Hochstenbach, Michiel E.; Sleijpen, Gerard L. G. Harmonic and refined Rayleigh-Ritz for the polynomial eigenvalue problem. (English) Zbl 1212.65150 Numer. Linear Algebra Appl. 15, No. 1, 35-54 (2008). Summary: After reviewing the harmonic Rayleigh-Ritz approach for the standard and generalized eigenvalue problem, we discuss several extraction processes for subspace methods for the polynomial eigenvalue problem. We generalize the harmonic and refined Rayleigh-Ritz approaches which lead to new approaches to extract promising approximate eigenpairs from a search space. We give theoretical as well as numerical results of the methods. In addition, we study the convergence of the Jacobi-Davidson method for polynomial eigenvalue problems with exact and inexact linear solves and discuss several algorithmic details. Cited in 15 Documents MSC: 65F15 Numerical computation of eigenvalues and eigenvectors of matrices 15A54 Matrices over function rings in one or more variables Keywords:polynomial eigenvalue problem; harmonic Rayleigh-Ritz; refined Rayleigh-Ritz; interior eigenvalues; Jacobi-Davidson; subspace extraction; subspace method; subspace expansion; Rayleigh-Ritz; convergence; numerical results PDFBibTeX XMLCite \textit{M. E. Hochstenbach} and \textit{G. L. G. Sleijpen}, Numer. Linear Algebra Appl. 15, No. 1, 35--54 (2008; Zbl 1212.65150) Full Text: DOI References: [1] Computational methods for large eigenvalue problems. Handbook of Numerical Analysis, vol. VIII. North-Holland: Amsterdam, 2002; 3–179. [2] Matrix Algorithms, vol. II. SIAM: Philadelphia, PA, 2001. · Zbl 0984.65031 [3] Sleijpen, Electronic Transactions on Numerical Analysis 7 pp 75– (1998) [4] Nested iteration methods for nonlinear matrix problems. Ph.D. Thesis, Utrecht University, 2003. [5] Paige, Numerical Linear Algebra with Applications 2 pp 115– (1995) [6] Numerical Methods for Large Eigenvalue Problems. Manchester University Press: Manchester, U.K., 1992. [7] The Symmetric Eigenvalue Problem. SIAM: Philadelphia, PA, 1998. [8] Jia, Mathematics of Computation 74 pp 1441– (2005) [9] Hochstenbach, Linear Algebra and its Applications 358 pp 145– (2003) [10] Lanczos, Journal of Research of the National Bureau of Standards 45 pp 255– (1950) [11] Sleijpen, BIT 36 pp 595– (1996) [12] Stathopoulos, SIAM Journal on Matrix Analysis and Applications 24 pp 238– (2002) [13] Hochstenbach, SIAM Journal on Scientific Computing 25 pp 591– (2003) [14] Tisseur, Linear Algebra and its Applications 309 pp 339– (2000) · Zbl 0969.37004 [15] Tisseur, SIAM Journal on Matrix Analysis and Applications 23 pp 187– (2001) [16] Jia, Linear Algebra and its Applications 259 pp 1– (1997) [17] Spellucci, BIT 40 pp 183– (2000) [18] . Homogeneous Jacobi–Davidson. CASA Report 06-36, Department of Mathematics, TU Eindhoven, The Netherlands, November 2006. Electronic Transactions on Numerical Analysis, in press. [19] Hwang, SIAM Journal on Scientific Computing 24 pp 1283– (2003) 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.