{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,10,3]],"date-time":"2026-10-03T17:16:44Z","timestamp":1791047804674,"version":"4.1.0"},"reference-count":49,"publisher":"Society for Industrial & Applied Mathematics (SIAM)","issue":"4","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["SIAM J. Matrix Anal. Appl."],"published-print":{"date-parts":[[2004,1]]},"abstract":"<jats:p>The performance of Krylov subspace eigenvalue algorithms for large matrices can be measured by the angle between a desired invariant subspace and the Krylov subspace. We develop general bounds for this convergence that include the effects of polynomial restarting and impose no restrictions concerning the diagonalizability of the matrix or its degree of nonnormality. Associated with a desired set of eigenvalues is a maximum \"reachable invariant subspace\" that can be developed from the given starting vector. Convergence for this distinguished subspace is bounded in terms involving a polynomial approximation problem. Elementary results from potential theory lead to convergence rate estimates and suggest restarting strategies based on optimal approximation points (e.g., Leja or Chebyshev points); exact shifts are evaluated within this framework. Computational examples illustrate the utility of these results. Origins of superlinear effects are also described.<\/jats:p>","DOI":"10.1137\/s0895479801398608","type":"journal-article","created":{"date-parts":[[2004,7,15]],"date-time":"2004-07-15T21:03:14Z","timestamp":1089925394000},"page":"1074-1109","source":"Crossref","is-referenced-by-count":33,"title":["Convergence of Restarted Krylov Subspaces to Invariant Subspaces"],"prefix":"10.1137","volume":"25","author":[{"given":"Christopher","family":"Beattie","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Mark","family":"Embree","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"John","family":"Rossi","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2006,7,31]]},"reference":[{"key":"R1","doi-asserted-by":"publisher","DOI":"10.1007\/BF02510405"},{"key":"R2","doi-asserted-by":"publisher","DOI":"10.1090\/qam\/42792"},{"key":"R3","first-page":"124","volume":"7","author":"Baglama J.","year":"1998","journal-title":"Electron. Trans. Numer. Anal."},{"key":"R4","doi-asserted-by":"publisher","DOI":"10.1137\/1.9780898719581"},{"key":"R5","doi-asserted-by":"crossref","unstructured":"A. Berman and R. J. Plemmons,\n                      Nonnegative Matrices in the Mathematical Sciences\n                      , Classics Appl. Math. 9, SIAM, Philadelphia, 1994.","DOI":"10.1137\/1.9781611971262"},{"key":"R6","first-page":"1","volume":"2","author":"Calvetti D.","year":"1994","journal-title":"Electron. Trans. Numer. Anal."},{"key":"R7","volume-title":"Eigenvalues of matrices","author":"Chatelin Fran\u00e7oise","year":"1993"},{"key":"R8","doi-asserted-by":"crossref","unstructured":"J. B. Conway,\n                      Functions of One Complex Variable\n                      , 2nd ed., Springer\u2010Verlag, New York, 1978.","DOI":"10.1007\/978-1-4612-6313-5"},{"key":"R9","doi-asserted-by":"publisher","DOI":"10.1023\/A:1019105012193"},{"key":"R10","doi-asserted-by":"publisher","DOI":"10.1137\/S0036144596305582"},{"key":"R11","unstructured":"N. Dunford and J. Schwartz,\n                      Linear Operators, Part I: General Theory\n                      , Wiley, New York, 1971."},{"key":"R12","unstructured":"M. Embree,\n                      Convergence of Krylov Subspace Methods for Nonnormal Matrices\n                      , D.Phil. Thesis, Oxford University, Oxford, UK, 2000."},{"key":"R13","volume-title":"The theory of matrices. Vol. 1","author":"Gantmacher F.","year":"1998"},{"key":"R14","volume-title":"Matrix computations","author":"Golub Gene","year":"1996"},{"key":"R15","unstructured":"A. Greenbaum,\n                      Using the Cauchy Integral Formula and the Partial Fractions Decomposition of the Resolvent to Estimate \u2016f(A)\u2016\n                      , manuscript, University of Washington, Seattle, WA, 2000."},{"key":"R16","doi-asserted-by":"publisher","DOI":"10.1007\/BF02141739"},{"key":"R17","doi-asserted-by":"publisher","DOI":"10.1007\/s002110050040"},{"key":"R18","first-page":"62","volume":"5","author":"Heuveline Vincent","year":"1997","journal-title":"Electron. Trans. Numer. Anal."},{"key":"R19","doi-asserted-by":"crossref","unstructured":"R. A. Horn and C. R. Johnson,\n                      Topics in Matrix Analysis\n                      , Cambridge University Press, Cambridge, UK, 1991.","DOI":"10.1017\/CBO9780511840371"},{"key":"R20","doi-asserted-by":"publisher","DOI":"10.1137\/S0895479893246753"},{"key":"R21","doi-asserted-by":"publisher","DOI":"10.1090\/S0025-5718-00-01208-4"},{"key":"R22","doi-asserted-by":"publisher","DOI":"10.1007\/BF01386205"},{"key":"R23","doi-asserted-by":"publisher","DOI":"10.1006\/jmaa.1999.6558"},{"key":"R24","doi-asserted-by":"publisher","DOI":"10.1016\/S0024-3795(99)00122-6"},{"key":"R25","doi-asserted-by":"publisher","DOI":"10.1137\/S0895479899358595"},{"key":"R26","doi-asserted-by":"publisher","DOI":"10.1007\/BF01263066"},{"key":"R27","doi-asserted-by":"publisher","DOI":"10.1090\/S0025-5718-96-00745-4"},{"key":"R28","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-0348-8547-8"},{"key":"R29","volume-title":"Completely bounded maps and dilations","author":"Paulsen Vern","year":"1986"},{"key":"R30","doi-asserted-by":"publisher","DOI":"10.1016\/0024-3795(92)90374-J"},{"key":"R31","volume-title":"Functional analysis","author":"Riesz Frigyes","year":"1955"},{"key":"R32","doi-asserted-by":"publisher","DOI":"10.1016\/0024-3795(80)90169-X"},{"key":"R33","doi-asserted-by":"crossref","unstructured":"Y. Saad,\n                      Projection methods for solving large sparse eigenvalue problems\n                      , in Matrix Pencils: Proceedings, Lecture Notes in Math. 973, B. K\u00e5gstr\u00f6m and A. Ruhe, eds., Springer\u2010Verlag, Berlin, 1983, pp. 121\u2013144.","DOI":"10.1007\/BFb0062098"},{"key":"R34","doi-asserted-by":"publisher","DOI":"10.1090\/S0025-5718-1984-0736453-8"},{"key":"R35","volume-title":"Numerical methods for large eigenvalue problems","author":"Saad Youcef","year":"1992"},{"key":"R36","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-662-03329-6"},{"key":"R37","doi-asserted-by":"publisher","unstructured":"V. Simoncini, Ritz and pseudo\u2010Ritz values using matrix polynomials, Proceedings of the Fourth Conference of the International Linear Algebra Society (Rotterdam, 1994), Vol. 241\/243, 1996, 787\u201380110.1016\/0024-3795(95)00682-697e:65034","DOI":"10.1016\/0024-3795(95)00682-6"},{"key":"R38","doi-asserted-by":"publisher","DOI":"10.1137\/1015095"},{"key":"R39","unstructured":"G. W. Stewart and J.\u2010G. Sun,\n                      Matrix Perturbation Theory\n                      , Academic Press, Boston, 1990."},{"key":"R40","doi-asserted-by":"publisher","DOI":"10.1137\/0917002"},{"key":"R41","doi-asserted-by":"publisher","DOI":"10.1137\/S0895479897324020"},{"key":"R42","doi-asserted-by":"crossref","unstructured":"L. Trefethen, Approximation theory and numerical linear algebra, Chapman and Hall, London, 1990, 336\u201336091j:65063","DOI":"10.1007\/978-1-4899-3442-0_30"},{"key":"R43","unstructured":"L. Trefethen, Pseudospectra of matrices, Pitman Res. Notes Math. Ser., Vol. 260, Longman Sci. Tech., Harlow, 1992, 234\u201326693e:65064"},{"key":"R44","doi-asserted-by":"crossref","unstructured":"Lloyd Trefethen, Computation of pseudospectra, Acta Numer., Vol. 8, Cambridge Univ. Press, Cambridge, 1999, 247\u20132952002b:65062","DOI":"10.1017\/S0962492900002932"},{"key":"R45","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-662-03972-4"},{"key":"R46","doi-asserted-by":"publisher","DOI":"10.1007\/BF01389450"},{"key":"R47","doi-asserted-by":"publisher","DOI":"10.1016\/0024-3795(87)90129-7"},{"key":"R48","doi-asserted-by":"publisher","DOI":"10.1016\/0377-0427(93)90028-A"},{"key":"R49","unstructured":"T. G. Wright,\n                      MATLAB Pseudospectra GUI\n                      , 2000\u20132001, http:\/\/www.comlab.ox.ac.uk\/pseudospectra\/psagui."}],"container-title":["SIAM Journal on Matrix Analysis and Applications"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/epubs.siam.org\/doi\/pdf\/10.1137\/S0895479801398608","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T17:35:01Z","timestamp":1787333701000},"score":1,"resource":{"primary":{"URL":"https:\/\/epubs.siam.org\/doi\/10.1137\/S0895479801398608"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2004,1]]},"references-count":49,"journal-issue":{"issue":"4","published-print":{"date-parts":[[2004,1]]}},"alternative-id":["10.1137\/S0895479801398608"],"URL":"https:\/\/doi.org\/10.1137\/s0895479801398608","relation":{},"ISSN":["0895-4798","1095-7162"],"issn-type":[{"value":"0895-4798","type":"print"},{"value":"1095-7162","type":"electronic"}],"subject":[],"published":{"date-parts":[[2004,1]]}}}