{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T17:34:48Z","timestamp":1787333688142,"version":"build-2736575974"},"reference-count":42,"publisher":"Society for Industrial & Applied Mathematics (SIAM)","issue":"4","funder":[{"DOI":"10.13039\/501100000038","name":"Natural Sciences and Engineering Research Council of Canada","doi-asserted-by":"publisher","award":["OGP0009236"],"award-info":[{"award-number":["OGP0009236"]}],"id":[{"id":"10.13039\/501100000038","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["SIAM J. Matrix Anal. Appl."],"published-print":{"date-parts":[[2019,1]]},"abstract":"<jats:p>In [ SIAM J. Matrix Anal. Appl., 31 (2010), pp. 2347--2359] it was shown that $k$ steps of the finite precision Lanczos process for tridiagonalizing an $n\\times n$ Hermitian matrix $A$ could be viewed as an exact Lanczos process for a $(k+n)\\times (k+n)$ augmented Hermitian matrix, producing exactly orthogonal vectors. Here we use this and related results to prove the highly accurate behavior of the finite precision Lanczos process when used for finding the eigensystem of $A$, or for solving linear systems $Ax=b$. It turns out that the finite precision process mimics the exact process in iterative rather than $n$-step ways and makes available backward stable results. These results are also complete, such as making available the complete eigensystem of an $A$ with distinct eigenvalues. The matrix $S_k$ used to obtain these results is shown to provide valuable theoretical information on the loss of orthogonality between any $k$ vectors, among its many other useful properties.<\/jats:p>","DOI":"10.1137\/17m1133725","type":"journal-article","created":{"date-parts":[[2019,11,20]],"date-time":"2019-11-20T09:44:40Z","timestamp":1574243080000},"page":"1371-1398","source":"Crossref","is-referenced-by-count":5,"title":["Accuracy of the Lanczos Process for the Eigenproblem and Solution of Equations"],"prefix":"10.1137","volume":"40","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-0844-0276","authenticated-orcid":true,"given":"Christopher C.","family":"Paige","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2019,11,19]]},"reference":[{"key":"atypb1","doi-asserted-by":"publisher","DOI":"10.1137\/0613015"},{"key":"atypb2","doi-asserted-by":"publisher","DOI":"10.1137\/140990735"},{"key":"atypb3","unstructured":"X.W. Chang,\n                      Personal communication\n                      , 2018."},{"key":"atypb4","doi-asserted-by":"publisher","DOI":"10.1137\/100787921"},{"key":"atypb5","doi-asserted-by":"publisher","DOI":"10.1090\/S0002-9904-1969-12330-X"},{"key":"atypb6","doi-asserted-by":"publisher","DOI":"10.1137\/10079687X"},{"key":"atypb7","doi-asserted-by":"publisher","DOI":"10.1137\/0702016"},{"key":"atypb8","unstructured":"G. H. Golub and C. F. Van Loan,\n                      Matrix Computations\n                      , 4th ed., The Johns Hopkins University Press, Baltimore, MD, 2013."},{"key":"atypb9","unstructured":"A. Greenbaum,\n                      Convergence Properties of the Conjugate Gradient Algorithm in Exact and Finite Precision Arithmetic\n                      , Ph.D. thesis, University of California Berkeley, Berkeley, CA, 1981."},{"key":"atypb10","doi-asserted-by":"publisher","DOI":"10.1016\/0024-3795(89)90285-1"},{"key":"atypb11","doi-asserted-by":"publisher","DOI":"10.1137\/0613011"},{"key":"atypb12","doi-asserted-by":"publisher","DOI":"10.1137\/15M1030078"},{"key":"atypb13","doi-asserted-by":"publisher","DOI":"10.1137\/S0895479897331862"},{"key":"atypb14","unstructured":"S. Hammarling and N. J. Higham,\n                      Wilkinson and Backward Error Analysis\n                      , Website of the Numerical Linear Algebra Group, School of Mathematics, University of Manchester, 2019,https:\/\/nla-group.org\/2019\/02\/18\/wilkinson-and-backward-error-analysis\/."},{"key":"atypb15","doi-asserted-by":"publisher","DOI":"10.6028\/jres.049.044"},{"key":"atypb16","doi-asserted-by":"publisher","DOI":"10.6028\/jres.045.026"},{"key":"atypb17","doi-asserted-by":"publisher","DOI":"10.6028\/jres.049.006"},{"key":"atypb18","doi-asserted-by":"crossref","unstructured":"G. Meurant,\n                      The Lanczos and Conjugate Gradient Algorithms\n                      , Software Environ. Tools 19, SIAM, Philadelphia, 2006,https:\/\/doi.org\/10.1137\/1.9780898718140.","DOI":"10.1137\/1.9780898718140"},{"key":"atypb19","doi-asserted-by":"publisher","DOI":"10.1017\/S096249290626001X"},{"key":"atypb20","doi-asserted-by":"publisher","DOI":"10.1007\/s00211-007-0078-x"},{"key":"atypb21","unstructured":"C. C. Paige,\n                      The Computation of Eigenvalues and Eigenvectors of Very Large Sparse Matrices\n                      , Ph.D. thesis, London University, London, England, 1971."},{"key":"atypb22","doi-asserted-by":"publisher","DOI":"10.1093\/imamat\/18.3.341"},{"key":"atypb23","doi-asserted-by":"publisher","DOI":"10.1016\/0024-3795(80)90167-6"},{"key":"atypb24","doi-asserted-by":"publisher","DOI":"10.1137\/080725167"},{"key":"atypb25","doi-asserted-by":"publisher","DOI":"10.1137\/090761343"},{"key":"atypb26","doi-asserted-by":"crossref","unstructured":"C. C. Paige,\n                      The effects of loss of orthogonality on large scale numerical computations\n                      , in Computational Science and Its Applications, ICCSA 2018, Lecture Notes in Comput. Sci. 10962, O. Gervasi et al., eds., Springer, Cham, pp. 429-439,https:\/\/doi.org\/10.1007\/978-3-319-95168-3_29.","DOI":"10.1007\/978-3-319-95168-3_29"},{"key":"atypb27","doi-asserted-by":"publisher","DOI":"10.1137\/100796285"},{"key":"atypb28","doi-asserted-by":"publisher","DOI":"10.1016\/j.laa.2013.05.009"},{"key":"atypb29","doi-asserted-by":"publisher","DOI":"10.1137\/050630416"},{"key":"atypb30","doi-asserted-by":"publisher","DOI":"10.1137\/0712047"},{"key":"atypb31","doi-asserted-by":"publisher","DOI":"10.1145\/355984.355989"},{"key":"atypb32","doi-asserted-by":"publisher","DOI":"10.1145\/355993.356000"},{"key":"atypb33","doi-asserted-by":"publisher","DOI":"10.1007\/s002110100314"},{"key":"atypb34","doi-asserted-by":"publisher","DOI":"10.1137\/120897687"},{"key":"atypb35","unstructured":"I. Panayotov,\n                      Eigenvalue Estimation with the Rayleigh-Ritz and Lanczos Methods\n                      , Ph.D. thesis, McGill University, Montr\u00e9al, Canada, 2010."},{"key":"atypb36","doi-asserted-by":"crossref","unstructured":"B. N. Parlett,\n                      The Symmetric Eigenvalue Problem\n                      , Classics Appl. Math. 20, SIAM, Philadelphia, 1998,https:\/\/doi.org\/10.1137\/1.9781611971163.","DOI":"10.1137\/1.9781611971163"},{"key":"atypb37","doi-asserted-by":"publisher","DOI":"10.1137\/0725052"},{"key":"atypb38","doi-asserted-by":"publisher","DOI":"10.1137\/1019104"},{"key":"atypb39","doi-asserted-by":"crossref","unstructured":"Z. Strako\u0161,\n                      On the real convergence rate of the conjugate gradient method\n                      , Linear Algebra Appl., 154-156 (1991), pp. 535-549,https:\/\/doi.org\/10.1016\/0024-3795(91)90393-B.","DOI":"10.1016\/0024-3795(91)90393-B"},{"key":"atypb40","unstructured":"Z. Strako\u0161,\n                      Convergence and numerical behavior of the Krylov space methods\n                      , in NATO ASI Institute, Algorithms for Large Sparse Linear Algebraic Systems: The State of the Art and Applications in Science and Engineering, G. Winter Althaus and E. Spedicato, eds., Kluwer, Dordrecht, The Netherlands, 1998, pp. 175-197."},{"key":"atypb41","unstructured":"J. H. Wilkinson,\n                      Rounding Errors in Algebraic Processes\n                      , Notes Appl. Sci. 32, Her Majesty's Stationery Office, London, 1963. (Also published by Prentice-Hall, Englewood Cliffs, NJ, 1964. Reprinted by Dover Publications, New York, 1994.)"},{"key":"atypb42","unstructured":"J. H. Wilkinson,\n                      The Algebraic Eigenvalue Problem\n                      , Clarendon Press, Oxford, UK, 1965."}],"container-title":["SIAM Journal on Matrix Analysis and Applications"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/epubs.siam.org\/doi\/pdf\/10.1137\/17M1133725","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T17:18:06Z","timestamp":1787332686000},"score":1,"resource":{"primary":{"URL":"https:\/\/epubs.siam.org\/doi\/10.1137\/17M1133725"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2019,1]]},"references-count":42,"journal-issue":{"issue":"4","published-print":{"date-parts":[[2019,1]]}},"alternative-id":["10.1137\/17M1133725"],"URL":"https:\/\/doi.org\/10.1137\/17m1133725","relation":{},"ISSN":["0895-4798","1095-7162"],"issn-type":[{"value":"0895-4798","type":"print"},{"value":"1095-7162","type":"electronic"}],"subject":[],"published":{"date-parts":[[2019,1]]}}}