def sym_eig2x2(A, dt):
"""Compute the eigenvalues and right eigenvectors (Av=lambda v) of a 2x2 real symmetric matrix.
Mathematical concept refers to https://en.wikipedia.org/wiki/Eigendecomposition_of_a_matrix.
Args:
A (ti.Matrix(2, 2)): input 2x2 symmetric matrix `A`.
dt (DataType): date type of elements in matrix `A`, typically accepts ti.f32 or ti.f64.
Returns:
eigenvalues (ti.Vector(2)): The eigenvalues. Each entry store one eigen value.
eigenvectors (ti.Matrix(2, 2)): The eigenvectors. Each column stores one eigenvector.
"""
tr = A.trace()
det = A.determinant()
gap = tr**2 - 4 * det
lambda1 = (tr + ops.sqrt(gap)) * 0.5
lambda2 = (tr - ops.sqrt(gap)) * 0.5
eigenvalues = Vector([lambda1, lambda2], dt=dt)
A1 = A - lambda1 * Matrix.identity(dt, 2)
A2 = A - lambda2 * Matrix.identity(dt, 2)
v1 = Vector.zero(dt, 2)
v2 = Vector.zero(dt, 2)
if all(A1 == Matrix.zero(dt, 2, 2)) and all(A1 == Matrix.zero(dt, 2, 2)):
v1 = Vector([0.0, 1.0]).cast(dt)
v2 = Vector([1.0, 0.0]).cast(dt)
else:
v1 = Vector([A2[0, 0], A2[1, 0]], dt=dt).normalized()
v2 = Vector([A1[0, 0], A1[1, 0]], dt=dt).normalized()
eigenvectors = Matrix.cols([v1, v2])
return eigenvalues, eigenvectors
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 eig2x2(A, dt):
"""Compute the eigenvalues and right eigenvectors (Av=lambda v) of a 2x2 real matrix.
Mathematical concept refers to https://en.wikipedia.org/wiki/Eigendecomposition_of_a_matrix.
Args:
A (ti.Matrix(2, 2)): input 2x2 matrix `A`.
dt (DataType): date type of elements in matrix `A`, typically accepts ti.f32 or ti.f64.
Returns:
eigenvalues (ti.Matrix(2, 2)): The eigenvalues in complex form. Each row stores one eigenvalue. The first number of the eigenvalue represents the real part and the second number represents the imaginary part.
eigenvectors: (ti.Matrix(4, 2)): The eigenvectors in complex form. Each column stores one eigenvector. Each eigenvector consists of 2 entries, each of which is represented by two numbers for its real part and imaginary part.
"""
tr = A.trace()
det = A.determinant()
gap = tr**2 - 4 * det
lambda1 = Vector.zero(dt, 2)
lambda2 = Vector.zero(dt, 2)
v1 = Vector.zero(dt, 4)
v2 = Vector.zero(dt, 4)
if gap > 0:
lambda1 = Vector([tr + ops.sqrt(gap), 0.0], dt=dt) * 0.5
lambda2 = Vector([tr - ops.sqrt(gap), 0.0], dt=dt) * 0.5
A1 = A - lambda1[0] * Matrix.identity(dt, 2)
A2 = A - lambda2[0] * Matrix.identity(dt, 2)
if all(A1 == Matrix.zero(dt, 2, 2)) and all(
A1 == Matrix.zero(dt, 2, 2)):
v1 = Vector([0.0, 0.0, 1.0, 0.0]).cast(dt)
v2 = Vector([1.0, 0.0, 0.0, 0.0]).cast(dt)
else:
v1 = Vector([A2[0, 0], 0.0, A2[1, 0], 0.0], dt=dt).normalized()
v2 = Vector([A1[0, 0], 0.0, A1[1, 0], 0.0], dt=dt).normalized()
else:
lambda1 = Vector([tr, ops.sqrt(-gap)], dt=dt) * 0.5
lambda2 = Vector([tr, -ops.sqrt(-gap)], dt=dt) * 0.5
A1r = A - lambda1[0] * Matrix.identity(dt, 2)
A1i = -lambda1[1] * Matrix.identity(dt, 2)
A2r = A - lambda2[0] * Matrix.identity(dt, 2)
A2i = -lambda2[1] * Matrix.identity(dt, 2)
v1 = Vector([A2r[0, 0], A2i[0, 0], A2r[1, 0], A2i[1, 0]],
dt=dt).normalized()
v2 = Vector([A1r[0, 0], A1i[0, 0], A1r[1, 0], A1i[1, 0]],
dt=dt).normalized()
eigenvalues = Matrix.rows([lambda1, lambda2])
eigenvectors = Matrix.cols([v1, v2])
return eigenvalues, eigenvectors
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
