def riesz_energy(points, s):
num_points = len(points)
if num_points < 2:
return gmpy2.zero()
if s == 1:
energy = gmpy2.rec_sqrt(squared_distance(points[0], points[1]))
for j in range(2, num_points):
energy += gmpy2.rec_sqrt(squared_distance(points[0], points[j]))
for i in range(1, num_points):
for j in range(i + 1, num_points):
energy += gmpy2.rec_sqrt(squared_distance(points[i], points[j]))
elif s == 2:
energy = 1 / squared_distance(points[0], points[1])
for j in range(2, num_points):
energy += 1 / squared_distance(points[0], points[j])
for i in range(1, num_points):
for j in range(i + 1, num_points):
energy += 1 / squared_distance(points[i], points[j])
else:
energy = squared_distance(points[0], points[1]) ** (-s/2)
for j in range(2, num_points):
energy += squared_distance(points[0], points[j]) ** (-s/2)
for i in range(1, num_points):
for j in range(i + 1, num_points):
energy += squared_distance(points[i], points[j]) ** (-s/2)
return energy
def riesz_force(points, s):
point_indices = range(len(points))
dim_indices = range(len(points[0]))
force = [[gmpy2.zero() for _ in dim_indices] for _ in point_indices]
if s == 2:
for i in point_indices:
for j in point_indices:
if i != j:
dist, disp = squared_difference(points[i], points[j])
dist = 2 / gmpy2.square(dist)
for k in dim_indices:
force[i][k] += dist * disp[k]
elif s == 1:
for i in point_indices:
for j in point_indices:
if i != j:
dist, disp = squared_difference(points[i], points[j])
dist = gmpy2.rec_sqrt(dist) / dist
for k in dim_indices:
force[i][k] += dist * disp[k]
else:
for i in point_indices:
for j in point_indices:
if i != j:
dist, disp = squared_difference(points[i], points[j])
dist = s * dist**(-s / 2 - 1)
for k in dim_indices:
force[i][k] += dist * disp[k]
return force
def mpf_to_mpfr(f):
assert isinstance(f, MPF)
if f.isNaN():
return gmpy2.nan()
elif f.isInfinite():
if f.isPositive():
return gmpy2.inf(1)
else:
return gmpy2.inf(-1)
elif f.isZero():
if f.isPositive():
return gmpy2.zero(1)
else:
return gmpy2.zero(-1)
else:
return gmpy2.mpfr(f.to_python_string())
def squared_distance(v, w):
assert len(v) == len(w)
if len(v) == 0:
return gmpy2.zero()
else:
dist = gmpy2.square(v[0] - w[0])
for i in range(1, len(v)):
dist += gmpy2.square(v[i] - w[i])
return dist
def double_matrix_multiply(a, b):
m = len(a)
n = len(a[0])
c = [[gmpy2.zero() for _ in range(n)] for _ in range(m)]
for i in range(m):
for j in range(n):
for k in range(m):
for l in range(n):
c[i][j] += a[i][j][k][l] * b[k][l]
return c
def squared_difference(v, w):
assert len(v) == len(w)
if len(v) == 0:
return gmpy2.zero(), ()
else:
diff = tuple(v[i] - w[i] for i in range(len(v)))
dist = gmpy2.square(diff[0])
for i in range(1, len(v)):
dist += gmpy2.square(diff[i])
return dist, diff
def mvmul(mat, vec):
m = len(mat)
n_set = set(map(len, mat))
assert len(n_set) == 1
n, = n_set
assert len(vec) == n
ans = [gmpy2.zero() for _ in range(m)]
for i in range(m):
for j in range(n):
ans[i] += mat[i][j] * vec[j]
return tuple(ans)
def centroid(points):
dim_set = set(map(len, points))
assert len(dim_set) == 1
dim, = dim_set
cent = [gmpy2.zero() for _ in range(dim)]
for point in points:
for i in range(dim):
cent[i] += point[i]
num_points = len(points)
for i in range(dim):
cent[i] /= num_points
return tuple(cent)
def icosahedron_vertices():
zero = gmpy2.zero()
one = gmpy2.mpfr(1)
phi = (gmpy2.sqrt(5) + 1) / 2
return (
(+one, +phi, zero),
(+one, -phi, zero),
(-one, +phi, zero),
(-one, -phi, zero),
(zero, +one, +phi),
(zero, +one, -phi),
(zero, -one, +phi),
(zero, -one, -phi),
(+phi, zero, +one),
(+phi, zero, -one),
(-phi, zero, +one),
(-phi, zero, -one))
def full_icosahedral_group():
zero = gmpy2.zero()
one = gmpy2.mpfr(1)
half = one / 2
psi = (gmpy2.sqrt(5) - 1) / 4
phi = (gmpy2.sqrt(5) + 1) / 4
return (
((+one, zero, zero), (zero, +one, zero), (zero, zero, +one)),
((+one, zero, zero), (zero, +one, zero), (zero, zero, -one)),
((+one, zero, zero), (zero, -one, zero), (zero, zero, +one)),
((+one, zero, zero), (zero, -one, zero), (zero, zero, -one)),
((-one, zero, zero), (zero, +one, zero), (zero, zero, +one)),
((-one, zero, zero), (zero, +one, zero), (zero, zero, -one)),
((-one, zero, zero), (zero, -one, zero), (zero, zero, +one)),
((-one, zero, zero), (zero, -one, zero), (zero, zero, -one)),
((zero, +one, zero), (zero, zero, +one), (+one, zero, zero)),
((zero, +one, zero), (zero, zero, +one), (-one, zero, zero)),
((zero, +one, zero), (zero, zero, -one), (+one, zero, zero)),
((zero, +one, zero), (zero, zero, -one), (-one, zero, zero)),
((zero, -one, zero), (zero, zero, +one), (+one, zero, zero)),
((zero, -one, zero), (zero, zero, +one), (-one, zero, zero)),
((zero, -one, zero), (zero, zero, -one), (+one, zero, zero)),
((zero, -one, zero), (zero, zero, -one), (-one, zero, zero)),
((zero, zero, +one), (+one, zero, zero), (zero, +one, zero)),
((zero, zero, +one), (+one, zero, zero), (zero, -one, zero)),
((zero, zero, +one), (-one, zero, zero), (zero, +one, zero)),
((zero, zero, +one), (-one, zero, zero), (zero, -one, zero)),
((zero, zero, -one), (+one, zero, zero), (zero, +one, zero)),
((zero, zero, -one), (+one, zero, zero), (zero, -one, zero)),
((zero, zero, -one), (-one, zero, zero), (zero, +one, zero)),
((zero, zero, -one), (-one, zero, zero), (zero, -one, zero)),
((+half, +psi, +phi), (+psi, +phi, -half), (+phi, -half, -psi)),
((+half, +psi, +phi), (+psi, +phi, -half), (-phi, +half, +psi)),
((+half, +psi, +phi), (-psi, -phi, +half), (+phi, -half, -psi)),
((+half, +psi, +phi), (-psi, -phi, +half), (-phi, +half, +psi)),
((+half, +psi, -phi), (+psi, +phi, +half), (+phi, -half, +psi)),
((+half, +psi, -phi), (+psi, +phi, +half), (-phi, +half, -psi)),
((+half, +psi, -phi), (-psi, -phi, -half), (+phi, -half, +psi)),
((+half, +psi, -phi), (-psi, -phi, -half), (-phi, +half, -psi)),
((+half, -psi, +phi), (+psi, -phi, -half), (+phi, +half, -psi)),
((+half, -psi, +phi), (+psi, -phi, -half), (-phi, -half, +psi)),
((+half, -psi, +phi), (-psi, +phi, +half), (+phi, +half, -psi)),
((+half, -psi, +phi), (-psi, +phi, +half), (-phi, -half, +psi)),
((+half, -psi, -phi), (+psi, -phi, +half), (+phi, +half, +psi)),
((+half, -psi, -phi), (+psi, -phi, +half), (-phi, -half, -psi)),
((+half, -psi, -phi), (-psi, +phi, -half), (+phi, +half, +psi)),
((+half, -psi, -phi), (-psi, +phi, -half), (-phi, -half, -psi)),
((-half, +psi, +phi), (+psi, -phi, +half), (+phi, +half, +psi)),
((-half, +psi, +phi), (+psi, -phi, +half), (-phi, -half, -psi)),
((-half, +psi, +phi), (-psi, +phi, -half), (+phi, +half, +psi)),
((-half, +psi, +phi), (-psi, +phi, -half), (-phi, -half, -psi)),
((-half, +psi, -phi), (+psi, -phi, -half), (+phi, +half, -psi)),
((-half, +psi, -phi), (+psi, -phi, -half), (-phi, -half, +psi)),
((-half, +psi, -phi), (-psi, +phi, +half), (+phi, +half, -psi)),
((-half, +psi, -phi), (-psi, +phi, +half), (-phi, -half, +psi)),
((-half, -psi, +phi), (+psi, +phi, +half), (+phi, -half, +psi)),
((-half, -psi, +phi), (+psi, +phi, +half), (-phi, +half, -psi)),
((-half, -psi, +phi), (-psi, -phi, -half), (+phi, -half, +psi)),
((-half, -psi, +phi), (-psi, -phi, -half), (-phi, +half, -psi)),
((-half, -psi, -phi), (+psi, +phi, -half), (+phi, -half, -psi)),
((-half, -psi, -phi), (+psi, +phi, -half), (-phi, +half, +psi)),
((-half, -psi, -phi), (-psi, -phi, +half), (+phi, -half, -psi)),
((-half, -psi, -phi), (-psi, -phi, +half), (-phi, +half, +psi)),
((+phi, +half, +psi), (+half, -psi, -phi), (+psi, -phi, +half)),
((+phi, +half, +psi), (+half, -psi, -phi), (-psi, +phi, -half)),
((+phi, +half, +psi), (-half, +psi, +phi), (+psi, -phi, +half)),
((+phi, +half, +psi), (-half, +psi, +phi), (-psi, +phi, -half)),
((+phi, +half, -psi), (+half, -psi, +phi), (+psi, -phi, -half)),
((+phi, +half, -psi), (+half, -psi, +phi), (-psi, +phi, +half)),
((+phi, +half, -psi), (-half, +psi, -phi), (+psi, -phi, -half)),
((+phi, +half, -psi), (-half, +psi, -phi), (-psi, +phi, +half)),
((+phi, -half, +psi), (+half, +psi, -phi), (+psi, +phi, +half)),
((+phi, -half, +psi), (+half, +psi, -phi), (-psi, -phi, -half)),
((+phi, -half, +psi), (-half, -psi, +phi), (+psi, +phi, +half)),
((+phi, -half, +psi), (-half, -psi, +phi), (-psi, -phi, -half)),
((+phi, -half, -psi), (+half, +psi, +phi), (+psi, +phi, -half)),
((+phi, -half, -psi), (+half, +psi, +phi), (-psi, -phi, +half)),
((+phi, -half, -psi), (-half, -psi, -phi), (+psi, +phi, -half)),
((+phi, -half, -psi), (-half, -psi, -phi), (-psi, -phi, +half)),
((-phi, +half, +psi), (+half, +psi, +phi), (+psi, +phi, -half)),
((-phi, +half, +psi), (+half, +psi, +phi), (-psi, -phi, +half)),
((-phi, +half, +psi), (-half, -psi, -phi), (+psi, +phi, -half)),
((-phi, +half, +psi), (-half, -psi, -phi), (-psi, -phi, +half)),
((-phi, +half, -psi), (+half, +psi, -phi), (+psi, +phi, +half)),
((-phi, +half, -psi), (+half, +psi, -phi), (-psi, -phi, -half)),
((-phi, +half, -psi), (-half, -psi, +phi), (+psi, +phi, +half)),
((-phi, +half, -psi), (-half, -psi, +phi), (-psi, -phi, -half)),
((-phi, -half, +psi), (+half, -psi, +phi), (+psi, -phi, -half)),
((-phi, -half, +psi), (+half, -psi, +phi), (-psi, +phi, +half)),
((-phi, -half, +psi), (-half, +psi, -phi), (+psi, -phi, -half)),
((-phi, -half, +psi), (-half, +psi, -phi), (-psi, +phi, +half)),
((-phi, -half, -psi), (+half, -psi, -phi), (+psi, -phi, +half)),
((-phi, -half, -psi), (+half, -psi, -phi), (-psi, +phi, -half)),
((-phi, -half, -psi), (-half, +psi, +phi), (+psi, -phi, +half)),
((-phi, -half, -psi), (-half, +psi, +phi), (-psi, +phi, -half)),
((+psi, +phi, +half), (+phi, -half, +psi), (+half, +psi, -phi)),
((+psi, +phi, +half), (+phi, -half, +psi), (-half, -psi, +phi)),
((+psi, +phi, +half), (-phi, +half, -psi), (+half, +psi, -phi)),
((+psi, +phi, +half), (-phi, +half, -psi), (-half, -psi, +phi)),
((+psi, +phi, -half), (+phi, -half, -psi), (+half, +psi, +phi)),
((+psi, +phi, -half), (+phi, -half, -psi), (-half, -psi, -phi)),
((+psi, +phi, -half), (-phi, +half, +psi), (+half, +psi, +phi)),
((+psi, +phi, -half), (-phi, +half, +psi), (-half, -psi, -phi)),
((+psi, -phi, +half), (+phi, +half, +psi), (+half, -psi, -phi)),
((+psi, -phi, +half), (+phi, +half, +psi), (-half, +psi, +phi)),
((+psi, -phi, +half), (-phi, -half, -psi), (+half, -psi, -phi)),
((+psi, -phi, +half), (-phi, -half, -psi), (-half, +psi, +phi)),
((+psi, -phi, -half), (+phi, +half, -psi), (+half, -psi, +phi)),
((+psi, -phi, -half), (+phi, +half, -psi), (-half, +psi, -phi)),
((+psi, -phi, -half), (-phi, -half, +psi), (+half, -psi, +phi)),
((+psi, -phi, -half), (-phi, -half, +psi), (-half, +psi, -phi)),
((-psi, +phi, +half), (+phi, +half, -psi), (+half, -psi, +phi)),
((-psi, +phi, +half), (+phi, +half, -psi), (-half, +psi, -phi)),
((-psi, +phi, +half), (-phi, -half, +psi), (+half, -psi, +phi)),
((-psi, +phi, +half), (-phi, -half, +psi), (-half, +psi, -phi)),
((-psi, +phi, -half), (+phi, +half, +psi), (+half, -psi, -phi)),
((-psi, +phi, -half), (+phi, +half, +psi), (-half, +psi, +phi)),
((-psi, +phi, -half), (-phi, -half, -psi), (+half, -psi, -phi)),
((-psi, +phi, -half), (-phi, -half, -psi), (-half, +psi, +phi)),
((-psi, -phi, +half), (+phi, -half, -psi), (+half, +psi, +phi)),
((-psi, -phi, +half), (+phi, -half, -psi), (-half, -psi, -phi)),
((-psi, -phi, +half), (-phi, +half, +psi), (+half, +psi, +phi)),
((-psi, -phi, +half), (-phi, +half, +psi), (-half, -psi, -phi)),
((-psi, -phi, -half), (+phi, -half, +psi), (+half, +psi, -phi)),
((-psi, -phi, -half), (+phi, -half, +psi), (-half, -psi, +phi)),
((-psi, -phi, -half), (-phi, +half, -psi), (+half, +psi, -phi)),
((-psi, -phi, -half), (-phi, +half, -psi), (-half, -psi, +phi)))