A Simultaneous Iteration Algorithm for Real Matrices

Abstract

A FORTRAN algorithm m presented for obtaining the set of elgenvalues of largest absolute magnitude of a matrix together with the corresponding left or right eigenvectors The method m particularly suitable for large sparse matrices.

A Fully Implicit Model of the Three-Dimensional Thermohaline Ocean Circulation

“…In [9], a combination of spectral transformations and the Arnoldi algorithm [33] is used and applied to determine the linear stability of steady (coating) flows. A variant of the methods in [9] was used in [13], being a combination of a spectral transformation and the Simultaneous Iteration Technique [38]. As in [9], the idea of the algorithm is to transform the eigenvalue problem in such a way that the most dangerous modes become the most dominant modes (i.e.…”

Limited Memory Block Krylov Subspace Optimization for Computing Dominant Singular Value Decompositions

“…which proves (26). Finally, when x T x = 1 and Bx = 0, we divide the left-hand side of (26) by x T B 2 x and the right-hand side by B 2 2 ≥ x T B 2 x, leading to (27).…”

“…The classic simple subspace, or simultaneous, iteration method, extends the idea of the power method which computes the largest eigenvalue and its eigenvector (see [16,17,24,26] for example), performing repeated matrix multiplication followed by orthogonalization. More elaborative algorithms including Arnoldi methods [14,13], Lanczos methods [21,12], Jacobi-Davidson methods [23,2], to cite a few for each type, are specifically designed for large-scale but sparse matrices or other types of structured matrices.…”

Voltage Stability Analysis of Muscat Power System During Summer Weather Conditions

“…So the eigenvalues of importance are the critical eigenvalues of the reduced Jacobian matrix J R . An algorithm for calculating the minimum eigenvalue and the corresponding left and right eigenvectors has been developed in Stewart and Jennings (1981).…”