The Role Eigenvalues Play in Forming GMRES Residual Norms with Non-Normal Matrices

Authors: G. Meurant, Jurjen Duintjer Tebbens
Publication year: 2015
Publication type: Článek v odborném periodiku
Periodical: Numerical Algorithms
Projects: Iterative Methods in Computational Mathematics: Analysis, Preconditioning, and Applications
Bibliographic citation: MEURANT, G., DUINTJER TEBBENS, Jurjen. The Role Eigenvalues Play in Forming GMRES Residual Norms with Non-Normal Matrices. Numerical Algorithms. 2015, 68(1), 143-165. ISSN 1017-1398. E-ISSN 1572-9265
Dostupné z: doi:10.1007/s11075-014-9891-3
DOI: 10.1007/s11075-014-9891-3
ID: 430466

Abstract

In this paper we give explicit expressions for the norms of the residual vectors generated by the GMRES algorithm applied to a non-normal matrix. They involve the right-hand side of the linear system, the eigenvalues, the eigenvectors and, in the non-diagonalizable case, the principal vectors. They give a complete description of how eigenvalues contribute in forming residual norms and offer insight in what quantities can prevent GMRES from being governed by eigenvalues.