def test_jacobi_method(self):
A = [[-1.0, -0.5, -0.2],
[-0.5, -2.0, -1.0],
[-0.2, -1.0, -3.0]]
lamda1, x1 = ea.jacobi_method (A)
self.assertEqual(len(A), len(lamda1))
self.assertEqual(len(A[0]), len(lamda1[0]))
self.assertEqual(len(A), len(x1))
self.assertEqual(len(A[0]), len(x1[0]))
Ax1 = la.multiply_matrix_matrix(A, x1)
# check A V = Lambda V
for k_pivot in xrange(len(A)):
# diagonal term
lambda_i = lamda1[k_pivot][k_pivot]
# off diagonal
for i_row in xrange(len(A)):
self.assertAlmostEqual(Ax1[i_row][k_pivot],lambda_i * x1[i_row][k_pivot])
# check VT A V = Lambda
x1TAx1 = la.multiply_matrix_matrix(zip(*x1), Ax1)
for i_row in xrange(0, len(A) - 1):
# check diagonal
self.assertAlmostEqual(x1TAx1[i_row][i_row], lamda1[i_row][i_row])
# check off-diagonal
for j_column in xrange(i_row + 1, len(A)):
self.assertAlmostEqual(x1TAx1[i_row][j_column], 0.0)
def test_jacobi_method(self):
A = [[-1.0, -0.5, -0.2], [-0.5, -2.0, -1.0], [-0.2, -1.0, -3.0]]
lamda1, x1 = ea.jacobi_method(A)
self.assertEqual(len(A), len(lamda1))
self.assertEqual(len(A[0]), len(lamda1[0]))
self.assertEqual(len(A), len(x1))
self.assertEqual(len(A[0]), len(x1[0]))
Ax1 = la.multiply_matrix_matrix(A, x1)
# check A V = Lambda V
for k_pivot in xrange(len(A)):
# diagonal term
lambda_i = lamda1[k_pivot][k_pivot]
# off diagonal
for i_row in xrange(len(A)):
self.assertAlmostEqual(Ax1[i_row][k_pivot],
lambda_i * x1[i_row][k_pivot])
# check VT A V = Lambda
x1TAx1 = la.multiply_matrix_matrix(zip(*x1), Ax1)
for i_row in xrange(0, len(A) - 1):
# check diagonal
self.assertAlmostEqual(x1TAx1[i_row][i_row], lamda1[i_row][i_row])
# check off-diagonal
for j_column in xrange(i_row + 1, len(A)):
self.assertAlmostEqual(x1TAx1[i_row][j_column], 0.0)
def general_eigenproblem_symmetric(A, B):
"""
Solve Az = lambda Bz using Cholesky decomposition
Let
B = L LT
and
z = LT**(-1)y
then
A LT**(-1)y = lambda L LT LT**(-1)y = lambda L y
Mutiplying L**(-1) gives
L**(-1) A LT**(-1)y = lambda L**(-1) L y = lambda y
So let
C = L**(-1) A LT**(-1)
and find eigenvalues and eigenvectors of C.
Later
Z = LT**(-1)Y
ref: Susan Blackford, Generalized Symmetric Definite Eigenproblems, http://www.netlib.org/lapack/lug/node54.html,
1999 Oct 01 (accessed 2015 Nov 30).
:param A: n x n matrix
:param B: n x n matrix
:return w: 1 x n eigenvalue vector
:return Z: n x n eigenvector matrix
"""
L = cholesky_decomposition(B)
LT = zip(*L)
L_inv = gj.gauss_jordan(L)
LT_inv = gj.gauss_jordan(LT)
del L[:], LT[:]
del L, LT
L_inv_A = la.multiply_matrix_matrix(L_inv, A)
del L_inv[:]
del L_inv
C = la.multiply_matrix_matrix(L_inv_A, LT_inv)
W, Y = jacobi_method(C)
del C[:]
del C
# diagonal elements
w = [Wi[i] for i, Wi in enumerate(W)]
del W[:]
del W
Z = la.multiply_matrix_matrix(LT_inv, Y)
del Y[:]
del Y
return w, Z
def general_eigenproblem_symmetric(A, B):
"""
Solve Az = lambda Bz using Cholesky decomposition
Let
B = L LT
and
z = LT**(-1)y
then
A LT**(-1)y = lambda L LT LT**(-1)y = lambda L y
Mutiplying L**(-1) gives
L**(-1) A LT**(-1)y = lambda L**(-1) L y = lambda y
So let
C = L**(-1) A LT**(-1)
and find eigenvalues and eigenvectors of C.
Later
Z = LT**(-1)Y
ref: Susan Blackford, Generalized Symmetric Definite Eigenproblems, http://www.netlib.org/lapack/lug/node54.html,
1999 Oct 01 (accessed 2015 Nov 30).
:param A: n x n matrix
:param B: n x n matrix
:return w: 1 x n eigenvalue vector
:return Z: n x n eigenvector matrix
"""
L = cholesky_decomposition(B)
LT = zip(*L)
L_inv = gj.gauss_jordan(L)
LT_inv = gj.gauss_jordan(LT)
del L[:], LT[:]
del L, LT
L_inv_A = la.multiply_matrix_matrix(L_inv, A)
del L_inv[:]
del L_inv
C = la.multiply_matrix_matrix(L_inv_A, LT_inv)
W, Y = jacobi_method(C)
del C[:]
del C
# diagonal elements
w = [Wi[i] for i, Wi in enumerate(W)]
del W[:]
del W
Z = la.multiply_matrix_matrix(LT_inv, Y)
del Y[:]
del Y
return w, Z
def test_multiply_matrix_matrix_02(self):
for angle_deg in xrange(361):
angle_rad = math.radians(angle_deg)
cos = math.cos(angle_rad)
sin = math.sin(angle_rad)
A = [[cos, -sin],
[sin, cos]]
angle2_deg = - angle_deg
angle2_rad = math.radians(angle2_deg)
cos2 = math.cos(angle2_rad)
sin2 = math.sin(angle2_rad)
B = [[cos2, -sin2],
[sin2, cos2]]
C = la.multiply_matrix_matrix (A, B)
expected = [[1.0, 0.0],
[0.0, 1.0]]
self.assertEqual(len(C), len(expected))
for k in xrange(len(expected)):
self.assertEqual(len(C[k]), len(expected[k]))
for j in xrange(len(expected[k])):
self.assertAlmostEqual(C[k][j], expected[k][j])
def test_multiply_matrix_matrix_01(self):
A = [[1.0, 1.0],
[1.0, 1.0]]
B = [[1.0, -1.0],
[-1.0, 1.0]]
C = la.multiply_matrix_matrix(A, B)
expected = [[0.0, 0.0],
[0.0, 0.0]]
self.assertSequenceEqual(C, expected)
def test_cholesky_decomposition_01(self):
L_expected = [[10.0, 0.0, 0.0, 0.0], [4.0, 9.0, 0.0, 0.0],
[3.0, 5.0, 8.0, 0.0], [1.0, 2.0, 6.0, 7.0]]
A = la.multiply_matrix_matrix(L_expected, zip(*L_expected))
L = ea.cholesky_decomposition(A)
A_expected = la.multiply_matrix_matrix(L, zip(*L))
# check size
self.assertEqual(len(A), len(L))
# check row
for i_row in xrange(0, len(A)):
self.assertEqual(len(A), len(L[i_row]))
for j_column in xrange(0, len(A)):
self.assertAlmostEqual(A[i_row][j_column],
A_expected[i_row][j_column])
self.assertAlmostEqual(L_expected[i_row][j_column],
L[i_row][j_column])
def test_cholesky_decomposition_01(self):
L_expected = [[10.0, 0.0, 0.0, 0.0],
[4.0, 9.0, 0.0, 0.0],
[3.0, 5.0, 8.0, 0.0],
[1.0, 2.0, 6.0, 7.0]]
A = la.multiply_matrix_matrix(L_expected, zip(*L_expected))
L = ea.cholesky_decomposition(A)
A_expected = la.multiply_matrix_matrix(L, zip(*L))
# check size
self.assertEqual(len(A), len(L))
# check row
for i_row in xrange(0, len(A)):
self.assertEqual(len(A), len(L[i_row]))
for j_column in xrange(0, len(A)):
self.assertAlmostEqual(A[i_row][j_column], A_expected[i_row][j_column])
self.assertAlmostEqual(L_expected[i_row][j_column], L[i_row][j_column])
def test_multiply_matrix_matrix_03(self):
A = [[11, 12, 13],
[21, 22, 23]]
B = [[111, 112],
[121, 122],
[131, 132]]
C = la.multiply_matrix_matrix(A, B)
BT = zip(*B)
expected = [[la.dot(A[0], BT[0]), la.dot(A[0], BT[1])],
[la.dot(A[1], BT[0]), la.dot(A[1], BT[1])]]
self.assertSequenceEqual(C, expected)
def test_gauss_jordan_01(self):
A = [[3, 2, 1],
[2, 3, 2],
[1, 2, 3]]
A_inverse = gj.gauss_jordan(A)
result = la.multiply_matrix_matrix(A, A_inverse)
self.assertEqual(len(A), len(A_inverse))
for i in xrange(len(A)):
self.assertEqual(len(A[i]), len(A_inverse[i]))
for j in xrange(len(A[i])):
if i == j:
self.assertAlmostEqual(result[i][j], 1.0)
else:
self.assertAlmostEqual(result[i][j], 0.0)
def test_cholesky_decomposition_00(self):
A = [[16.0, 12.0, 4.0,],
[12.0, 34.0, 13.0,],
[ 4.0, 13.0, 41.0,]]
L_expected = ((4, 0, 0),
(3, 5, 0),
(1, 2, 6),)
L = ea.cholesky_decomposition(A)
A_expected = la.multiply_matrix_matrix(L, zip(*L))
# check size
self.assertEqual(len(A), len(L))
# check row
for i_row in xrange(0, len(A)):
self.assertEqual(len(A), len(L[i_row]))
for j_column in xrange(0, len(A)):
self.assertAlmostEqual(A[i_row][j_column], A_expected[i_row][j_column])
self.assertAlmostEqual(L_expected[i_row][j_column], L[i_row][j_column])
def test_cholesky_decomposition_00(self):
A = [[
16.0,
12.0,
4.0,
], [
12.0,
34.0,
13.0,
], [
4.0,
13.0,
41.0,
]]
L_expected = (
(4, 0, 0),
(3, 5, 0),
(1, 2, 6),
)
L = ea.cholesky_decomposition(A)
A_expected = la.multiply_matrix_matrix(L, zip(*L))
# check size
self.assertEqual(len(A), len(L))
# check row
for i_row in xrange(0, len(A)):
self.assertEqual(len(A), len(L[i_row]))
for j_column in xrange(0, len(A)):
self.assertAlmostEqual(A[i_row][j_column],
A_expected[i_row][j_column])
self.assertAlmostEqual(L_expected[i_row][j_column],
L[i_row][j_column])
# 행 반복문
for j_row in xrange(0, n_row):
if j_row != i_pivot:
ratio = -AI[j_row][i_pivot]
# 열 반복문
for k_column in xrange(n_column * 2):
AI[j_row][k_column] += ratio * AI[i_pivot][k_column]
# 이 반복문이 끝나고 나면 주 대각선 이외의 요소는 모두 0
print "After Gauss Jordan"
pprint(AI)
# 오른쪽의 행렬을 떼어냄
result = []
for i_row in xrange(n_row):
result.append(AI[i_row][n_column:])
return result
if "__main__" == __name__:
A = [[3, 2, 1], [2, 3, 2], [1, 2, 3]]
A_inverse = gauss_jordan(A)
print "A inverse"
pprint(A_inverse)
I_expected = la.multiply_matrix_matrix(A, A_inverse)
print "I expected"
pprint(I_expected)
if j_row != i_pivot:
ratio = -AI[j_row][i_pivot]
# 열 반복문
for k_column in xrange(n_column * 2):
AI[j_row][k_column] += ratio * AI[i_pivot][k_column]
# 이 반복문이 끝나고 나면 주 대각선 이외의 요소는 모두 0
print "After Gauss Jordan"
pprint (AI)
# 오른쪽의 행렬을 떼어냄
result = []
for i_row in xrange(n_row):
result.append(AI[i_row][n_column:])
return result
if "__main__" == __name__:
A = [[3, 2, 1],
[2, 3, 2],
[1, 2, 3]]
A_inverse = gauss_jordan(A)
print "A inverse"
pprint(A_inverse)
I_expected = la.multiply_matrix_matrix(A, A_inverse)
print "I expected"
pprint(I_expected)
pprint(X)
counter += 1
print("Jacobi method counter = %d" % counter)
return A0, X
if "__main__" == __name__:
A = [[2.0, 1.0],
[1.0, 3.0]]
lamda1, x1 = power_method(A)
print "lambda =", lamda1
print "x =", x1
lamda, x = jacobi_method(A)
print "lambda ="
pprint(lamda, width=30)
print "x ="
pprint(x)
import random
n = 64
A = [[random.randint(1, 127)/256.0 for j_column in xrange(n)] for i_row in xrange(n)]
# symmetric
A = la.multiply_matrix_matrix(A, zip(*A))
import cProfile
cProfile.run('jacobi_method(A)')
