Jia, ZhongXiao; Wang, Zhen A convergence analysis of the inexact Rayleigh quotient iteration and simplified Jacobi-Davidson method for the large Hermitian matrix eigenproblem. (English) Zbl 1179.65039 Sci. China, Ser. A 51, No. 12, 2205-2216 (2008). Summary: The inexact Rayleigh quotient iteration (RQI) is used for computing the smallest eigenpair of a large Hermitian matrix. Under certain condition, the method was proved to converge quadratically in literature. However, it is shown in this paper that under the original given condition the inexact RQI may not quadratically converge to the desired eigenpair and even may misconverge to some other undesired eigenpair. A new condition, called the uniform positiveness condition, is given that can fix the misconvergence problem and ensure the quadratic convergence of the inexact RQI. An alternative to the inexact RQI is the Jacobi-Davidson method without subspace acceleration. A new proof of its linear convergence is presented and a sharper bound is established in the paper. All the results are verified and analyzed by numerical experiments. Cited in 3 Documents MSC: 65F15 Numerical computation of eigenvalues and eigenvectors of matrices Keywords:eigenvalue; eigenvector; inexact Rayleigh quotient iteration; smallest eigenpair; large Hermitian matrix; uniform positiveness condition; convergence; misconvergence; Jacobi-Davidson method Software:JDCG; Harwell-Boeing sparse matrix collection; JDQZ PDFBibTeX XMLCite \textit{Z. Jia} and \textit{Z. Wang}, Sci. China, Ser. A 51, No. 12, 2205--2216 (2008; Zbl 1179.65039) Full Text: DOI References: [1] Golub G H, Loan C V. Matrix Computations. Baltimore-London: The John Hopkins University Press, 1996 [2] Parlett B N. The Symmetric Eigenvalue Problem. Philadelphia: SIAM, 1998 · Zbl 0885.65039 [3] Berns-Müller J, Spence A. Inexact inverse iteration with variable shift for nonsymmetric generalized eigenvalue problems. SIAM J Matrix Anal Appl, 28: 1069–1082 (2006) · Zbl 1134.65025 [4] Hochstenbach M, Sleijpen G L G. Two-sided and alternating Jacobi-Davidson. Linear Algebra Appl, 358: 145–172 (2003) · Zbl 1087.65035 [5] Notay Y. Convergence analysis of inexact Rayleigh quotient iteration. SIAM J Matrix Anal Appl, 24: 627–644 (2003) · Zbl 1045.65032 [6] Simoncini V, Elden L. Inexact Rayleigh quotient-type methods for eigenvalue computations. BIT, 42: 159–182 (2002) · Zbl 1003.65033 [7] Smit P, Paardekooper M. The effects of inexact solvers in algorithms for symmetric eigenvalue problems. Linear Algebra Appl, 287: 337–357 (1999) · Zbl 0943.65048 [8] Eshof J. The convergence of Jacobi-Davidson iterations for Hermitian eigenproblems. Numer Linear Algebra Appl, 9: 163–179 (2002) · Zbl 1071.65518 [9] Stewart G W. Matrix Algorithms Vol. II. Philadelphia: SIAM, 2001 [10] Davidson E. The iterative calculation of a few of the lowest eigenvalues and corresponding eigenvectors of large real-symmetric matrices. J Comput Phys, 17: 87–94 (1975) · Zbl 0293.65022 [11] Sleijpen G L G, Vorst H A. A Jacobi-Davidson iteration method for linear eigenvalue problems. SIAM J Matrix Anal Appl, 17: 401–425 (1996) · Zbl 0860.65023 [12] Saad Y. Iterative Methods for Sparse Linear Systems. Philadelphia: SIAM, 2003 · Zbl 1031.65046 [13] Notay Y. Combination of Jacobi-Davidson and conjugate gradients for the partial symmetric eigenproblem. Numer Linear Algebra Appl, 9: 21–44 (2002) · Zbl 1071.65516 [14] Notay Y. Is Jacobi-Davidson faster than Davidson? SIAM J Matrix Anal Appl, 36: 522–543 (2005) [15] Duff I S, Grimes R, Lewis J. Users’ Guide for the Harwell-Boeing Sparse Matrix Collection (Release I). Technical Report TR/PA/92/86, CERFACS, 1992 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.