def matrix_QR_factorization_householder(matrix):
matrix_R = [[matrix[j][i] for i in range(len(matrix[0]))]
for j in range(len(matrix))] #HA starts by being A
matrix_Q = [[0 for j in range(len(matrix))] for k in range(len(matrix))
] #Q starts by being the identity matrix
for j in range(len(matrix)):
matrix_Q[j][j] = 1
for i in range(len(matrix[0]) - 1):
a = [
matrix_R[j][i] for j in range(len(matrix_R))
] #a is the i'th column of matrix_R, with 0s inserted above the diagonal
for j in range(i):
a[j] = 0
r = [0] * len(
a
) #r is just a 0 vector, with the i'th column being the magnitude of a
r[i] = vector_2norm(a)
u = vector_add(a, vector_scal_mult(-1, r))
i_ = [[0 for j in range(len(matrix))] for k in range(len(matrix))]
for j in range(len(matrix)):
i_[j][j] = 1
if vector_dot(u, u) != 0:
h = matrix_add(i_,
matrix_scal_mult(-2 / vector_dot(u, u),
matrix_outer(u, u))) #computer h_i
else: #if u ends up being the 0 matrix, then h is just the identity matrix
h = i_
matrix_Q = matrix_mult(matrix_Q, h)
matrix_R = matrix_mult(h, matrix_R)
return [matrix_Q, matrix_R]
def matrix_rayleigh_quotient_iteration(matrix,
tol,
max_iter,
getIterCount=False):
error = tol * 10
count = 0
v_i = matrix[0]
v_i = vector_scal_mult(1 / vector_2norm(v_i), v_i)
eigenvaluenew = matrix_mult(matrix_transpose(convert_vec_mat(v_i)),
matrix_mult(matrix,
convert_vec_mat(v_i)))[0][0]
I = [[int(i == j) for i in range(len(matrix))] for j in range(len(matrix))]
while error > tol and count < max_iter:
eigenvalueold = eigenvaluenew
v_i = matrix_solve_LU(
matrix_add(matrix, matrix_scal_mult(-eigenvaluenew, I)), v_i)
v_i = vector_scal_mult(1 / vector_2norm(v_i), v_i)
eigenvaluenew = matrix_mult(matrix_transpose(convert_vec_mat(v_i)),
matrix_mult(matrix,
convert_vec_mat(v_i)))[0][0]
error = abs(eigenvaluenew - eigenvalueold)
count += 1
if getIterCount == True:
return eigenvaluenew, count
else:
return eigenvaluenew
def matrix_inverse_power_iteration(matrix,
alpha,
tol,
max_iter,
getIterCount=False):
error = tol * 10
count = 0
eigenvaluenew = 0
I = [[-int(i == j) * alpha for i in range(len(matrix))]
for j in range(len(matrix))]
v_i = matrix[0]
while error > tol and count < max_iter:
eigenvalueold = eigenvaluenew
v_i = matrix_solve_LU(matrix_add(matrix, I), v_i)
v_i = vector_scal_mult(1 / vector_2norm(v_i), v_i)
eigenvaluenew = matrix_mult(matrix_transpose(convert_vec_mat(v_i)),
matrix_mult(matrix,
convert_vec_mat(v_i)))[0][0]
count += 1
error = abs(eigenvaluenew - eigenvalueold)
if getIterCount == True:
return eigenvaluenew, count
else:
return eigenvaluenew
