sympy"))
def _f(i, j):
ldaDICT = dict((k, self.lda[k][j]) for k in self.wijnTerms)
ldaDICT[-1] = mpf(0)
def g(n):
return ldaDICT[n]
_g = np.vectorize(g)
return _g(i)
def test_mpmath_lambda():
f = lambdify(x, sin(x), "mpmath")
prec = 1e-49 # mpmath precision is around 50 decimal places
assert -prec < f(mpmath.mpf("0.2")) - sin02 < prec
try:
f(x) # if this succeeds, it can't be a mpmath function
assert False
except TypeError:
pass
def update_maximum_error_points(list_a):
n = len(list_a) - 1
extreme_points = search_extreme_points(list_a)
if len(extreme_points) == n + 1:
return [sm.mpf(0.0)] + extreme_points
else:
raise Exception('[ERROR]number of extreme point ' + \
str(n+2) + '->' + str(len(extreme_points)))
def test_number_precision():
dps = mpmath.mp.dps
mpmath.mp.dps = 50
try:
sin02 = mpmath.mpf("0.19866933079506121545941262711838975037020672954020")
f = lambdify(x, sin02, "mpmath")
prec = 1e-49 # mpmath precision is around 50 decimal places
assert -prec < f(0) - sin02 < prec
finally:
mpmath.mp.dps = dps
def probability(n):
""" Calculates the probability of an n-sided die to have all numbers
in the list from 1 through n in a trial of n throws
Input n : n-sided die
Output prob : probability
"""
if not isinstance(n, int):
raise ValueError('Input not of integer instance')
if n <= 0:
raise ValueError('Input not a positive integer')
exclusions_list = [(nC(n, r) * ((n - r)**n)) for r in range(n)]
nr = reduce(operator.add,
[x * y for x, y in izip(exclusions_list, cycle([1, -1]))])
dr = n**n
prob = mpmath.mpf(nr) / mpmath.mpf(dr)
return prob
def arithmeticalOperationX(self, i2, alpha, operation):
""" Wrapper class for performing Fuzzy Arithmetic on X-mu Functions using SymPy/MPI. Primarily used as a private class, but could be used publicly.
@param i2 the target sympy formula.
@param alpha the level on which to perform the operation.
@param operation One of the following: +, -, *, /, **
"""
res = self.get_xequals().subs(ALPHA, float(alpha))
res2 = i2.get_xequals().subs(ALPHA, float(alpha))
if (not res.is_EmptySet) and (not res2.is_EmptySet):
if (type(res) != FiniteSet) and (type(res2) != FiniteSet):
result = eval("res.to_mpi() " + operation + " res2.to_mpi()")
else:
res_mpi = mpmath.mpi(res.inf, res.sup)
res2_mpi = mpmath.mpi(res2.inf, res2.sup)
result = eval("res_mpi " + operation + " res2_mpi")
return BasicXmu(self.u, Interval(float(mpmath.mpf(result.a)), float(mpmath.mpf(result.b))))
else:
return None
def __init__(self, input_parameters, V, Q, T):
self.V, self.Q, self.T = V, mpf(Q), mpf(T)
########## Process input # Perform Sanity Checks ###########
for i, j in input_parameters.items():
if not isinstance(j, list):
raise ValueError
self.input_parameters[i] = np.asarray(j)
if len(set(map(len, input_parameters.values()))) > 1:
raise ValueError
if not isinstance(self.V, np.ndarray) or not isinstance(self.V, list):
self.V = np.array([self.V]).ravel()
self.isZeroT = mp.almosteq(self.T, mpf(0))
########### Restructure distance and voltage, if necessary
if len(self.input_parameters["x"].shape) == 1:
self.input_parameters["x"] = self.input_parameters["x"][:,newaxis]
########### Generate parameters
self.gtot = mp.fsum(self.input_parameters["g"])
self.genParameters()
self.genScaledVoltage()
def _eval_evalf(self, prec):
"""Evaluate this complex root to the given precision. """
with workprec(prec):
g = self.poly.gen
if not g.is_Symbol:
d = Dummy('x')
func = lambdify(d, self.expr.subs(g, d))
else:
func = lambdify(g, self.expr)
interval = self._get_interval()
if not self.is_real:
# For complex intervals, we need to keep refining until the
# imaginary interval is disjunct with other roots, that is,
# until both ends get refined.
ay = interval.ay
by = interval.by
while interval.ay == ay or interval.by == by:
interval = interval.refine()
while True:
if self.is_real:
x0 = mpf(str(interval.center))
else:
x0 = mpc(*map(str, interval.center))
try:
root = findroot(func, x0, verify=False)
# If the (real or complex) root is not in the 'interval',
# then keep refining the interval. This happens if findroot
# accidentally finds a different root outside of this
# interval because our initial estimate 'x0' was not close
# enough.
if self.is_real:
a = mpf(str(interval.a))
b = mpf(str(interval.b))
if a == b:
root = a
break
if not (a < root < b):
raise ValueError("Root not in the interval.")
else:
ax = mpf(str(interval.ax))
bx = mpf(str(interval.bx))
ay = mpf(str(interval.ay))
by = mpf(str(interval.by))
if ax == bx and ay == by:
root = ax + S.ImaginaryUnit * by
break
if not (ax < root.real < bx and ay < root.imag < by):
raise ValueError("Root not in the interval.")
except ValueError:
interval = interval.refine()
continue
else:
break
return Float._new(root.real._mpf_,
prec) + I * Float._new(root.imag._mpf_, prec)
def __pow__(self, exponent):
"""
Se calcula la potencia de un intervalo; operador '**'
UNDER TESTING
"""
if isinstance(exponent, Intervalo): # exponent is an interval
if exponent.lo == exponent.hi: # exponent is a thin interval
return self**exponent.lo
else: # exponent is a generic interval
return (exponent * self.log()).exp()
else: # exponent is a number (int, float, mpf, ...)
if exponent == int(exponent): # exponent is an integer
if exponent >= 0:
if exponent % 2 == 0: # even exponent
return Intervalo((self.mig())**exponent,
(self.mag())**exponent)
else: # odd exponent
return Intervalo(self.lo**exponent, self.hi**exponent)
else: # exponent < 0
return (self**(-exponent)).reciprocal()
else:
# exponent is a generic float
if exponent >= 0:
if 0 in self:
domainNatural = Intervalo(0, mpf('inf'))
intervalRestricted = self.intersection(domainNatural)
txt_warning = "\nWARNING: Interval {} contains 0.\n".format(
self)
print txt_warning
txt_warning = "Restricting to the intersection "\
"with the natural domain of **, i.e. {}\n".format(intervalRestricted)
print txt_warning
return Intervalo(0, self.hi**exponent)
elif 0 > self:
raise ValueError(
"negative interval can not be raised to a fractional power"
)
else:
return Intervalo(self.lo**exponent, self.hi**exponent)
else:
return (self**(-exponent)).reciprocal()
def _eval_evalf(self, prec):
"""Evaluate this complex root to the given precision. """
_prec, mp.prec = mp.prec, prec
try:
func = lambdify(self.poly.gen, self.expr)
interval = self._get_interval()
if not self.is_real:
# For complex intervals, we need to keep refining until the
# imaginary interval is disjunct with other roots, that is,
# until both ends get refined.
ay = interval.ay
by = interval.by
while interval.ay == ay or interval.by == by:
interval = interval.refine()
while True:
if self.is_real:
x0 = mpf(str(interval.center))
else:
x0 = mpc(*map(str, interval.center))
try:
root = findroot(func, x0)
# If the (real or complex) root is not in the 'interval',
# then keep refining the interval. This happens if findroot
# accidentally finds a different root outside of this
# interval because our initial estimate 'x0' was not close
# enough.
if self.is_real:
a = mpf(str(interval.a))
b = mpf(str(interval.b))
# This is needed due to the bug #3364:
a, b = min(a, b), max(a, b)
if not (a < root < b):
raise ValueError("Root not in the interval.")
else:
ax = mpf(str(interval.ax))
bx = mpf(str(interval.bx))
ay = mpf(str(interval.ay))
by = mpf(str(interval.by))
# This is needed due to the bug #3364:
ax, bx = min(ax, bx), max(ax, bx)
ay, by = min(ay, by), max(ay, by)
if not (ax < root.real < bx and ay < root.imag < by):
raise ValueError("Root not in the interval.")
except ValueError:
interval = interval.refine()
continue
else:
break
finally:
mp.prec = _prec
return Float._new(root.real._mpf_,
prec) + I * Float._new(root.imag._mpf_, prec)
def sympy2mpmath(value, prec=None):
if prec is None:
from mathics.builtin.numeric import machine_precision
prec = machine_precision
value = value.n(dps(prec))
if value.is_real:
return mpmath.mpf(value)
elif value.is_number:
return mpmath.mpc(*value.as_real_imag())
else:
return None
def test_lambdify():
mpmath.mp.dps = 16
sin02 = mpmath.mpf("0.198669330795061215459412627")
f = lambdify(x, sin(x), "numpy")
prec = 1e-15
assert -prec < f(0.2) - sin02 < prec
try:
f(x) # if this succeeds, it can't be a numpy function
assert False
except AttributeError:
pass
def test_number_precision():
dps = mpmath.mp.dps
mpmath.mp.dps = 50
try:
sin02 = mpmath.mpf(
"0.19866933079506121545941262711838975037020672954020")
f = lambdify(x, sin02, "mpmath")
prec = 1e-49 # mpmath precision is around 50 decimal places
assert -prec < f(0) - sin02 < prec
finally:
mpmath.mp.dps = dps
def confidence(s, p):
"""Return a symmetric (p*100)% confidence interval. For example,
p=0.95 gives a 95% confidence interval. Currently this function
only handles numerical values except in the trivial case p=1.
For example, one standard deviation:
>>> from sympy.statistics import Normal
>>> N = Normal(0, 1)
>>> N.confidence(0.68)
(-0.994457883209753, 0.994457883209753)
>>> N.probability(*_).evalf()
0.680000000000000
Two standard deviations:
>>> N = Normal(0, 1)
>>> N.confidence(0.95)
(-1.95996398454005, 1.95996398454005)
>>> N.probability(*_).evalf()
0.950000000000000
"""
if p == 1:
return (-oo, oo)
if p > 1:
raise ValueError("p cannot be greater than 1")
# In terms of n*sigma, we have n = sqrt(2)*ierf(p). The inverse
# error function is not yet implemented in SymPy but can easily be
# computed numerically
from sympy.mpmath import mpf, erfinv
# calculate y = ierf(p) by solving erf(y) - p = 0
y = erfinv(mpf(p))
t = Float(str(mpf(float(s.sigma)) * mpf(2)**0.5 * y))
mu = s.mu.evalf()
return (mu - t, mu + t)
def __init__(self, input_parameters, V, Q, T):
self.V, self.Q, self.T = V, mpf(Q), mpf(T)
########## Process input # Perform Sanity Checks ###########
for i, j in input_parameters.items():
if not isinstance(j, list):
raise ValueError
self.input_parameters[i] = np.asarray(j)
if len(set(map(len, input_parameters.values()))) > 1:
raise ValueError
if not isinstance(self.V, np.ndarray) or not isinstance(self.V, list):
self.V = np.array([self.V]).ravel()
self.isZeroT = mp.almosteq(self.T, mpf(0))
########### Restructure distance and voltage, if necessary
if len(self.input_parameters["x"].shape) == 1:
self.input_parameters["x"] = self.input_parameters["x"][:, newaxis]
########### Generate parameters
self.gtot = mp.fsum(self.input_parameters["g"])
self.genParameters()
self.genScaledVoltage()
def f(t):
re, im, re_acc, im_acc = evalf(func, mp.prec, {'subs': {x: t}})
have_part[0] = re or have_part[0]
have_part[1] = im or have_part[1]
max_real_term[0] = max(max_real_term[0], fastlog(re))
max_imag_term[0] = max(max_imag_term[0], fastlog(im))
if im:
return mpc(re or fzero, im)
return mpf(re or fzero)