Pseudospectra computation of large matrices

IMA Journal of Numerical Analysis, 2005

Two useful measures of the robust stability of the discrete-time dynamical system x k+1 = Ax k are the -pseudospectral radius and the numerical radius of A. The -pseudospectral radius of A is the largest of the moduli of the points in the -pseudospectrum of A, while the numerical radius is the largest of the moduli of the points in the field of values. We present globally convergent algorithms for computing the -pseudospectral radius and the numerical radius. For the former algorithm, we discuss conditions under which it is quadratically convergent and provide a detailed accuracy analysis giving conditions under which the algorithm is backward stable. The algorithms are inspired by methods of Byers, Boyd-Balakrishnan, He-Watson and Burke-Lewis-Overton for related problems and depend on computing eigenvalues of symplectic pencils and Hamiltonian matrices.

Systems & Control Letters, 2010

SIAM Journal on Matrix Analysis and Applications, 2013

We consider the real ε-pseudospectrum of a real square matrix, which is the set of eigenvalues of all real matrices that are ε-close to the given matrix, where closeness is measured in either the 2-norm or the Frobenius norm. We characterize extremal points and compare the situation with that for the complex ε-pseudospectrum. We present differential equations for rank-1 and rank-2 matrices for the computation of the real pseudospectral abscissa and radius. Discretizations of the differential equations yield algorithms that are fast and well suited for sparse large matrices. Based on these low-rank differential equations, we further obtain an algorithm for drawing boundary sections of the real pseudospectrum with respect to both the 2-norm and the Frobenius norm.

Numerical Linear Algebra with Applications, 2014

The isospectral reduction of matrix, which is closely related to its Schur complement, allows to reduce the size of a matrix while maintaining its eigenvalues up to a known set. Here we generalize this procedure by increasing the number of possible ways a matrix can be isospectrally reduced. The reduced matrix has rational functions as entries. We show that the notion of pseudospectrum can be extended to this class of matrices and that the pseudospectrum of a matrix shrinks as the matrix is reduced. Hence the eigenvalues of a reduced matrix are more robust to entry-wise perturbations than the eigenvalues of the original matrix. We also introduce the notion of inverse pseudospectrum (or pseudoresonances), which indicates how stable the poles of a matrix with rational function entries are to certain matrix perturbations. A mass spring system is used to illustrate and give a physical interpretation to both pseudospectra and inverse pseudospectra.

2001

Abstract Given a matrix A, the computation of its pseudospectrum A∈(A) is a far more expensive task than the computation of characteristics such as the condition number and the matrix spectrum. As research of the last 15 years has shown, however, the matrix pseudospectrum provides valuable information that is not included in other indicators. So, we ask how to compute it efficiently and build a tool that would facilitate engineers and scientists to make such analyses?

Linear Algebra and its Applications, 2020

We approximate the pseudospectra of a block triangular matrix with two blocks on the diagonal in terms of the pseudospectra of its diagonal blocks.

Journal of Mathematical Analysis and Applications, 2017

Given a Hilbert space operator T , the level sets of function Ψ T (z) = (T − z) −1 −1 determine the so-called pseudospectra of T. We set Ψ T to be zero on the spectrum of T. After giving some elementary properties of Ψ T (which, as it seems, were not noticed before), we apply them to the study of the approximation. We prove that for any operator T , there is a sequence {T n } of finite matrices such that Ψ Tn (z) tends to Ψ T (z) uniformly on C. In this proof, quasitriangular operators play a special role. This is merely an existence result, we do not give a concrete construction of this sequence of matrices. One of our main points is to show how to use infinite-dimensional operator models in order to produce examples and counterexamples in the set of finite matrices of large size. In particular, we get a result, which means, in a sense, that the pseudospectrum of a nilpotent matrix can be anything one can imagine. We also study the norms of the multipliers in the context of Cowen-Douglas class operators. We use these results to show that, to the opposite to the function Ψ S , the function √ S − z for certain finite matrices S may oscillate arbitrarily fast even far away from the spectrum. 2010 Mathematics Subject Classification. 47B35 (primary).

Journal of Computational and Applied Mathematics

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1999

2016

We present a new iterative algorithm to compute the pseudospectral abscissa for a class of nonlinear eigenvalue problems, which includes the delay and the polynomial eigenvalue problem. Two methods are developed that exploit real valued matrix perturbations, for the Frobenius and the spectral norm. In both cases, it is proved that the pseudospectral abscissa can be obtained when restricting to at most rank-2 perturbations and that the critical perturbations on the coefficient share their column and row spaces. This is a surprising property, giving that the combined additive perturbation on the characteristic matrix is highly structured. The proposed iterative algorithms can be interpreted as the discretization of a gradient ow, where properties of critical perturbations are fully exploited. Even though the main contributions concern the nonlinear eigenvalue problem, the paper contributes to the special case of the standard eigenvalue problem in the following way. For the Frobenius norm perturbations we provide differential equations generating a path in the space of perturbations of rank smaller or equal to two (instead of rank exactly two in the present literature), allowing us to treat the case of complex and real rightmost eigenvalues simultaneously. For the spectral norm case we derive explicit optimality conditions in terms of the compact singular value decomposition. Finally we provide two extensions for the Frobenius norm pseudospectrum. The first one consists of using a combined measure on the perturbations, allowing us to assess from the critical perturbations which coefficient matrices the pseudospectral abscissa is most sensitive to. The second extension consists of considering structured perturbations on the coefficient matrices. In this way our algorithm exploits structure at three levels: the structure of the nonlinear eigenvalue problem (instead of studying unstructured pseudospectra of some linearization of the eigenvalue problem), the property that perturbations are assumed to be real valued, and additional structure on the perturbations of individual matrices. For this case we show that the pseudospectral abscissa cannot always be reached when restricting to perturbations of maximum size.