def lu(M, info=None):
if not isinstance(M, Matrix):
return None
rows, cols = M.size()
if rows != cols:
raise ArithmeticError("LU factorization requires a square matrix (rows == cols)")
A = copy.deepcopy(M)
L = Matrix(rows, cols, mtype='i')
U = Matrix(rows, cols)
P = Matrix(rows, cols, mtype='i')
# Adjust rows of A so largest element is in the pivot of each row
l = 0
swaps = 0
for i in range(rows):
max_l = abs(A[i][l])
swap = i
for j in range(i + 1, rows):
if abs(A[j][l]) > max_l:
max_l = abs(A[j][l])
swap = j
if i != swap:
P.swap(i, swap)
swaps += 1
l += 1
# Swap rows to get larger pivots
A = P * A
# Keep track of the current pivot we are at to determine side of diagonal
l = 0
for i in range(rows):
for j in range(cols):
# Upper diagonal of U, including diagonal
if j >= l:
U[i][j] = A[i][j] - __ludot(i, U, L, i, j)
# Lower diagonal of L, not including the diagonal
else:
L[i][j] = 1 / U[j][j] * (A[i][j] - __ludot(j, U, L, i, j))
l += 1
if info and info == 's':
return [L, U, P, swaps]
return [L, U, P]
def test_det(self):
# Test property that determinant of identity matrix is 1
A = Matrix(10, 10, mtype="i")
self.assertAlmostEqual(lalg.det(A), 1, delta=0.001)
# Test property row exchanges case that determinant is (-1)^swaps
A.swap(1, 2)
self.assertAlmostEqual(lalg.det(A), -1, delta=0.001)
A.swap(3, 4)
self.assertAlmostEqual(lalg.det(A), 1, delta=0.001)
# Multiplying one row of a matrix by t results in the determinant being multiplied by t
# | t*a t*b | = t * |a b|
# | c d | |c d|
A = Matrix([4 * 2, 4 * 8], [3, 9])
B = Matrix([2, 8], [3, 9])
self.assertAlmostEqual(lalg.det(A), 4 * lalg.det(B), delta=0.001)
# Adding to one row of a matrix results in a linear combination of the matrix and addition
# | a+da b+db | = |a b| + |da db|
# | c d | |c d| |c d |
A = Matrix([8 + 9, -2 + 7], [-5, 12])
B = Matrix([8, -2], [-5, 12])
C = Matrix([9, 7], [-5, 12])
self.assertAlmostEqual(lalg.det(A), lalg.det(B) + lalg.det(C), delta=0.001)
# If A has a row of all zeros the determinant should be 0
A = Matrix(12, 12)
for i in range(12):
for j in range(12):
A[i][j] = random.randint(-1000, 1000)
A.set_row(3, [0] * 12)
self.assertAlmostEqual(lalg.det(A), 0)
# If two rows are equal the determinant is 0
A = Matrix(12, 12)
for i in range(12):
for j in range(12):
A[i][j] = random.randint(-1000, 1000)
A.set_row(random.randint(0, 5), [i for i in range(12)])
A.set_row(random.randint(6, 11), [i for i in range(12)])
self.assertAlmostEqual(lalg.det(A), 0)
def test_swap(self):
m = Matrix([1, 2, 3], [4, 5, 6])
m.swap(0, 1)
self.assertEqual(m, Matrix([4, 5, 6], [1, 2, 3]))
m = Matrix([1, 2], [3, 4], [5, 6], [7, 8], [9, 10])
m.swap(0, 4)
m.swap(1, 3)
# Middle row shouldn't move
m.swap(2, 2)
self.assertEqual(m, Matrix([9, 10], [7, 8], [5, 6], [3, 4], [1, 2]))
