def inverse_iteration_for_eigenvector_int(A, eigenvalue):
"""Performs a series of inverse iteration steps with a known
eigenvalue to produce its eigenvector.
:param A: The 3x3 matrix to which the eigenvalue belongs.
:type A: :py:class:`numpy.ndarray`
:param eigenvalue: One approximate eigenvalue of the matrix A.
:type eigenvalue: int
:return: The eigenvector of this matrix and eigenvalue combination.
:rtype: :py:class:`numpy.ndarray`
"""
A = np.array(A, 'int64')
# Subtract the eigenvalue from the diagonal entries of the matrix.
for k in xrange(A.shape[0]):
A[k, k] -= eigenvalue
A, scale = fp.scale_64bit_matrix(A)
# Obtain the inverse of the matrix.
adj_A = mo.inverse_3by3_int64(A.flatten(), False)
eigenvector = adj_A.reshape((3, 3)).sum(1)
eigenvector, scale = fp.scale_64bit_vector(eigenvector)
e_norm = int(np.sqrt((eigenvector ** 2).sum()))
if (eigenvector[0] < 0) and (eigenvector[2] < 0):
eigenvector = -eigenvector
return eigenvector, e_norm
def inverse_iteration_for_eigenvector_int(A, eigenvalue):
"""Performs a series of inverse iteration steps with a known
eigenvalue to produce its eigenvector.
:param A: The 3x3 matrix to which the eigenvalue belongs.
:type A: :py:class:`numpy.ndarray`
:param eigenvalue: One approximate eigenvalue of the matrix A.
:type eigenvalue: int
:return: The eigenvector of this matrix and eigenvalue combination.
:rtype: :py:class:`numpy.ndarray`
"""
A = np.array(A, 'int64')
# Subtract the eigenvalue from the diagonal entries of the matrix.
for k in xrange(A.shape[0]):
A[k, k] -= eigenvalue
A, scale = fp.scale_64bit_matrix(A)
# Obtain the inverse of the matrix.
adj_A = mo.inverse_3by3_int64(A.flatten(), False)
eigenvector = adj_A.reshape((3, 3)).sum(1)
eigenvector, scale = fp.scale_64bit_vector(eigenvector)
e_norm = int(np.sqrt((eigenvector**2).sum()))
if (eigenvector[0] < 0) and (eigenvector[2] < 0):
eigenvector = -eigenvector
return eigenvector, e_norm
def QR_algorithm_shift_Givens_int(A):
"""The QR algorithm with largest value shift for finding eigenvalues, in integer precision.
Using Givens rotations for finding RQ.
:param A: The square matrix to find eigenvalues of.
:type A: :py:class:`numpy.ndarray`
:return: The eigenvalues.
:rtype: list
"""
A, scale = fp.scale_64bit_matrix(A.copy())
# First, tridiagonalize the input matrix to a Hessenberg matrix.
T = ma.convert_to_Hessenberg_Givens_int(A)
m, n = T.shape
if n != m:
raise np.linalg.LinAlgError("Array must be square.")
convergence_measure = []
eigenvalues = np.zeros((m, ), dtype='int64')
m -= 1
while m > 0:
# Obtain the shift from the lower right corner of the matrix.
mu_matrix = np.eye(T.shape[0], dtype='int64') * T[m, m]
# Perform Givens QR step (which returns RQ) on the shifted
# matrix and then shift it back.
T = Givens_QR_step_int(T - mu_matrix) + mu_matrix
# Add convergence information and extract eigenvalue if close enough.
convergence_measure.append(np.abs(T[m, m - 1]))
if convergence_measure[-1] <= 1:
eigenvalues[m] = T[m, m]
T = T[:m, :m]
m -= 1
eigenvalues[0] = T
return eigenvalues << scale, convergence_measure
def QR_algorithm_shift_Givens_int(A):
"""The QR algorithm with largest value shift for finding eigenvalues, in integer precision.
Using Givens rotations for finding RQ.
:param A: The square matrix to find eigenvalues of.
:type A: :py:class:`numpy.ndarray`
:return: The eigenvalues.
:rtype: list
"""
A, scale = fp.scale_64bit_matrix(A.copy())
# First, tridiagonalize the input matrix to a Hessenberg matrix.
T = ma.convert_to_Hessenberg_Givens_int(A)
m, n = T.shape
if n != m:
raise np.linalg.LinAlgError("Array must be square.")
convergence_measure = []
eigenvalues = np.zeros((m, ), dtype='int64')
m -= 1
while m > 0:
# Obtain the shift from the lower right corner of the matrix.
mu_matrix = np.eye(T.shape[0], dtype='int64') * T[m, m]
# Perform Givens QR step (which returns RQ) on the shifted
# matrix and then shift it back.
T = Givens_QR_step_int(T - mu_matrix) + mu_matrix
# Add convergence information and extract eigenvalue if close enough.
convergence_measure.append(np.abs(T[m, m - 1]))
if convergence_measure[-1] <= 1:
eigenvalues[m] = T[m, m]
T = T[:m, :m]
m -= 1
eigenvalues[0] = T
return eigenvalues << scale, convergence_measure