def QR_algorithm(A):
"""The straight QR algorithm for finding eigenvalues.
N_POLYPOINTS.B. Has very slow convergence. Use shifted versions instead.
.. code-block:: matlab
function [myeig tmvec] = QRalgorithm(A)
%------ Phase 1 ------
T = mytridiag(A);
%------ Phase 2 ------
m = length(A);
myeig = zeros(m,1);
% Size of tmvec is not known in advance.
% Estimating an initial size.
tmvec = zeros(m^2,1);
counter = 0;
while (m > 1)
counter = counter + 1;
[Q,R] = myQR(T);
T = R*Q;
tmvec(counter) = abs(T(m,m-1));
if (tmvec(counter) < 1e-8)
myeig(m) = T(m,m);
m = m - 1;
T = T(1:m,1:m);
end
end
myeig(1) = T;
tmvec = tmvec(1:counter);
end
:param A: A square matrix to find eigenvalues in.
:type A: :py:class:`numpy.ndarray`
:return: The eigenvalues.
:rtype: list
"""
# First, tridiagonalize the input matrix to a Hessenberg matrix.
T = ma.convert_to_Hessenberg_double(A.copy())
m, n = T.shape
if n != m:
raise np.linalg.LinAlgError("Array must be square.")
convergence_measure = []
eigenvalues = np.zeros((n, ), dtype='float')
n -= 1
while n > 0:
# Perform QR decomposition.
Q, R = ma.QR_factorisation_Householder_double(T)
# Multiply R and Q.
T = np.dot(R, Q)
# Add convergence information and extract eigenvalue if close enough.
convergence_measure.append(np.abs(T[n, n - 1]))
if convergence_measure[-1] < QR_ALGORITHM_TOLERANCE:
eigenvalues[n] = T[n, n]
T = T[:n, :n]
n -= 1
eigenvalues[0] = T
return eigenvalues, convergence_measure
def QR_algorithm_Wilkinson_shift(A):
"""The QR algorithm with Wilkinson shift for finding eigenvalues.
.. code-block:: matlab
function [myeig tmvec] = QRwilkinson(A)
%------ Phase 1 ------
T = mytridiag(A);
%------ Phase 2 ------
m = length(A);
myeig = zeros(m,1);
% Size of tmvec is not known in advance.
% Estimating an initial size.
tmvec = zeros(m*8,1);
mu = 0;
counter = 0;
while (m > 1)
counter = counter + 1;
muMat = diag(mu * ones(m,1));
[Q,R] = myQR(T - muMat);
T = R*Q + muMat;
tmvec(counter) = abs(T(m,m-1));
delta = (T(m-1,m-1) - T(m,m)) / 2;
mu = T(m,m) - sign(delta) * T(m,m-1) / ...
(abs(delta) + norm([delta T(m,m-1)]));
if (tmvec(counter) < 1e-8)
myeig(m) = T(m,m);
m = m - 1;
T = T(1:m,1:m);
end
end
myeig(1) = T;
tmvec = tmvec(1:counter);
end
:param A: The matrix to find eigenvalues in.
:type A: :py:class:`numpy.ndarray`
:return: The eigenvalues.
:rtype: list
"""
# First, tridiagonalize the input matrix to a Hessenberg matrix.
T = ma.convert_to_Hessenberg_double(A.copy())
m, n = T.shape
convergence_measure = []
eigenvalues = np.zeros((m, ), dtype='float')
m -= 1
mu = 0
while m > 0:
mu_matrix = np.eye(T.shape[0]) * mu
Q, R = ma.QR_factorisation_Householder_double(T - mu_matrix)
T = np.dot(R, Q) + mu_matrix
convergence_measure.append(np.abs(T[m, m - 1]))
delta = (T[m - 1, m - 1] - T[m, m]) / 2
mu = T[m, m] - np.sign(delta) * T[m, m - 1] / \
(np.abs(delta) + np.linalg.norm([delta, T[m, m - 1]]))
if convergence_measure[-1] < QR_ALGORITHM_TOLERANCE:
eigenvalues[m] = T[m, m]
T = T[:m, :m]
m -= 1
eigenvalues[0] = T
return eigenvalues, convergence_measure
def QR_algorithm_shift(A):
"""The QR algorithm with largest value shift for finding eigenvalues.
Using Householder reflection QR decomposition for finding RQ.
.. code-block:: matlab
function [myeig tmvec] = QRshift(A)
%------ Phase 1 ------
T = mytridiag(A);
%------ Phase 2 ------
m = length(A);
myeig = zeros(m,1);
% Size of tmvec is not known in advance.
% Estimating an initial size.
tmvec = zeros(m*8,1);
mu = 0;
counter = 0;
while (m > 1)
counter = counter + 1;
muMat = diag(mu * ones(m,1));
[Q,R] = myQR(T - muMat);
T = R*Q + muMat;
tmvec(counter) = abs(T(m,m-1));
mu = T(m,m)
if (tmvec(counter) < 1e-8)
myeig(m) = T(m,m);
m = m - 1;
T = T(1:m,1:m);
end
end
myeig(1) = T;
tmvec = tmvec(1:counter);
end
:param A:
:type A:
:return: The eigenvalues.
:rtype: list
"""
# First, tridiagonalize the input matrix to a Hessenberg matrix.
T = ma.convert_to_Hessenberg_double(A.copy())
m, n = T.shape
if n != m:
raise np.linalg.LinAlgError("Array must be square.")
convergence_measure = []
eigenvalues = np.zeros((n, ), dtype='float')
n -= 1
while n > 0:
# Obtain the shift from the lower right corner of the matrix.
mu_matrix = np.eye(T.shape[0]) * T[n, n]
# Perform QR decomposition on the shifted matrix.
Q, R = ma.QR_factorisation_Householder_double(T - mu_matrix)
# Multiply R and Q and shift the matrix back.
T = np.dot(R, Q) + mu_matrix
# Add convergence information and extract eigenvalue if close enough.
convergence_measure.append(np.abs(T[n, n - 1]))
if convergence_measure[-1] < QR_ALGORITHM_TOLERANCE:
eigenvalues[n] = T[n, n]
T = T[:n, :n]
n -= 1
eigenvalues[0] = T
return eigenvalues, convergence_measure
def QR_algorithm(A):
"""The straight QR algorithm for finding eigenvalues.
N_POLYPOINTS.B. Has very slow convergence. Use shifted versions instead.
.. code-block:: matlab
function [myeig tmvec] = QRalgorithm(A)
%------ Phase 1 ------
T = mytridiag(A);
%------ Phase 2 ------
m = length(A);
myeig = zeros(m,1);
% Size of tmvec is not known in advance.
% Estimating an initial size.
tmvec = zeros(m^2,1);
counter = 0;
while (m > 1)
counter = counter + 1;
[Q,R] = myQR(T);
T = R*Q;
tmvec(counter) = abs(T(m,m-1));
if (tmvec(counter) < 1e-8)
myeig(m) = T(m,m);
m = m - 1;
T = T(1:m,1:m);
end
end
myeig(1) = T;
tmvec = tmvec(1:counter);
end
:param A: A square matrix to find eigenvalues in.
:type A: :py:class:`numpy.ndarray`
:return: The eigenvalues.
:rtype: list
"""
# First, tridiagonalize the input matrix to a Hessenberg matrix.
T = ma.convert_to_Hessenberg_double(A.copy())
m, n = T.shape
if n != m:
raise np.linalg.LinAlgError("Array must be square.")
convergence_measure = []
eigenvalues = np.zeros((n, ), dtype='float')
n -= 1
while n > 0:
# Perform QR decomposition.
Q, R = ma.QR_factorisation_Householder_double(T)
# Multiply R and Q.
T = np.dot(R, Q)
# Add convergence information and extract eigenvalue if close enough.
convergence_measure.append(np.abs(T[n, n - 1]))
if convergence_measure[-1] < QR_ALGORITHM_TOLERANCE:
eigenvalues[n] = T[n, n]
T = T[:n, :n]
n -= 1
eigenvalues[0] = T
return eigenvalues, convergence_measure
def QR_algorithm_Wilkinson_shift(A):
"""The QR algorithm with Wilkinson shift for finding eigenvalues.
.. code-block:: matlab
function [myeig tmvec] = QRwilkinson(A)
%------ Phase 1 ------
T = mytridiag(A);
%------ Phase 2 ------
m = length(A);
myeig = zeros(m,1);
% Size of tmvec is not known in advance.
% Estimating an initial size.
tmvec = zeros(m*8,1);
mu = 0;
counter = 0;
while (m > 1)
counter = counter + 1;
muMat = diag(mu * ones(m,1));
[Q,R] = myQR(T - muMat);
T = R*Q + muMat;
tmvec(counter) = abs(T(m,m-1));
delta = (T(m-1,m-1) - T(m,m)) / 2;
mu = T(m,m) - sign(delta) * T(m,m-1) / ...
(abs(delta) + norm([delta T(m,m-1)]));
if (tmvec(counter) < 1e-8)
myeig(m) = T(m,m);
m = m - 1;
T = T(1:m,1:m);
end
end
myeig(1) = T;
tmvec = tmvec(1:counter);
end
:param A: The matrix to find eigenvalues in.
:type A: :py:class:`numpy.ndarray`
:return: The eigenvalues.
:rtype: list
"""
# First, tridiagonalize the input matrix to a Hessenberg matrix.
T = ma.convert_to_Hessenberg_double(A.copy())
m, n = T.shape
convergence_measure = []
eigenvalues = np.zeros((m, ), dtype='float')
m -= 1
mu = 0
while m > 0:
mu_matrix = np.eye(T.shape[0]) * mu
Q, R = ma.QR_factorisation_Householder_double(T - mu_matrix)
T = np.dot(R, Q) + mu_matrix
convergence_measure.append(np.abs(T[m, m - 1]))
delta = (T[m - 1, m - 1] - T[m, m]) / 2
mu = T[m, m] - np.sign(delta) * T[m, m - 1] / \
(np.abs(delta) + np.linalg.norm([delta, T[m, m - 1]]))
if convergence_measure[-1] < QR_ALGORITHM_TOLERANCE:
eigenvalues[m] = T[m, m]
T = T[:m, :m]
m -= 1
eigenvalues[0] = T
return eigenvalues, convergence_measure
def QR_algorithm_shift(A):
"""The QR algorithm with largest value shift for finding eigenvalues.
Using Householder reflection QR decomposition for finding RQ.
.. code-block:: matlab
function [myeig tmvec] = QRshift(A)
%------ Phase 1 ------
T = mytridiag(A);
%------ Phase 2 ------
m = length(A);
myeig = zeros(m,1);
% Size of tmvec is not known in advance.
% Estimating an initial size.
tmvec = zeros(m*8,1);
mu = 0;
counter = 0;
while (m > 1)
counter = counter + 1;
muMat = diag(mu * ones(m,1));
[Q,R] = myQR(T - muMat);
T = R*Q + muMat;
tmvec(counter) = abs(T(m,m-1));
mu = T(m,m)
if (tmvec(counter) < 1e-8)
myeig(m) = T(m,m);
m = m - 1;
T = T(1:m,1:m);
end
end
myeig(1) = T;
tmvec = tmvec(1:counter);
end
:param A:
:type A:
:return: The eigenvalues.
:rtype: list
"""
# First, tridiagonalize the input matrix to a Hessenberg matrix.
T = ma.convert_to_Hessenberg_double(A.copy())
m, n = T.shape
if n != m:
raise np.linalg.LinAlgError("Array must be square.")
convergence_measure = []
eigenvalues = np.zeros((n, ), dtype='float')
n -= 1
while n > 0:
# Obtain the shift from the lower right corner of the matrix.
mu_matrix = np.eye(T.shape[0]) * T[n, n]
# Perform QR decomposition on the shifted matrix.
Q, R = ma.QR_factorisation_Householder_double(T - mu_matrix)
# Multiply R and Q and shift the matrix back.
T = np.dot(R, Q) + mu_matrix
# Add convergence information and extract eigenvalue if close enough.
convergence_measure.append(np.abs(T[n, n - 1]))
if convergence_measure[-1] < QR_ALGORITHM_TOLERANCE:
eigenvalues[n] = T[n, n]
T = T[:n, :n]
n -= 1
eigenvalues[0] = T
return eigenvalues, convergence_measure
