def _gauss_elimination_3x3(Ab, dt):
for i in static(range(3)):
max_row = i
max_v = ops.abs(Ab[i, i])
for j in static(range(i + 1, 3)):
if ops.abs(Ab[j, i]) > max_v:
max_row = j
max_v = ops.abs(Ab[j, i])
assert max_v != 0.0, "Matrix is singular in linear solve."
if i != max_row:
if max_row == 1:
for col in static(range(4)):
Ab[i, col], Ab[1, col] = Ab[1, col], Ab[i, col]
else:
for col in static(range(4)):
Ab[i, col], Ab[2, col] = Ab[2, col], Ab[i, col]
assert Ab[i, i] != 0.0, "Matrix is singular in linear solve."
for j in static(range(i + 1, 3)):
scale = Ab[j, i] / Ab[i, i]
Ab[j, i] = 0.0
for k in static(range(i + 1, 4)):
Ab[j, k] -= Ab[i, k] * scale
# Back substitution
x = Vector.zero(dt, 3)
for i in static(range(2, -1, -1)):
x[i] = Ab[i, 3]
for k in static(range(i + 1, 3)):
x[i] -= Ab[i, k] * x[k]
x[i] = x[i] / Ab[i, i]
return x
def _gauss_elimination_2x2(Ab, dt):
if ops.abs(Ab[0, 0]) < ops.abs(Ab[1, 0]):
Ab[0, 0], Ab[1, 0] = Ab[1, 0], Ab[0, 0]
Ab[0, 1], Ab[1, 1] = Ab[1, 1], Ab[0, 1]
Ab[0, 2], Ab[1, 2] = Ab[1, 2], Ab[0, 2]
assert Ab[0, 0] != 0.0, "Matrix is singular in linear solve."
scale = Ab[1, 0] / Ab[0, 0]
Ab[1, 0] = 0.0
for k in static(range(1, 3)):
Ab[1, k] -= Ab[0, k] * scale
x = Vector.zero(dt, 2)
# Back substitution
x[1] = Ab[1, 2] / Ab[1, 1]
x[0] = (Ab[0, 2] - Ab[0, 1] * x[1]) / Ab[0, 0]
return x
def sym_eig3x3(A, dt):
"""Compute the eigenvalues and right eigenvectors (Av=lambda v) of a 3x3 real symmetric matrix using Cardano's method.
Mathematical concept refers to https://www.mpi-hd.mpg.de/personalhomes/globes/3x3/.
Args:
A (ti.Matrix(3, 3)): input 3x3 symmetric matrix `A`.
dt (DataType): date type of elements in matrix `A`, typically accepts ti.f32 or ti.f64.
Returns:
eigenvalues (ti.Vector(3)): The eigenvalues. Each entry store one eigen value.
eigenvectors (ti.Matrix(3, 3)): The eigenvectors. Each column stores one eigenvector.
"""
M_SQRT3 = 1.73205080756887729352744634151
m = A.trace()
dd = A[0, 1] * A[0, 1]
ee = A[1, 2] * A[1, 2]
ff = A[0, 2] * A[0, 2]
c1 = A[0, 0] * A[1, 1] + A[0, 0] * A[2, 2] + A[1, 1] * A[2, 2] - (dd + ee +
ff)
c0 = A[2, 2] * dd + A[0, 0] * ee + A[1, 1] * ff - A[0, 0] * A[1, 1] * A[
2, 2] - 2.0 * A[0, 2] * A[0, 1] * A[1, 2]
p = m * m - 3.0 * c1
q = m * (p - 1.5 * c1) - 13.5 * c0
sqrt_p = ops.sqrt(ops.abs(p))
phi = 27.0 * (0.25 * c1 * c1 * (p - c1) + c0 * (q + 6.75 * c0))
phi = (1.0 / 3.0) * ops.atan2(ops.sqrt(ops.abs(phi)), q)
c = sqrt_p * ops.cos(phi)
s = (1.0 / M_SQRT3) * sqrt_p * ops.sin(phi)
eigenvalues = Vector([0.0, 0.0, 0.0], dt=dt)
eigenvalues[2] = (1.0 / 3.0) * (m - c)
eigenvalues[1] = eigenvalues[2] + s
eigenvalues[0] = eigenvalues[2] + c
eigenvalues[2] = eigenvalues[2] - s
t = ops.abs(eigenvalues[0])
u = ops.abs(eigenvalues[1])
if u > t:
t = u
u = ops.abs(eigenvalues[2])
if u > t:
t = u
if t < 1.0:
u = t
else:
u = t * t
Q = Matrix.zero(dt, 3, 3)
Q[0, 1] = A[0, 1] * A[1, 2] - A[0, 2] * A[1, 1]
Q[1, 1] = A[0, 2] * A[0, 1] - A[1, 2] * A[0, 0]
Q[2, 1] = A[0, 1] * A[0, 1]
Q[0, 0] = Q[0, 1] + A[0, 2] * eigenvalues[0]
Q[1, 0] = Q[1, 1] + A[1, 2] * eigenvalues[0]
Q[2, 0] = (A[0, 0] - eigenvalues[0]) * (A[1, 1] - eigenvalues[0]) - Q[2, 1]
norm = Q[0, 0] * Q[0, 0] + Q[1, 0] * Q[1, 0] + Q[2, 0] * Q[2, 0]
norm = ops.sqrt(1.0 / norm)
Q[0, 0] *= norm
Q[1, 0] *= norm
Q[2, 0] *= norm
Q[0, 1] = Q[0, 1] + A[0, 2] * eigenvalues[1]
Q[1, 1] = Q[1, 1] + A[1, 2] * eigenvalues[1]
Q[2, 1] = (A[0, 0] - eigenvalues[1]) * (A[1, 1] - eigenvalues[1]) - Q[2, 1]
norm = Q[0, 1] * Q[0, 1] + Q[1, 1] * Q[1, 1] + Q[2, 1] * Q[2, 1]
norm = ops.sqrt(1.0 / norm)
Q[0, 1] *= norm
Q[1, 1] *= norm
Q[2, 1] *= norm
Q[0, 2] = Q[1, 0] * Q[2, 1] - Q[2, 0] * Q[1, 1]
Q[1, 2] = Q[2, 0] * Q[0, 1] - Q[0, 0] * Q[2, 1]
Q[2, 2] = Q[0, 0] * Q[1, 1] - Q[1, 0] * Q[0, 1]
return eigenvalues, Q
def __abs__(self):
return ops.abs(self)
