def f(x):
degree = 7
n = 5
u = x[0]
v = x[1]
B = x[2:]
data = [
(B[0], z(n)),
(B[1], fsd(n, (u, 1))),
(B[2], fsd(n, (v, 1))),
(B[3], fsd(n, (u, 2))),
(B[4], fsd(n, (v, 2))),
(B[5], fsd(n, (u, 3))),
]
points, weights = untangle(data)
exponents = get_all_exponents(n, degree)
# flatten list
exponents = numpy.array([item for sublist in exponents for item in sublist])
def evaluate_all_monomials(x):
return numpy.prod(x[..., None] ** exponents.T[:, None], axis=0).T
flt = numpy.vectorize(float)
exact_vals = flt([integrate_monomial_over_unit_nball(k) for k in exponents])
A = evaluate_all_monomials(points.T)
out = numpy.dot(A, weights)
out -= exact_vals
norm_v = numpy.sqrt(numpy.dot(out, out))
print(norm_v)
return norm_v
def f(x):
degree = 19
data = [
(x[0], _f((math.sqrt(1.0 / 3.0), 3))),
(x[1], _f2(x[6])),
(x[2], _f2(x[7])),
(x[3], _f2(x[8])),
(x[4], _f2(x[9])),
(x[5], _f11(x[10], x[11])),
]
points, weights = untangle(data)
azimuthal, polar = cartesian_to_spherical(points).T
out = orthopy.sphere.tree_sph(
polar, azimuthal, degree, standardization="quantum mechanic"
)
A = numpy.array([row for level in out for row in level])
out = numpy.dot(A, weights)
out[0] -= 1.0 / (2 * numpy.sqrt(numpy.pi))
v = numpy.sqrt(out.real ** 2 + out.imag ** 2)
norm_v = numpy.sqrt(numpy.vdot(v, v))
print(norm_v)
return norm_v
def f(x):
degree = 11
n = 3
u = 0.871_740_148_509_601
v = 0.591_700_181_433_148
w = 0.209_299_217_902_484
# u = x[0]
# v = x[1]
# w = x[2]
# B = x[3:]
B = x
data = [
(B[0], z(n)),
(B[1], fsd(n, (u, 1))),
(B[2], fsd(n, (v, 1))),
(B[3], fsd(n, (w, 1))),
(B[4], fsd(n, (u, 2))),
(B[5], fsd(n, (v, 2))),
(B[6], fsd(n, (w, 2))),
(B[7], fsd(n, (u, 1), (v, 1))),
(B[8], fsd(n, (u, 1), (w, 1))),
(B[9], fsd(n, (u, 3))),
(B[10], fsd(n, (v, 3))),
(B[11], fsd(n, (w, 3))),
(B[12], fsd(n, (u, 2), (v, 1))),
]
points, weights = untangle(data)
exponents = get_all_exponents(n, degree)
# flatten list
exponents = numpy.array(
[item for sublist in exponents for item in sublist])
def evaluate_all_monomials(x):
return numpy.prod(x[..., None]**exponents.T[:, None], axis=0).T
flt = numpy.vectorize(float)
exact_vals = flt(
[integrate_monomial_over_unit_nball(k) for k in exponents])
A = evaluate_all_monomials(points.T)
out = numpy.dot(A, weights)
out -= exact_vals
norm_v = numpy.sqrt(numpy.dot(out, out))
# print()
print(norm_v)
# print()
# for xx in x:
# print(f"{xx:.15e}")
return norm_v
def f(x):
degree = 11
n = 5
u = x[0]
v = x[1]
w = x[2]
B = x[3:]
data = [
(B[0], z(n)),
(B[1], fsd(n, (u, 1))),
(B[2], fsd(n, (v, 1))),
(B[3], fsd(n, (w, 1))),
(B[4], fsd(n, (u, 2))),
(B[5], fsd(n, (v, 2))),
(B[6], fsd(n, (w, 2))),
(B[7], fsd(n, (u, 1), (v, 1))),
(B[8], fsd(n, (u, 1), (w, 1))),
(B[9], fsd(n, (u, 3))),
(B[10], fsd(n, (v, 3))),
(B[11], fsd(n, (w, 3))),
(B[12], fsd(n, (u, 2), (v, 1))),
]
if n > 3:
data += [(B[13], fsd(n, (u, 4))), (B[14], fsd(n, (v, 4)))]
if n > 4:
data += [(B[15], fsd(n, (u, 5)))]
points, weights = untangle(data)
exponents = get_all_exponents(n, degree)
# flatten list
exponents = numpy.array(
[item for sublist in exponents for item in sublist])
def evaluate_all_monomials(x):
return numpy.prod(x[..., None]**exponents.T[:, None], axis=0).T
flt = numpy.vectorize(float)
exact_vals = flt(
[integrate_monomial_over_unit_nball(k) for k in exponents])
A = evaluate_all_monomials(points.T)
out = numpy.dot(A, weights)
out -= exact_vals
norm_v = numpy.sqrt(numpy.dot(out, out))
print()
print(norm_v)
print()
for xx in x:
print(f"{xx:.15e}")
return norm_v
def f(x):
degree = 5
n = 7
lmbda, xi, mu, gamma = x
eta = 0
A = 1 / 9
B = 1 / 72
C = B
# data = [
# (B, rd(n, [(+lmbda, 1), (+xi, n - 1)])),
# (B, rd(n, [(-lmbda, 1), (-xi, n - 1)])),
# (C, rd(n, [(+mu, 2), (+gamma, n - 2)])),
# (C, rd(n, [(-mu, 2), (-gamma, n - 2)])),
# (2 * A, numpy.full((1, n), eta)),
# ]
# points, weights = untangle(data)
# weights *= numpy.sqrt(numpy.pi) ** n
data = [
(B, rd(n, [(+lmbda, 1), (+xi, n - 1)])),
(B, rd(n, [(-lmbda, 1), (-xi, n - 1)])),
(C, rd(n, [(+mu, 2), (+gamma, n - 2)])),
(C, rd(n, [(-mu, 2), (-gamma, n - 2)])),
(2 * A, numpy.full((1, n), eta)),
]
points, weights = untangle(data)
weights *= numpy.sqrt(numpy.pi)**n
A = numpy.concatenate(
orthopy.enr2.tree(points.T, degree, symbolic=False))
out = numpy.dot(A, weights)
out[0] -= numpy.sqrt(numpy.sqrt(numpy.pi))**n
norm_v = numpy.sqrt(numpy.vdot(out, out))
return norm_v
def f(x):
degree = 7
data = [
(x[0], fsd(2, (x[3], 1))),
(x[1], fsd(2, (x[4], 1))),
(x[2], fsd(2, (x[3], 1), (x[4], 1))),
]
points = numpy.array(
[
[0.0, +x[3]],
[0.0, -x[3]],
[+x[3], 0.0],
[-x[3], 0.0],
#
[0.0, +x[4]],
[0.0, -x[4]],
[+x[4], 0.0],
[-x[4], 0.0],
#
#
[+x[3], +x[4]],
[+x[3], -x[4]],
[-x[3], +x[4]],
[-x[3], -x[4]],
]
)
points, weights = untangle(data)
A = numpy.concatenate(orthopy.e2r2.tree(points.T, degree, symbolic=False))
out = numpy.dot(A, weights)
out[0] -= numpy.sqrt(numpy.pi)
norm_v = numpy.sqrt(numpy.vdot(out, out))
# print(norm_v)
return norm_v
def f(x):
degree = 13
data = [
(x[0], [[x[8], x[9]]]),
(x[1], _s40(x[10])),
(x[2], _s40(x[11])),
(x[3], _s4(x[12])),
(x[4], _s4(x[13])),
(x[5], _s4(x[14])),
(x[6], _s8(x[15], x[16])),
(x[7], _s8(x[17], x[18])),
]
points, weights = untangle(data)
exponents = numpy.concatenate([partition(2, d) for d in range(degree + 1)])
exact_vals = numpy.array([integrate_monomial_over_enr(k) for k in exponents])
def fun(x):
k = exponents.T
# <https://stackoverflow.com/a/46689653/353337>
s = x.shape[1:] + k.shape[1:]
return (
(x.reshape(x.shape[0], -1, 1) ** k.reshape(k.shape[0], 1, -1))
.prod(0)
.reshape(s)
)
A = fun(points.T).T
print(A)
print(exact_vals)
print(sum(weights))
out = numpy.dot(A, weights) - exact_vals
nrm = numpy.sqrt(numpy.dot(out, out))
print(nrm)
exit(1)
return nrm
