def test_levin_3():
mp.dps = 17
z = mp.mpf(2)
eps = mp.mpf(mp.eps)
with mp.extraprec(
7 * mp.prec
): # we need copious amount of precision to sum this highly divergent series
L = mp.levin(method="levin", variant="t")
n, s = 0, 0
while 1:
s += (-z)**n * mp.fac(4 * n) / (mp.fac(n) * mp.fac(2 * n) * (4**n))
n += 1
v, e = L.step_psum(s)
if e < eps:
break
if n > 1000: raise RuntimeError("iteration limit exceeded")
eps = mp.exp(0.8 * mp.log(eps))
exact = mp.quad(lambda x: mp.exp(-x * x / 2 - z * x**4),
[0, mp.inf]) * 2 / mp.sqrt(2 * mp.pi)
# there is also a symbolic expression for the integral:
# exact = mp.exp(mp.one / (32 * z)) * mp.besselk(mp.one / 4, mp.one / (32 * z)) / (4 * mp.sqrt(z * mp.pi))
err = abs(v - exact)
assert err < eps
w = mp.nsum(lambda n: (-z)**n * mp.fac(4 * n) /
(mp.fac(n) * mp.fac(2 * n) * (4**n)), [0, mp.inf],
method="levin",
levin_variant="t",
workprec=8 * mp.prec,
steps=[2] + [1 for x in xrange(1000)])
err = abs(v - w)
assert err < eps
def test_levin_2():
# [2] A. Sidi - "Pratical Extrapolation Methods" p.373
mp.dps = 17
z = mp.mpf(10)
eps = mp.mpf(mp.eps)
with mp.extraprec(2 * mp.prec):
L = mp.levin(method="sidi", variant="t")
n = 0
while 1:
s = (-1)**n * mp.fac(n) * z**(-n)
v, e = L.step(s)
n += 1
if e < eps:
break
if n > 1000: raise RuntimeError("iteration limit exceeded")
eps = mp.exp(0.9 * mp.log(eps))
exact = mp.quad(lambda x: mp.exp(-x) / (1 + x / z), [0, mp.inf])
# there is also a symbolic expression for the integral:
# exact = z * mp.exp(z) * mp.expint(1,z)
err = abs(v - exact)
assert err < eps
w = mp.nsum(lambda n: (-1)**n * mp.fac(n) * z**(-n), [0, mp.inf],
method="sidi",
levin_variant="t")
assert err < eps
def run(qtype, FW, R, alpha=0, beta=0):
X, W = mp.gauss_quadrature(n, qtype, alpha=alpha, beta=beta)
a = 0
for i in xrange(len(X)):
a += W[i] * F(X[i])
b = mp.quad(lambda x: FW(x) * F(x), R)
c = mp.fabs(a - b)
if verbose:
print(qtype, c, a, b)
assert c < 1e-5
def run(qtype, FW, R, alpha = 0, beta = 0):
X, W = mp.gauss_quadrature(n, qtype, alpha = alpha, beta = beta)
a = 0
for i in xrange(len(X)):
a += W[i] * F(X[i])
b = mp.quad(lambda x: FW(x) * F(x), R)
c = mp.fabs(a - b)
if verbose:
print(qtype, c, a, b)
assert c < 1e-5
def test_levin_3():
mp.dps = 17
z=mp.mpf(2)
eps = mp.mpf(mp.eps)
with mp.extraprec(7*mp.prec): # we need copious amount of precision to sum this highly divergent series
L = mp.levin(method = "levin", variant = "t")
n, s = 0, 0
while 1:
s += (-z)**n * mp.fac(4 * n) / (mp.fac(n) * mp.fac(2 * n) * (4 ** n))
n += 1
v, e = L.step_psum(s)
if e < eps:
break
if n > 1000: raise RuntimeError("iteration limit exceeded")
eps = mp.exp(0.8 * mp.log(eps))
exact = mp.quad(lambda x: mp.exp( -x * x / 2 - z * x ** 4), [0,mp.inf]) * 2 / mp.sqrt(2 * mp.pi)
# there is also a symbolic expression for the integral:
# exact = mp.exp(mp.one / (32 * z)) * mp.besselk(mp.one / 4, mp.one / (32 * z)) / (4 * mp.sqrt(z * mp.pi))
err = abs(v - exact)
assert err < eps
w = mp.nsum(lambda n: (-z)**n * mp.fac(4 * n) / (mp.fac(n) * mp.fac(2 * n) * (4 ** n)), [0, mp.inf], method = "levin", levin_variant = "t", workprec = 8*mp.prec, steps = [2] + [1 for x in xrange(1000)])
err = abs(v - w)
assert err < eps
def test_levin_2():
# [2] A. Sidi - "Pratical Extrapolation Methods" p.373
mp.dps = 17
z=mp.mpf(10)
eps = mp.mpf(mp.eps)
with mp.extraprec(2 * mp.prec):
L = mp.levin(method = "sidi", variant = "t")
n = 0
while 1:
s = (-1)**n * mp.fac(n) * z ** (-n)
v, e = L.step(s)
n += 1
if e < eps:
break
if n > 1000: raise RuntimeError("iteration limit exceeded")
eps = mp.exp(0.9 * mp.log(eps))
exact = mp.quad(lambda x: mp.exp(-x)/(1+x/z),[0,mp.inf])
# there is also a symbolic expression for the integral:
# exact = z * mp.exp(z) * mp.expint(1,z)
err = abs(v - exact)
assert err < eps
w = mp.nsum(lambda n: (-1) ** n * mp.fac(n) * z ** (-n), [0, mp.inf], method = "sidi", levin_variant = "t")
assert err < eps
def integral_bounded_mp(fn, lb, ub):
integral, _ = mp.quad(fn, [lb, ub], error=True)
return integral
def integral_inf_mp(fn):
integral, _ = mp.quad(fn, [-mp.inf, mp.inf], error=True)
return integral
def integral_bounded_mp(fn, lb, ub): integral, _ = mp.quad(fn, [lb, ub], error=True) return integral
def integral_inf_mp(fn): integral, _ = mp.quad(fn, [-mp.inf, mp.inf], error=True) return integral
