def matrix_solve_steepest_descent(matrix,
vector_b,
tol,
max_iter,
getIterCount=False):
error = tol * 10
count = 0
x_i = vector_b.copy() #x_0
while error > tol and count < max_iter:
#print(count)# used for testing
#print(error)# used for testing
count += 1
residual = vector_add(
vector_b,
vector_scal_mult(
-1, convert_vec_mat(matrix_mult(matrix,
convert_vec_mat(x_i)))))
d1 = vector_dot(residual, residual)
d2 = vector_dot(
residual,
convert_vec_mat(matrix_mult(matrix, convert_vec_mat(residual))))
alpha = d1 / d2
x_i = vector_add(x_i, vector_scal_mult(alpha, residual))
error = vector_2norm(residual)
if getIterCount == True:
return x_i, count
else:
return x_i
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_solve_conjugate_gradient(matrix,
vector_b,
tol,
max_iter,
getIterCount=False):
error = tol * 10
count = 0
x_i = vector_b.copy() # x_0
residual = vector_add(
vector_b,
vector_scal_mult(
-1, convert_vec_mat(matrix_mult(matrix, convert_vec_mat(x_i)))))
direction = residual
d1new = vector_dot(residual, residual)
while error > tol and count < max_iter:
s = convert_vec_mat(matrix_mult(matrix, convert_vec_mat(direction)))
d1old = d1new
alpha = d1old / vector_dot(direction, s)
x_i = vector_add(x_i, vector_scal_mult(alpha, direction))
residual = vector_add(residual, vector_scal_mult(-alpha, s))
d1new = vector_dot(residual, residual)
direction = vector_add(residual,
vector_scal_mult(d1new / d1old, direction))
count += 1
error = vector_2norm(residual)
if getIterCount == True:
return x_i, count
else:
return x_i
def matrix_QR_factorization(matrix):
matrix_columns = [[matrix[i][j] for i in range(len(matrix))] for j in range(len(matrix[0]))]
u_vectors = [[matrix[i][j] for i in range(len(matrix))] for j in range(len(matrix[0]))]
for j in range(len(matrix[0])):
for i in range(j):
u_vectors[j] = vector_add(u_vectors[j],vector_scal_mult(-1 * vector_dot(matrix_columns[j], u_vectors[i]), u_vectors[i]))
u_vectors[j] = vector_scal_mult(1 / vector_2norm(u_vectors[j]), u_vectors[j])
matrix_Q = [[u_vectors[i][j] for i in range(len(u_vectors))] for j in range(len(u_vectors[0]))]
matrix_R = matrix_mult(matrix_transpose(matrix_Q), matrix)
return [matrix_Q,matrix_R]