def woodbury(X, Ainv, m, up=True):
'''
Does a rank-k adjustment of Ainv using the woodbury matrix identity
(A + XX^H)^(-1) = A^(-1) - (A^(-1)X ( I + X^(H)A^(-1)X )^(-1)X^(H)A^(-1))
If up == True: A rank update is performed,
otherwise we do a rank degrade
NB! Not finished
'''
factor = 1
if up != True:
factor = -1
XH = mynp.conjugatetranspose(X)
XHAinv = mynp.dot(XH, Ainv)
AinvX = mynp.dot(Ainv, XH)
XHAinvX = mynp.dot(XHAinv, X)
IXHAinvX = mynp.eye(m) + (factor * XHAinvX)
# TODO: inv_IXHAinvX = we need to invert IXHAinvX!!!
AinvX_IXHAinvX_XHAinv = mynp.dot(mynp.dot(AinvX, IXHAinvX), XHAinv)
newAinv = Ainv + (-factor * AinvX_IXHAinvX_XHAinv)
return newAinv
def woodbury(X, Ainv, m, up = True):
'''
Does a rank-k adjustment of Ainv using the woodbury matrix identity
(A + XX^H)^(-1) = A^(-1) - (A^(-1)X ( I + X^(H)A^(-1)X )^(-1)X^(H)A^(-1))
If up == True: A rank update is performed,
otherwise we do a rank degrade
NB! Not finished
'''
factor = 1
if up != True:
factor = -1
XH = mynp.conjugatetranspose(X)
XHAinv = mynp.dot(XH, Ainv)
AinvX = mynp.dot(Ainv, XH)
XHAinvX = mynp.dot(XHAinv, X)
IXHAinvX = mynp.eye(m) + (factor * XHAinvX)
# TODO: inv_IXHAinvX = we need to invert IXHAinvX!!!
AinvX_IXHAinvX_XHAinv = mynp.dot(mynp.dot(AinvX, IXHAinvX), XHAinv)
newAinv = Ainv + (-factor * AinvX_IXHAinvX_XHAinv)
return newAinv
def testSolvingComplexAxbWithCholeskyAndForwardBackwardSolve(self):
R = md.cholesky(self.complexA, self.n)
for i in range(self.n):
for j in range(self.n):
self.assertAlmostEqual(
self.complexR[i][j].tolist(), R[i][j].tolist(), 14,
"expected " + str(self.complexR[i][j]) + " was " +
str(R[i][j]))
RT = mynp.conjugatetranspose(R)
y = ls.forwardSolve(RT, self.complexb, self.n)
for i in range(self.n):
self.assertAlmostEqual(
self.complexy[i].tolist(), y[i].tolist(), 14,
"expected " + str(self.complexy[i]) + " was " + str(y[i]))
x = ls.backtrackSolve(R, y, self.n)
for i in range(self.n):
self.assertAlmostEqual(
self.complexx[i].tolist(), x[i].tolist(), 14,
"expected " + str(self.complexx[i]) + " was " + str(x[i]))
def solveCholeskyCpp(A, b, n): ''' Solves the Hermitian positive definite system Ax = b using cholesky decomposition''' U = md.cholesky(A, n) UT = np.conjugatetranspose(U) y = forwardSolve(UT, b, n) x = backtrackSolve(U, y, n)
def solveCholeskyCpp(A, b, n):
''' Solves the Hermitian positive definite system Ax = b using cholesky decomposition'''
U = md.cholesky(A, n)
UT = np.conjugatetranspose(U)
y = forwardSolve(UT, b, n)
x = backtrackSolve(U, y, n)
def testRTransposedTimesRIsEqualA(self):
R = md.cholesky(self.A, self.n)
RT = mynp.conjugatetranspose(R)
A = mynp.dot(RT, R)
for i in range(self.n):
for j in range(self.n):
self.assertAlmostEqual(self.A[i][j].tolist(), A[i][j].tolist(), 15,
"expected " + str(self.A[i][j]) + " was " + str(A[i][j]))
def solveUHDU(A, b, n): ''' Solves the Hermitian positive definite system Ax = b using U'DU decomposition''' UD = md.uhdu(A, n); UH = np.conjugatetranspose(UD[0]) y = forwardSolve(UH, b, n) z = diagonalSolve(UD[1], y, n) x = backtrackSolve(UD[0], z, n) return x
def solveUHDU(A, b, n):
''' Solves the Hermitian positive definite system Ax = b using U'DU decomposition'''
UD = md.uhdu(A, n)
UH = np.conjugatetranspose(UD[0])
y = forwardSolve(UH, b, n)
z = diagonalSolve(UD[1], y, n)
x = backtrackSolve(UD[0], z, n)
return x
def testRTransposedTimesRIsEqualA(self):
R = md.cholesky(self.A, self.n)
RT = mynp.conjugatetranspose(R)
A = mynp.dot(RT, R)
for i in range(self.n):
for j in range(self.n):
self.assertAlmostEqual(
self.A[i][j].tolist(), A[i][j].tolist(), 15,
"expected " + str(self.A[i][j]) + " was " + str(A[i][j]))
def testSolvingAxbWithCholeskyAndForwardBackwardSolve(self):
R = md.cholesky(self.A, self.n)
RT = mynp.conjugatetranspose(R)
y = ls.forwardSolve(RT, self.b1, self.n)
x = ls.backtrackSolve(R, y, self.n)
for i in range(self.n):
self.assertAlmostEqual(self.x1[i].tolist(), x[i].tolist(), 14,
"expected " + str(self.x1[i]) + " was " + str(x[i]))
y = ls.forwardSolve(RT, self.b2, self.n)
x = ls.backtrackSolve(R, y, self.n)
for i in range(self.n):
self.assertAlmostEqual(self.x2[i].tolist(), x[i].tolist(), 14,
"expected " + str(self.x2[i]) + " was " + str(x[i]))
def testSolvingAxbWithCholeskyAndForwardBackwardSolve(self):
R = md.cholesky(self.A, self.n)
RT = mynp.conjugatetranspose(R)
y = ls.forwardSolve(RT, self.b1, self.n)
x = ls.backtrackSolve(R, y, self.n)
for i in range(self.n):
self.assertAlmostEqual(
self.x1[i].tolist(), x[i].tolist(), 14,
"expected " + str(self.x1[i]) + " was " + str(x[i]))
y = ls.forwardSolve(RT, self.b2, self.n)
x = ls.backtrackSolve(R, y, self.n)
for i in range(self.n):
self.assertAlmostEqual(
self.x2[i].tolist(), x[i].tolist(), 14,
"expected " + str(self.x2[i]) + " was " + str(x[i]))
def testSolvingComplexAxbWithCholeskyAndForwardBackwardSolve(self):
R = md.cholesky(self.complexA, self.n)
for i in range(self.n):
for j in range(self.n):
self.assertAlmostEqual(self.complexR[i][j].tolist(), R[i][j].tolist(), 14,
"expected " + str(self.complexR[i][j]) + " was " + str(R[i][j]))
RT = mynp.conjugatetranspose(R)
y = ls.forwardSolve(RT, self.complexb, self.n)
for i in range(self.n):
self.assertAlmostEqual(self.complexy[i].tolist(), y[i].tolist(), 14,
"expected " + str(self.complexy[i]) + " was " + str(y[i]))
x = ls.backtrackSolve(R, y, self.n)
for i in range(self.n):
self.assertAlmostEqual(self.complexx[i].tolist(), x[i].tolist(), 14,
"expected " + str(self.complexx[i]) + " was " + str(x[i]))
def testComplexConjugateTransposed(self):
AT = mynp.conjugatetranspose(self.complexA)
self.assertListEqual(self.complexA.tolist(), AT.tolist())
def solveBiCG(A, b, x0, tol, itr):
''' Solves the complex linear system Ax = b using the complex-biconjugate-gradient method
Arguments:
A -- Coefficient matrix
b -- Right-side vector
x0 -- Initial solution (default: 0-vector)
tol -- Stop solver when error is less than tol (default: 0)
itr -- Stop solver after itr iterations -- default
'''
# Check arguments
m,n = len(A),len(A[0])
if m != n:
raise Exception('Coefficient matrix is not square')
if len(b) != n:
raise Exception('Dimension mismatch between A and b')
x = x0 # make origin as starting point if no x is passed
# Calculate initial residual r = b - Ax0 and search direction p
r0 = b - np.dot(A, x)
r = r0
normr0 = np.euclidnorm(r0)
if (np.euclidnorm(r) / normr0) <= tol: # break if error is less-than tolerance.
# TODO: Should also test bir
return x
p = np.array(r)
# init biresidual and bidirection
bir = r.conjugate()
bip = p.conjugate()
numerator,denominator = 0,0
alpha,betha = 0,0
AH = np.conjugatetranspose(A) # Simplification: since A is Hermitian and semi-positive definite in our case, we can avoid this step...
if itr <= 0 or itr > m:
itr = m
for i in range(itr):
# Calculate common matrix-vector products
Ap = np.dot(A, p)
AHbip = np.dot(AH, bip)
# Calculate step-length parameter
numerator = np.vdot(bir, r)
denominator = np.vdot(bip, Ap)
alpha = numerator / denominator
# Obtain new solution
x = x + (alpha * p)
# Calculate new residual and biresidual
# r = r - (alpha * Ap)
for i in range(m):
r[i] = r[i] - alpha * Ap[i] # Calculate new residual and biresidual
if np.euclidnorm(r) / normr0 <= tol:
# TODO: Should also test bir
return x
bir = bir - (alpha.conjugate() * AHbip)
# Calculate the biconjugacy coefficient
numerator = np.vdot(AHbip, r)
denominator = np.vdot(bip, Ap)
betha = -1.0 * (numerator / denominator)
# Obtain new search and bi-search-direction
p = r + (betha * p)
bip = bir + (betha.conjugate() * bip)
return x
def solveBiCG(A, b, x0, tol, itr):
''' Solves the complex linear system Ax = b using the complex-biconjugate-gradient method
Arguments:
A -- Coefficient matrix
b -- Right-side vector
x0 -- Initial solution (default: 0-vector)
tol -- Stop solver when error is less than tol (default: 0)
itr -- Stop solver after itr iterations -- default
'''
# Check arguments
m, n = len(A), len(A[0])
if m != n:
raise Exception('Coefficient matrix is not square')
if len(b) != n:
raise Exception('Dimension mismatch between A and b')
x = x0 # make origin as starting point if no x is passed
# Calculate initial residual r = b - Ax0 and search direction p
r0 = b - np.dot(A, x)
r = r0
normr0 = np.euclidnorm(r0)
if (np.euclidnorm(r) /
normr0) <= tol: # break if error is less-than tolerance.
# TODO: Should also test bir
return x
p = np.array(r)
# init biresidual and bidirection
bir = r.conjugate()
bip = p.conjugate()
numerator, denominator = 0, 0
alpha, betha = 0, 0
AH = np.conjugatetranspose(
A
) # Simplification: since A is Hermitian and semi-positive definite in our case, we can avoid this step...
if itr <= 0 or itr > m:
itr = m
for i in range(itr):
# Calculate common matrix-vector products
Ap = np.dot(A, p)
AHbip = np.dot(AH, bip)
# Calculate step-length parameter
numerator = np.vdot(bir, r)
denominator = np.vdot(bip, Ap)
alpha = numerator / denominator
# Obtain new solution
x = x + (alpha * p)
# Calculate new residual and biresidual
# r = r - (alpha * Ap)
for i in range(m):
r[i] = r[i] - alpha * Ap[
i] # Calculate new residual and biresidual
if np.euclidnorm(r) / normr0 <= tol:
# TODO: Should also test bir
return x
bir = bir - (alpha.conjugate() * AHbip)
# Calculate the biconjugacy coefficient
numerator = np.vdot(AHbip, r)
denominator = np.vdot(bip, Ap)
betha = -1.0 * (numerator / denominator)
# Obtain new search and bi-search-direction
p = r + (betha * p)
bip = bir + (betha.conjugate() * bip)
return x
def testCholeskyInPlace(self):
L = md.choleskyInPlace(self.complexA, self.n)
R = mynp.conjugatetranspose(L)
self.assertMatrixAlmosteEqual(self.complexR.tolist(), R.tolist(), 14)
def testForwardSolve(self):
x = ls.forwardSolve(mynp.conjugatetranspose(self.C), self.b1c, self.n)
self.assertListEqual(self.x1cT.tolist(), x.tolist(), "Error in forward solve")
def testComplexConjugateTransposed(self):
AT = mynp.conjugatetranspose(self.complexA)
self.assertListEqual(self.complexA.tolist(), AT.tolist())
def testCholeskyInPlace(self):
L = md.choleskyInPlace(self.complexA, self.n)
R = mynp.conjugatetranspose(L)
self.assertMatrixAlmosteEqual(self.complexR.tolist(), R.tolist(), 14)
