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 check_degree(quadrature, exact, dim, max_degree, tol):
exponents = get_all_exponents(dim, max_degree)
# flatten list
exponents = numpy.array(
[item for sublist in exponents for item in sublist])
flt = numpy.vectorize(float)
exact_vals = flt([exact(k) for k in exponents])
def evaluate_all_monomials(x):
# Evaluate monomials.
# There's a more complex, faster implementation using matmul, exp, log.
# However, this only works for strictly positive `x`, and requires some
# tinkering. See below and
# <https://stackoverflow.com/a/45421128/353337>.
return numpy.prod(x[..., None]**exponents.T[:, None], axis=0).T
vals = quadrature(evaluate_all_monomials)
# check relative error
err = abs(exact_vals - vals)
is_smaller = err < (1 + abs(exact_vals)) * tol
if numpy.all(is_smaller):
return max_degree, numpy.max(err / (1 + abs(exact_vals)))
k = numpy.where(~is_smaller)[0]
# Return the max error for all exponents that are one smaller than the max_degree.
# This is because this functions is usually called with target_degree + 1.
idx = numpy.sum(exponents, axis=1) < max_degree
return (
numpy.sum(exponents[k[0]]) - 1,
numpy.max(err[idx] / (1 + abs(exact_vals[idx]))),
)
def check_degree(quadrature, exact, dim, max_degree, tol=1.0e-14):
exponents = get_all_exponents(dim, max_degree)
# flatten list
exponents = numpy.array(
[item for sublist in exponents for item in sublist])
flt = numpy.vectorize(float)
exact_vals = flt([exact(k) for k in exponents])
def evaluate_all_monomials(x):
# Evaluate monomials.
# There's a more complex, faster implementation using matmul, exp, log.
# However, this only works for strictly positive `x`, and requires some
# tinkering. See below and
# <https://stackoverflow.com/a/45421128/353337>.
return numpy.prod(x[..., None]**exponents.T[:, None], axis=0).T
vals = quadrature(evaluate_all_monomials)
# print(exact_vals)
# print(vals)
# check relative error
# The allowance is quite large here, 1e5 over machine precision.
# Some tests fail if lowered, though.
# TODO increase precision
eps = numpy.finfo(float).eps
mytol = abs(exact_vals) * tol + (1.0e5 + tol + exact_vals) * eps
is_smaller = abs(exact_vals - vals) < mytol
if numpy.all(is_smaller):
return max_degree
k = numpy.where(numpy.logical_not(is_smaller))[0]
return numpy.sum(exponents[k[0]]) - 1 # = degree
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