Computation of Pseudospectra by Continuation

Abstract

The concept of pseudospectrum was introduced by Trefethen to explain the behavior of nonnormal operators. Many phenomena (for example, hydrodynamic instability and convergence of iterative methods for linear systems) cannot be accounted for by eigenvalue analysis but are more understandable by examining the pseudospectra. The straightforward way to compute pseudospectra involves many applications of the singular value decomposition (SVD). This paper presents several fast continuation methods to calculate pseudospectra of different types of matrices.

Estimation of the Linear Transient Growth of Perturbations of Cellular Flames

“…The direct Matlab implementation of SVD worked well in our case, though a few inverse iterations with continuation in z, suggested in [16], appeared to be as accurate and, on average, about six times faster. An alternative algorithm is based on projection to the interesting subspace through the Schur factorization followed by the Lanczos iterations.…”

Parallel computation of pseudospectra of large sparse matrices

“…The first attempt in this direction was done by Br€ u uhl [6]. Based on continuation with a predictor-corrector scheme, the process may fail in the case of angular discontinuities along the level curve [4,[12][13][14]22]. In [26], Wright and Trefethen use the upper Hessenberg matrix constructed after successive iterations of the implicitly restarted Arnoldi algorithm to cheaply compute an approximation of the pseudospectrum in a region near the interesting eigenvalues.…”

Computing the field of values and pseudospectra using the Lanczos method with continuation

“…If, at a given grid point, we obtain A1 and a corresponding eigenvector x1, then as the Lanczos starting vector for Algorithm 3 .1 at the next grid point we take u = x1 . Some justification for this continuation strategy is given by Lui [19] . For the field of values computation, if, for a given angle 0, we obtain extremal eigenvalues A1 and a,, and corresponding eigenvectors x1 and x,,, then our starting vector for the next value of 0 is u = ( x1 + x n )/IIx1 + x,,11 2 .…”

“…They obtain plots of comparable quality to those from the SVD method . Lui [19] presents several related methods . For large, possibly sparse matrices, which are our interest here, he proposes two algorithms .…”

“…The first applies inverse iteration to (A -zI)(A -zI)* . The second applies the Lanczos method to the same matrix (the use of reorthogonalization is not mentioned in [19]) . In both cases continuation is used and the iteration at each grid point is terminated when (an estimate of) the relative error in the computed eigenvalue of interest is at most 0 .1 .…”