def ldu_factorization(matrix):
lower, upper = lu_factorization(matrix)
diagonal = identity_matrix(num_rows(matrix))
for i in range(num_rows(diagonal)):
diagonal[i][i] = deepcopy(upper[i][i])
upper = row_op_mult(upper, i, 1 / upper[i][i])
return lower, diagonal, upper
def permute_matrix(matrix):
permuted = identity_matrix(num_rows(matrix))
permute_from = deepcopy(matrix)
for i in range(num_rows(permute_from)):
if permute_from[i][i] == 0:
s = i + 1
while s < num_rows(permute_from):
if permute_from[s][i] != 0:
break
else:
s += 1
if s < num_rows(permute_from):
permuted = el_matrix_swap(permuted, i, s)
permute_from = el_matrix_swap(permute_from, i, s)
return permuted
def lu_factorization(matrix):
upper_pieces = upper_triangular(matrix, inverse_history=True)
upper = upper_pieces[0]
lower = identity_matrix(num_rows(matrix))
for i in range(len(upper_pieces[2])):
lower = lower * upper_pieces[2][i]
return lower, upper
def inverse(matrix):
reduced, traceback, _ = gaussian_elimination(matrix, history=True)
inverted = identity_matrix(num_rows(matrix))
if is_identity(reduced):
for el in traceback:
inverted = el * inverted
return inverted
else:
return None
def gaussian_elimination(matrix,
reduced=True,
history=False,
inverse_history=False):
matrix_c = deepcopy(matrix)
traceback = []
inverse_traceback = []
i = 0
while i < num_rows(matrix_c):
lead = leading_term_index(matrix_c[i])
if lead < 0:
i += 1
continue
r = 1 / matrix_c[i][lead]
matrix_c = row_op_mult(matrix_c, i, r)
if history:
traceback.append(el_matrix_mult(matrix_c, i, r))
if inverse_history:
inverse_traceback.append(el_matrix_mult(matrix_c, i, 1 / r))
j = 0 if reduced else i + 1
while j < num_rows(matrix_c):
if j == i:
j += 1
continue
sign = 1 if matrix_c[i][lead] < 0 == matrix_c[j][lead] < 0 else -1
m = sign * matrix_c[j][lead] / matrix_c[i][lead]
matrix_c = row_op_add(matrix_c, i, j, m)
if history:
traceback.append(el_matrix_add(matrix_c, j, i, m))
if inverse_history:
inverse_traceback.append(el_matrix_add(matrix_c, j, i, -m))
j += 1
i += 1
for j in range(num_rows(matrix_c)):
if is_zero_vector(matrix_c[j]):
for h in range(j + 1, num_rows(matrix_c)):
matrix_c = row_op_swap(matrix_c, h - 1, h)
if history:
traceback.append(el_matrix_swap(matrix_c, h - 1, h))
if inverse_history:
inverse_traceback.append(el_matrix_swap(
matrix_c, h - 1, h))
return matrix_c, traceback, inverse_traceback
def lower_triangular(matrix, history=False, inverse_history=False):
matrix_c = deepcopy(matrix)
traceback = []
inverse_traceback = []
i = num_rows(matrix_c) - 1
while i >= 0:
j = i - 1
while j >= 0:
s = 1 if matrix_c[i][i] < 0 == matrix_c[j][i] < 0 else -1
m = s * matrix_c[j][i] / matrix_c[i][i]
matrix_c = row_op_add(matrix_c, i, j, m)
if history:
t = identity_matrix(num_rows(matrix))
t[j][i] = deepcopy(m)
traceback.append(t)
if inverse_history:
t = identity_matrix(num_rows(matrix))
t[j][i] = deepcopy(-m)
inverse_traceback.append(t)
j -= 1
i -= 1
return matrix_c, traceback, inverse_traceback
def el_matrix_add(matrix, i, j, k):
t = identity_matrix(num_rows(matrix))
t[i][j] = deepcopy(k)
return t
def el_matrix_swap(matrix, i, j):
t = identity_matrix(num_rows(matrix))
t[i], t[j] = t[j], t[i]
return t
def el_matrix_mult(matrix, i, k):
t = identity_matrix(num_rows(matrix))
t[i][i] = deepcopy(k)
return t
