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**Subspace methods for computing the pseudospectral abscissa and the stability radius.**
*(English)*
Zbl 1306.65187

The goal of the paper is to combine the linearly converging iterative method for computing the pseudospectral abscissa proposed by N. Guglielmi and M. L. Overton [SIAM J. Matrix Anal. Appl. 32, No. 4, 1166–1192 (2011; Zbl 1248.65034)], and its variants with subspace acceleration. The basic idea is to collect the essential information from all the previous iterates in a subspace and obtain a more accurate iterate by extracting a quasi-best approximation from this subspace.

The first section is an introduction into the nature of the subject.

The second section briefly summarizes the basic algorithm by Guglielmi and Overton together with some results.

In the third section, the authors develop the fundamental ideas behind the proposed subspace method and prove monotonicity as well as stability of the extraction procedure. Each extraction step computes the pseudospectral abscissa of a small rectangular matrix pencil.

In the fourth section, two different variants of the subspace are proposed and analysed from the point of view of convergence. The authors observe local quadratic convergence and prove local superlinear convergence of the resulting subspace methods.

The implementation details of the proposed subspace algorithms and some numerical experiments for a suite of dense and sparse test problems are discussed in the fifth section, in order to demonstrate the robustness and efficiency of the above algorithms.

The sixth section is concerned with extending the proposed methods to the computation of the stability radius.

The main conclusions are exposed in the last section.

The first section is an introduction into the nature of the subject.

The second section briefly summarizes the basic algorithm by Guglielmi and Overton together with some results.

In the third section, the authors develop the fundamental ideas behind the proposed subspace method and prove monotonicity as well as stability of the extraction procedure. Each extraction step computes the pseudospectral abscissa of a small rectangular matrix pencil.

In the fourth section, two different variants of the subspace are proposed and analysed from the point of view of convergence. The authors observe local quadratic convergence and prove local superlinear convergence of the resulting subspace methods.

The implementation details of the proposed subspace algorithms and some numerical experiments for a suite of dense and sparse test problems are discussed in the fifth section, in order to demonstrate the robustness and efficiency of the above algorithms.

The sixth section is concerned with extending the proposed methods to the computation of the stability radius.

The main conclusions are exposed in the last section.

Reviewer: Romulus Militaru (Craiova)

### MSC:

65F15 | Numerical computation of eigenvalues and eigenvectors of matrices |