Hochstenbach, Michiel E.; Sleijpen, Gerard L. G. Two-sided and alternating Jacobi-Davidson. (English) Zbl 1087.65035 Linear Algebra Appl. 358, No. 1-3, 145-172 (2003). Summary: We discuss two variants of a two-sided Jacobi-Davidson (JD) method, which have asymptotically cubic convergence for nonnormal matrices, and aim to find both right and left eigenvectors. These methods can be seen as Jacobi-Davidson analogs of A. M. Qstrowki’s two-sided Rayleigh quotient iteration (RQI) [Arch. Ration. Mech. Anal. 3 325–340, 341–347 (1959; Zbl 0089.11902); ibid. 3, 472–481 (1959); 4, 153–165 (1960; Zbl 0090.33901)]. Some relations between (exact and inexact) two-sided Jacobi-Davidson and (exact and inexact) two-sided Rayleigh quotient iteration are given, together with convergence rates. Furthermore, we introduce an alternating Jacobi-Davidson process that can be seen as the Jacobi-Davidson analog of B. N. Parlett’s alternating Rayleigh quotient iteration [Math. Comput. 28, 679–693 (1974; Zbl 0293.65023)]. The methods are extended to the generalized and polynomial eigenproblem. Advantages of the methods are illustrated by numerical examples. Cited in 15 Documents MSC: 65F15 Numerical computation of eigenvalues and eigenvectors of matrices Keywords:Jacobi-Davidson method; Ostrowski’s two-sided Rayleigh quotient iteration; Parlett’s alternating Rayleigh quotient iteration; Two-sided Lanczos method; correction equation; nonnormal matrix; inexact accelerated Newton method; generalized eigenproblem; polynomial eigenproblem; convergence; Rayleigh quotient iteration; numerical examples Citations:Zbl 0089.11902; Zbl 0090.33901; Zbl 0293.65023 PDFBibTeX XMLCite \textit{M. E. Hochstenbach} and \textit{G. L. G. Sleijpen}, Linear Algebra Appl. 358, No. 1--3, 145--172 (2003; Zbl 1087.65035) Full Text: DOI References: [2] Lancaster, P., A generalized Rayleigh quotient iteration for lambda-matrices, Arch. Ration. Mech. Anal., 8, 309-322 (1961) · Zbl 0105.31705 [3] Ostrowski, A. M., On the convergence of the Rayleigh quotient iteration for the computation of the characteristic roots and vectors. III (Generalized Rayleigh quotient characteristic roots with linear elementary divisors), Arch. Ration. Mech. Anal., 3, 325-340 (1959) · Zbl 0089.11902 [5] Parlett, B. N., The Rayleigh quotient iteration and some generalizations for nonnormal matrices, Math. Comput., 28, 679-693 (1974) · Zbl 0293.65023 [6] Parlett, B. N., The Symmetric Eigenvalue Problem (1998), Society for Industrial and Applied Mathematics (SIAM): Society for Industrial and Applied Mathematics (SIAM) Philadelphia, PA, corrected reprint of the 1980 original · Zbl 0885.65039 [8] Sleijpen, G. L.G.; van der Vorst, H. A., A Jacobi-Davidson iteration method for linear eigenvalue problems, SIAM J. Matrix Anal. Appl., 17, 401-425 (1996) · Zbl 0860.65023 [10] Smit, P.; Paardekooper, M. H.C., The effects of inexact solvers in algorithms for symmetric eigenvalue problems, Linear Algebra Appl., 287, 337-357 (1999), special issue celebrating the 60th birthday of Ludwig Elsner · Zbl 0943.65048 [12] Stewart, G. W.; Sun, J.-G., Matrix Perturbation Theory (1990), Academic Press: Academic Press San Diego, CA [14] van der Sluis, A.; van der Vorst, H. A., The convergence behavior of Ritz values in the presence of close eigenvalues, Linear Algebra Appl., 88-89, 651-694 (1987) · Zbl 0632.65035 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.