Page 69 - Textos de Matemática Vol. 44

NUMERICAL APPROXIMATION OF THE FIELD OF VALUES OF THE INVERSE OF A LARGE MATRIX
MICHIEL E. HOCHSTENBACH, DAVID A. SINGER AND PAUL F. ZACHLIN
Abstract. We consider the approximation of the field of values of the inverse of a large sparse matrix, without explicitly computing the inverse or using its action (i.e., accurately solving a linear system with this matrix). We review results by Manteuffel and Starke and give an alternative that may yield better approximations in practice. We give connections with the harmonic Rayleigh– Ritz approach. Several properties and applications of the studied concepts as well as numerical examples are provided.
1. Introduction
Let A be a large sparse nonsingular n × n matrix. The field of values
W ( A ) = { x ∗A x : ∥ x ∥ = 1 } ,
where ∥ · ∥ denotes the 2-norm, is a convex set in the complex plane that is of interest for many problems and applications. For instance, it is useful to obtain (approximate) eigenvalue inclusion regions for large matrices (see, e.g., [6, 4]).
In this paper we are interested in the numerical approximation of W(A−1), the field of values of the inverse of A, without computing or approximating A−1. In Section 2 we first focus on the numerical approximation of W(A), both for small and large matrices. Section 3 reviews some applications where knowledge of W (A−1) may be useful. We give some relations between W (A) and W (A−1) in Section 4.
In Section 5 we propose two approximation methods for W(A−1). We review an approach by Manteuffel and Starke and give relations with the harmonic Rayleigh–Ritz method. We also present an alternative approach that comes with no inclusion guarantee, but may work better in practice. We provide some analysis and numerical examples. We end with some conclusions in Section 6.
2010 Mathematics Subject Classification. 65F15, 65F50, 65F35, 47A12, 65F30, 65F10.
Key words and phrases. Field of values, numerical range, matrix inverse, large sparse matrix, Ritz values, harmonic Rayleigh–Ritz, harmonic Ritz values, GMRES convergence, Arnoldi, numerical radius, numerical abscissa, inner numerical radius, inclusion region.
The first author was supported by a Vidi research grant from the Netherlands Organization for Scientific Research (NWO)
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