Hochstenbach, Michiel E. Fields of values and inclusion regions for matrix pencils. (English) Zbl 1287.65025 ETNA, Electron. Trans. Numer. Anal. 38, 98-112 (2011). Summary: We are interested in (approximate) eigenvalue inclusion regions for matrix pencils \((A, B)\), in particular of large dimension, based on certain fields of values. We show how the usual field of values may be efficiently approximated for large Hermitian positive definite \(B\), but also point out limitations of this set. We introduce four field of values based inclusion regions, which may effectively be approximated, also for large pencils. Furthermore, we show that these four sets are special members of two families of inclusion regions, of which we study several properties. Connections with the usual harmonic Rayleigh-Ritz method and a new variant are shown, and we propose an automated algorithm which gives an approximated inclusion region. The results are illustrated by several numerical examples. Cited in 7 Documents MSC: 65F15 Numerical computation of eigenvalues and eigenvectors of matrices 15A42 Inequalities involving eigenvalues and eigenvectors 15A22 Matrix pencils 65F50 Computational methods for sparse matrices 15A60 Norms of matrices, numerical range, applications of functional analysis to matrix theory Keywords:inclusion region; exclusion region; matrix pencil; numerical range; field of values; generalized eigenvalue problem; large sparse matrix; harmonic Rayleigh-Ritz; harmonic Ritz values; Krylov space; numerical examples PDFBibTeX XMLCite \textit{M. E. Hochstenbach}, ETNA, Electron. Trans. Numer. Anal. 38, 98--112 (2011; Zbl 1287.65025) Full Text: EMIS