van den Eshof, Jasper; Hochbruck, Marlis Preconditioning Lanczos approximations to the matrix exponential. (English) Zbl 1105.65051 SIAM J. Sci. Comput. 27, No. 4, 1438-1457 (2006). The concept of preconditioning for linear systems is generalized to compute the matrix exponential \(\exp(-\tau A),\) for a symmetric and positive semidefinite \(A.\) The exponential function is a quickly decaying function. Hence the vector \(\exp(-\tau A)v\) is mostly determined by the smallest eigenvalues and their corresponding invariant subspaces. In this paper, the Lanczos method is applied to the inverse of \((I+\gamma A)\) with \(\gamma > 0,\) to emphasize the first eigenvalues and their eigenvectors using an inner-outer iteration. The convergence speed is independent of the norm of the matrix. In addition, a strategy for choosing the tolerances for the errors in the solution of the shifted system is proposed. The numerical results indicate the effectiveness of the method. Reviewer: Keehwan Kim (Kyongsan) Cited in 1 ReviewCited in 80 Documents MSC: 65F35 Numerical computation of matrix norms, conditioning, scaling 65F30 Other matrix algorithms (MSC2010) 65F10 Iterative numerical methods for linear systems Keywords:matrix exponential; iterative methods; restricted rational approximation; matrix-free time integration methods; preconditioning; Lanczos method; inverse; eigenvalues; eigenvectors; inner-outer iteration; convergence; numerical results PDFBibTeX XMLCite \textit{J. van den Eshof} and \textit{M. Hochbruck}, SIAM J. Sci. Comput. 27, No. 4, 1438--1457 (2006; Zbl 1105.65051) Full Text: DOI Link