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 test_levin_nsum():
mp.dps = 17
with mp.extraprec(mp.prec):
z = mp.mpf(10) ** (-10)
a = mp.nsum(lambda n: n**(-(1+z)), [1, mp.inf], method = "l") - 1 / z
assert abs(a - mp.euler) < 1e-10
eps = mp.exp(0.8 * mp.log(mp.eps))
a = mp.nsum(lambda n: (-1)**(n-1) / n, [1, mp.inf], method = "sidi")
assert abs(a - mp.log(2)) < eps
z = 2 + 1j
f = lambda n: mp.rf(2 / mp.mpf(3), n) * mp.rf(4 / mp.mpf(3), n) * z**n / (mp.rf(1 / mp.mpf(3), n) * mp.fac(n))
v = mp.nsum(f, [0, mp.inf], method = "levin", steps = [10 for x in xrange(1000)])
exact = mp.hyp2f1(2 / mp.mpf(3), 4 / mp.mpf(3), 1 / mp.mpf(3), z)
assert abs(exact - v) < eps
def test_levin_nsum():
mp.dps = 17
with mp.extraprec(mp.prec):
z = mp.mpf(10)**(-10)
a = mp.nsum(lambda n: n**(-(1 + z)), [1, mp.inf], method="l") - 1 / z
assert abs(a - mp.euler) < 1e-10
eps = mp.exp(0.8 * mp.log(mp.eps))
a = mp.nsum(lambda n: (-1)**(n - 1) / n, [1, mp.inf], method="sidi")
assert abs(a - mp.log(2)) < eps
z = 2 + 1j
f = lambda n: mp.rf(2 / mp.mpf(3), n) * mp.rf(4 / mp.mpf(3), n) * z**n / (
mp.rf(1 / mp.mpf(3), n) * mp.fac(n))
v = mp.nsum(f, [0, mp.inf],
method="levin",
steps=[10 for x in xrange(1000)])
exact = mp.hyp2f1(2 / mp.mpf(3), 4 / mp.mpf(3), 1 / mp.mpf(3), z)
assert abs(exact - v) < eps
def test_levin_1():
mp.dps = 17
eps = mp.mpf(mp.eps)
with mp.extraprec(2 * mp.prec):
L = mp.levin(method = "levin", variant = "v")
A, n = [], 1
while 1:
s = mp.mpf(n) ** (2 + 3j)
n += 1
A.append(s)
v, e = L.update(A)
if e < eps:
break
if n > 1000: raise RuntimeError("iteration limit exceeded")
eps = mp.exp(0.9 * mp.log(eps))
err = abs(v - mp.zeta(-2-3j))
assert err < eps
w = mp.nsum(lambda n: n ** (2 + 3j), [1, mp.inf], method = "levin", levin_variant = "v")
err = abs(v - w)
assert err < eps
def test_levin_0():
mp.dps = 17
eps = mp.mpf(mp.eps)
with mp.extraprec(2 * mp.prec):
L = mp.levin(method = "levin", variant = "u")
S, s, n = [], 0, 1
while 1:
s += mp.one / (n * n)
n += 1
S.append(s)
v, e = L.update_psum(S)
if e < eps:
break
if n > 1000: raise RuntimeError("iteration limit exceeded")
eps = mp.exp(0.9 * mp.log(eps))
err = abs(v - mp.pi ** 2 / 6)
assert err < eps
w = mp.nsum(lambda n: 1/(n * n), [1, mp.inf], method = "levin", levin_variant = "u")
err = abs(v - w)
assert err < eps
def test_levin_1():
mp.dps = 17
eps = mp.mpf(mp.eps)
with mp.extraprec(2 * mp.prec):
L = mp.levin(method="levin", variant="v")
A, n = [], 1
while 1:
s = mp.mpf(n)**(2 + 3j)
n += 1
A.append(s)
v, e = L.update(A)
if e < eps:
break
if n > 1000: raise RuntimeError("iteration limit exceeded")
eps = mp.exp(0.9 * mp.log(eps))
err = abs(v - mp.zeta(-2 - 3j))
assert err < eps
w = mp.nsum(lambda n: n**(2 + 3j), [1, mp.inf],
method="levin",
levin_variant="v")
err = abs(v - w)
assert err < eps
def test_levin_0():
mp.dps = 17
eps = mp.mpf(mp.eps)
with mp.extraprec(2 * mp.prec):
L = mp.levin(method="levin", variant="u")
S, s, n = [], 0, 1
while 1:
s += mp.one / (n * n)
n += 1
S.append(s)
v, e = L.update_psum(S)
if e < eps:
break
if n > 1000: raise RuntimeError("iteration limit exceeded")
eps = mp.exp(0.9 * mp.log(eps))
err = abs(v - mp.pi**2 / 6)
assert err < eps
w = mp.nsum(lambda n: 1 / (n * n), [1, mp.inf],
method="levin",
levin_variant="u")
err = abs(v - w)
assert err < eps
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