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 _gwr_no_memo(fn: Callable[[float], Any], time: float, M: int = 32, precin: int = 0) -> float:
"""
GWR alorithm without memoization. This is a near 1:1 translation from
Mathematica.
"""
tau = mp.log(2.0) / mp.mpf(time)
fni: List[float] = [0.0] * M
for i, n in enumerate(fni):
if i == 0:
continue
fni[i] = fn(n * tau)
G0: List[float] = [0.0] * M
Gp: List[float] = [0.0] * M
M1 = M
for n in range(1, M + 1):
try:
n_fac = mp.fac(n - 1)
G0[n - 1] = tau * mp.fac(2 * n) / (n * n_fac * n_fac)
s = 0.0
for i in range(n + 1):
s += mp.binomial(n, i) * (-1) ** i * fni[n + i]
G0[n - 1] *= s
except:
M1 = n - 1
break
best = G0[M1 - 1]
Gm: List[float] = [0.0] * M1
broken = False
for k in range(M1 - 1):
for n in range(M1 - 1 - k)[::-1]:
try:
expr = G0[n + 1] - G0[n]
except:
expr = 0.0
broken = True
break
expr = Gm[n + 1] + (k + 1) / expr
Gp[n] = expr
if k % 2 == 1 and n == M1 - 2 - k:
best = expr
if broken:
break
for n in range(M1 - k):
Gm[n] = G0[n]
G0[n] = Gp[n]
return best
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_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 fac(n: int, precin: int) -> float:
return mp.fac(n)
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 place_children(dim, c, use_sp, sp, sample_from, sb, precision):
"""
Algorithm to place a set of points uniformly on the n-dimensional unit
sphere. The idea is to try to tile the surface of the sphere with
hypercubes. What was done was to build it once with a large number of
points and then sample for this set of points.
Input:
* dim: Integer. Embedding dimensions
* c: Integer. Number of children
* use_sp: Boolean. Condition to rotate points or not.
* sp: the grandparent node
* sample_from: Boolean. Condition to sample from sb
* sb: array of coordinates in the embedding
* precision: precision used in mpmath library
Output:
* points: array of coordinates on the hypershpere for each child node
"""
mp.prec = precision
N = dim
K = c
if sample_from:
_, K_large = sb.shape
points = [mp.mpf(0) for i in range(0, N * K)]
points = np.array(points)
points = points.reshape((N, K))
for i in range(0, K - 2 + 1):
points[:, i] = sb[:, int(np.floor(K_large / (K - 1))) * i]
points[:, K - 1] = sb[:, K_large - 1]
min_d_ds = 2
for i in range(0, K):
for j in range(i + 1, K):
dist = np.linalg.norm(points[:, i] - points[:, j])
if dist < min_d_ds:
min_d_ds = dist
else:
if N % 2 == 1:
AN = N * (2**N) * mp.power(mp.mpf(np.pi), (N - 1) / 2) * mp.mpf(
mp.fac((N - 1) // 2) / (mp.fac(N)))
else:
AN = N * mp.power(mp.mpf(np.pi), mp.mpf(N / 2)) / (mp.fac(
(N // 2)))
delta = mp.power(mp.mpf(AN / K), (mp.mpf(1 / (N - 1))))
true_k = 0
while true_k < K:
points, true_k = place_on_sphere(delta, N, K, False, precision)
delta = delta * mp.power(mp.mpf(true_k / K), mp.mpf(1 / (N - 1)))
points, true_k = place_on_sphere(delta, N, K, True, precision)
if use_sp:
points = rotate_points(points, sp, N, K)
return np.array(points)