Hochstenbach, Michiel E.; Singer, David A.; Zachlin, Paul F. Numerical approximation of the field of values of the inverse of a large matrix. (English) Zbl 1299.65063 Fonseca, Carlos (ed.) et al., The Natália Bebiano anniversary volume. Special issue on the occasion of her 60th birthday. Coimbra: Universidade de Coimbra, Departamento de Matemática (ISBN 978-972-8564-48-3/pbk). Textos de Matemática 44, 59-71 (2013). The authors are interested in the numerical approximation of the field of values \(W(A^{-1})\), for a given large, sparse, nonsingular matrix \(A\), where \(W(A) = \{ x^* A x, \| x \| = 1 \}\). Starting from an approach considered by T. A. Manteuffel and G. Starke [Numer. Math. 73, No. 4, 489–506 (1996; Zbl 0864.65014)], they provide an alternative based on Krylov subspace method. Some numerical experiments are also provided.For the entire collection see [Zbl 1278.00015]. Reviewer: Constantin Popa (Constanţa) Cited in 4 Documents MSC: 65F15 Numerical computation of eigenvalues and eigenvectors of matrices 65F50 Computational methods for sparse matrices 65F35 Numerical computation of matrix norms, conditioning, scaling 47A12 Numerical range, numerical radius 65F30 Other matrix algorithms (MSC2010) 65F10 Iterative numerical methods for linear systems Keywords:field of values; numerical range; matrix inverse; large sparse matrix; Ritz values; harmonic Rayleigh-Ritz; harmonic Ritz values; Arnoldi; numerical radius; numerical abscissa; inner numerical radius; inclusion region Citations:Zbl 0864.65014 PDFBibTeX XMLCite \textit{M. E. Hochstenbach} et al., Textos Mat. 44, 59--71 (2013; Zbl 1299.65063)