{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2023,3,31]],"date-time":"2023-03-31T15:53:47Z","timestamp":1680278027413},"reference-count":27,"publisher":"Walter de Gruyter GmbH","issue":"2","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"published-print":{"date-parts":[[2020,4,1]]},"abstract":"<jats:title>Abstract<\/jats:title>\n               <jats:p>This paper focuses on the inner iteration that arises in inexact inverse subspace iteration for computing a small deflating subspace of a large matrix pencil.\nFirst, it is shown that the method achieves linear rate of convergence if the inner iteration is performed with increasing accuracy.\nThen, as inner iteration, block-GMRES is used with preconditioners generalizing the one\nby Robb\u00e9, Sadkane and Spence\n[Inexact inverse subspace iteration with preconditioning applied to non-Hermitian eigenvalue problems,\nSIAM J. Matrix Anal. Appl. 31 2009, 1, 92\u2013113].\nIt is shown that the preconditioners help to maintain the number of iterations needed by block-GMRES to approximately a small constant.\nThe efficiency of the preconditioners is illustrated by numerical examples.<\/jats:p>","DOI":"10.1515\/cmam-2018-0212","type":"journal-article","created":{"date-parts":[[2019,6,19]],"date-time":"2019-06-19T09:02:40Z","timestamp":1560934960000},"page":"343-359","source":"Crossref","is-referenced-by-count":1,"title":["Convergence and Preconditioning of Inexact Inverse Subspace Iteration for Generalized Eigenvalue Problems"],"prefix":"10.1515","volume":"20","author":[{"given":"Rayan","family":"Nasser","sequence":"first","affiliation":[{"name":"Universit\u00e9 de Brest , CNRS \u2013 UMR 6205, Laboratoire de Math\u00e9matiques de Bretagne Atlantique. 6, Av. Le Gorgeu. 29238 Brest Cedex 3 , France"}]},{"given":"Miloud","family":"Sadkane","sequence":"additional","affiliation":[{"name":"Universit\u00e9 de Brest , CNRS \u2013 UMR 6205, Laboratoire de Math\u00e9matiques de Bretagne Atlantique. 6, Av. Le Gorgeu. 29238 Brest Cedex 3 , France"}]}],"member":"374","published-online":{"date-parts":[[2019,6,19]]},"reference":[{"key":"2023033110443914106_j_cmam-2018-0212_ref_001_w2aab3b7e2084b1b6b1ab2b1b1Aa","doi-asserted-by":"crossref","unstructured":"Z.  Bai, J.  Demmel, J.  Dongarra, A.  Ruhe and H.  van der Vorst,\nTemplates for the Solution of Algebraic Eigenvalue Problems,\nSoftware Environ. Tools 11,\nSociety for Industrial and Applied Mathematics (SIAM), Philadelphia, 2000.","DOI":"10.1137\/1.9780898719581"},{"key":"2023033110443914106_j_cmam-2018-0212_ref_002_w2aab3b7e2084b1b6b1ab2b1b2Aa","doi-asserted-by":"crossref","unstructured":"J.  Berns-M\u00fcller, I. G.  Graham and A.  Spence,\nInexact inverse iteration for symmetric matrices,\nLinear Algebra Appl. 416 (2006), no. 2\u20133, 389\u2013413.","DOI":"10.1016\/j.laa.2005.11.019"},{"key":"2023033110443914106_j_cmam-2018-0212_ref_003_w2aab3b7e2084b1b6b1ab2b1b3Aa","doi-asserted-by":"crossref","unstructured":"J.  Berns-M\u00fcller and A.  Spence,\nInexact inverse iteration with variable shift for nonsymmetric generalized eigenvalue problems,\nSIAM J. Matrix Anal. Appl. 28 (2006), no. 4, 1069\u20131082.","DOI":"10.1137\/050623255"},{"key":"2023033110443914106_j_cmam-2018-0212_ref_004_w2aab3b7e2084b1b6b1ab2b1b4Aa","doi-asserted-by":"crossref","unstructured":"J.  Brandts,\nThe Riccati algorithm for eigenvalues and invariant subspaces of matrices with inexpensive action,\nLinear Algebra Appl. 358 (2003), 335\u2013365.","DOI":"10.1016\/S0024-3795(02)00392-0"},{"key":"2023033110443914106_j_cmam-2018-0212_ref_005_w2aab3b7e2084b1b6b1ab2b1b5Aa","doi-asserted-by":"crossref","unstructured":"M. A.  Freitag and P.  K\u00fcrschner,\nTuned preconditioners for inexact two-sided inverse and Rayleigh quotient iteration,\nNumer. Linear Algebra Appl. 22 (2015), no. 1, 175\u2013196.","DOI":"10.1002\/nla.1945"},{"key":"2023033110443914106_j_cmam-2018-0212_ref_006_w2aab3b7e2084b1b6b1ab2b1b6Aa","doi-asserted-by":"crossref","unstructured":"M. A.  Freitag, P.  K\u00fcrschner and J.  Pestana,\nGMRES convergence bounds for eigenvalue problems,\nComput. Methods Appl. Math. 18 (2018), no. 2, 203\u2013222.","DOI":"10.1515\/cmam-2017-0017"},{"key":"2023033110443914106_j_cmam-2018-0212_ref_007_w2aab3b7e2084b1b6b1ab2b1b7Aa","doi-asserted-by":"crossref","unstructured":"M. A.  Freitag and A.  Spence,\nConvergence of inexact inverse iteration with application to preconditioned iterative solves,\nBIT 47 (2007), no. 1, 27\u201344.","DOI":"10.1007\/s10543-006-0100-1"},{"key":"2023033110443914106_j_cmam-2018-0212_ref_008_w2aab3b7e2084b1b6b1ab2b1b8Aa","unstructured":"M. A.  Freitag and A.  Spence,\nConvergence theory for inexact inverse iteration applied to the generalised nonsymmetric eigenproblem,\nElectron. Trans. Numer. Anal. 28 (2007\/08), 40\u201364."},{"key":"2023033110443914106_j_cmam-2018-0212_ref_009_w2aab3b7e2084b1b6b1ab2b1b9Aa","doi-asserted-by":"crossref","unstructured":"M. A.  Freitag and A.  Spence,\nA tuned preconditioner for inexact inverse iteration applied to Hermitian eigenvalue problems,\nIMA J. Numer. Anal. 28 (2008), no. 3, 522\u2013551.","DOI":"10.1093\/imanum\/drm036"},{"key":"2023033110443914106_j_cmam-2018-0212_ref_010_w2aab3b7e2084b1b6b1ab2b1c10Aa","doi-asserted-by":"crossref","unstructured":"M. A.  Freitag and A.  Spence,\nRayleigh quotient iteration and simplified Jacobi\u2013Davidson method with preconditioned iterative solves,\nLinear Algebra Appl. 428 (2008), no. 8\u20139, 2049\u20132060.","DOI":"10.1016\/j.laa.2007.11.013"},{"key":"2023033110443914106_j_cmam-2018-0212_ref_011_w2aab3b7e2084b1b6b1ab2b1c11Aa","unstructured":"G. H.  Golub and C. F.  Van Loan,\nMatrix Computations, 3rd ed.,\nJohns Hopkins University, Baltimore, 1996."},{"key":"2023033110443914106_j_cmam-2018-0212_ref_012_w2aab3b7e2084b1b6b1ab2b1c12Aa","doi-asserted-by":"crossref","unstructured":"G. H.  Golub and Q.  Ye,\nInexact inverse iteration for generalized eigenvalue problems,\nBIT 40 (2000), no. 4, 671\u2013684.","DOI":"10.1023\/A:1022388317839"},{"key":"2023033110443914106_j_cmam-2018-0212_ref_013_w2aab3b7e2084b1b6b1ab2b1c13Aa","doi-asserted-by":"crossref","unstructured":"M.  Hochbruck and C.  Lubich,\nOn Krylov subspace approximations to the matrix exponential operator,\nSIAM J. Numer. Anal. 34 (1997), no. 5, 1911\u20131925.","DOI":"10.1137\/S0036142995280572"},{"key":"2023033110443914106_j_cmam-2018-0212_ref_014_w2aab3b7e2084b1b6b1ab2b1c14Aa","doi-asserted-by":"crossref","unstructured":"R. A.  Horn and C. R.  Johnson,\nMatrix Analysis,\nCambridge University, Cambridge, 1985.","DOI":"10.1017\/CBO9780511810817"},{"key":"2023033110443914106_j_cmam-2018-0212_ref_015_w2aab3b7e2084b1b6b1ab2b1c15Aa","doi-asserted-by":"crossref","unstructured":"Y.-L.  Lai, K.-Y.  Lin and W.-W.  Lin,\nAn inexact inverse iteration for large sparse eigenvalue problems,\nNumer. Linear Algebra Appl. 4 (1997), no. 5, 425\u2013437.","DOI":"10.1002\/(SICI)1099-1506(199709\/10)4:5<425::AID-NLA117>3.0.CO;2-G"},{"key":"2023033110443914106_j_cmam-2018-0212_ref_016_w2aab3b7e2084b1b6b1ab2b1c16Aa","doi-asserted-by":"crossref","unstructured":"R. B.  Lehoucq and K.  Meerbergen,\nUsing generalized Cayley transformations within an inexact rational Krylov sequence method,\nSIAM J. Matrix Anal. Appl. 20 (1999), no. 1, 131\u2013148.","DOI":"10.1137\/S0895479896311220"},{"key":"2023033110443914106_j_cmam-2018-0212_ref_017_w2aab3b7e2084b1b6b1ab2b1c17Aa","doi-asserted-by":"crossref","unstructured":"Y.  Notay,\nConvergence analysis of inexact Rayleigh quotient iteration,\nSIAM J. Matrix Anal. Appl. 24 (2003), no. 3, 627\u2013644.","DOI":"10.1137\/S0895479801399596"},{"key":"2023033110443914106_j_cmam-2018-0212_ref_018_w2aab3b7e2084b1b6b1ab2b1c18Aa","doi-asserted-by":"crossref","unstructured":"M.  Robb\u00e9 and M.  Sadkane,\nRiccati-based preconditioner for computing invariant subspaces of large matrices,\nNumer. Math. 92 (2002), no. 1, 129\u2013159.","DOI":"10.1007\/s002110100355"},{"key":"2023033110443914106_j_cmam-2018-0212_ref_019_w2aab3b7e2084b1b6b1ab2b1c19Aa","doi-asserted-by":"crossref","unstructured":"M.  Robb\u00e9 and M.  Sadkane,\nExact and inexact breakdowns in the block GMRES method,\nLinear Algebra Appl. 419 (2006), no. 1, 265\u2013285.","DOI":"10.1016\/j.laa.2006.04.018"},{"key":"2023033110443914106_j_cmam-2018-0212_ref_020_w2aab3b7e2084b1b6b1ab2b1c20Aa","doi-asserted-by":"crossref","unstructured":"M.  Robb\u00e9, M.  Sadkane and A.  Spence,\nInexact inverse subspace iteration with preconditioning applied to non-Hermitian eigenvalue problems,\nSIAM J. Matrix Anal. Appl. 31 (2009), no. 1, 92\u2013113.","DOI":"10.1137\/060673795"},{"key":"2023033110443914106_j_cmam-2018-0212_ref_021_w2aab3b7e2084b1b6b1ab2b1c21Aa","doi-asserted-by":"crossref","unstructured":"Y.  Saad,\nIterative Methods for Sparse Linear systems, 2nd ed.,\nSociety for Industrial and Applied Mathematics, Philadelphia, 2003.","DOI":"10.1137\/1.9780898718003"},{"key":"2023033110443914106_j_cmam-2018-0212_ref_022_w2aab3b7e2084b1b6b1ab2b1c22Aa","doi-asserted-by":"crossref","unstructured":"V.  Simoncini and L.  Eld\u00e9n,\nInexact Rayleigh quotient-type methods for eigenvalue computations,\nBIT 42 (2002), no. 1, 159\u2013182.","DOI":"10.1023\/A:1021930421106"},{"key":"2023033110443914106_j_cmam-2018-0212_ref_023_w2aab3b7e2084b1b6b1ab2b1c23Aa","doi-asserted-by":"crossref","unstructured":"G. L. G.  Sleijpen and H. A.  Van der Vorst,\nA Jacobi\u2013Davidson iteration method for linear eigenvalue problems,\nSIAM J. Matrix Anal. Appl. 17 (1996), no. 2, 401\u2013425.","DOI":"10.1137\/S0895479894270427"},{"key":"2023033110443914106_j_cmam-2018-0212_ref_024_w2aab3b7e2084b1b6b1ab2b1c24Aa","unstructured":"G. W.  Stewart and J. G.  Sun,\nMatrix Perturbation Theory,\nAcademic Press, Boston, 1990."},{"key":"2023033110443914106_j_cmam-2018-0212_ref_025_w2aab3b7e2084b1b6b1ab2b1c25Aa","doi-asserted-by":"crossref","unstructured":"F.  Xue and H. C.  Elman,\nConvergence analysis of iterative solvers in inexact Rayleigh quotient iteration,\nSIAM J. Matrix Anal. Appl. 31 (2009), no. 3, 877\u2013899.","DOI":"10.1137\/080712908"},{"key":"2023033110443914106_j_cmam-2018-0212_ref_026_w2aab3b7e2084b1b6b1ab2b1c26Aa","doi-asserted-by":"crossref","unstructured":"F.  Xue and H. C.  Elman,\nFast inexact subspace iteration for generalized eigenvalue problems with spectral transformation,\nLinear Algebra Appl. 435 (2011), no. 3, 601\u2013622.","DOI":"10.1016\/j.laa.2010.06.021"},{"key":"2023033110443914106_j_cmam-2018-0212_ref_027_w2aab3b7e2084b1b6b1ab2b1c27Aa","doi-asserted-by":"crossref","unstructured":"Q.  Ye and P.  Zhang,\nInexact inverse subspace iteration for generalized eigenvalue problems,\nLinear Algebra Appl. 434 (2011), no. 7, 1697\u20131715.","DOI":"10.1016\/j.laa.2010.08.001"}],"container-title":["Computational Methods in Applied Mathematics"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.degruyter.com\/view\/journals\/cmam\/20\/2\/article-p343.xml","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.degruyter.com\/document\/doi\/10.1515\/cmam-2018-0212\/xml","content-type":"application\/xml","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.degruyter.com\/document\/doi\/10.1515\/cmam-2018-0212\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2023,3,31]],"date-time":"2023-03-31T12:50:59Z","timestamp":1680267059000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.degruyter.com\/document\/doi\/10.1515\/cmam-2018-0212\/html"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2019,6,19]]},"references-count":27,"journal-issue":{"issue":"2","published-online":{"date-parts":[[2019,8,14]]},"published-print":{"date-parts":[[2020,4,1]]}},"alternative-id":["10.1515\/cmam-2018-0212"],"URL":"https:\/\/doi.org\/10.1515\/cmam-2018-0212","relation":{},"ISSN":["1609-9389","1609-4840"],"issn-type":[{"value":"1609-9389","type":"electronic"},{"value":"1609-4840","type":"print"}],"subject":[],"published":{"date-parts":[[2019,6,19]]}}}