{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,5,11]],"date-time":"2026-05-11T10:11:32Z","timestamp":1778494292895,"version":"3.51.4"},"reference-count":30,"publisher":"Springer Science and Business Media LLC","issue":"3","license":[{"start":{"date-parts":[[2018,11,20]],"date-time":"2018-11-20T00:00:00Z","timestamp":1542672000000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"},{"start":{"date-parts":[[2018,11,20]],"date-time":"2018-11-20T00:00:00Z","timestamp":1542672000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"}],"funder":[{"DOI":"10.13039\/501100001824","name":"Grantov\u00e1 Agentura Cesk\u00e9 Republiky","doi-asserted-by":"publisher","award":["17-04150J"],"award-info":[{"award-number":["17-04150J"]}],"id":[{"id":"10.13039\/501100001824","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Numer Algor"],"published-print":{"date-parts":[[2019,11]]},"DOI":"10.1007\/s11075-018-0634-8","type":"journal-article","created":{"date-parts":[[2018,11,20]],"date-time":"2018-11-20T04:12:35Z","timestamp":1542687155000},"page":"937-968","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":12,"title":["Approximating the extreme Ritz values and upper bounds for the A-norm of the error in CG"],"prefix":"10.1007","volume":"82","author":[{"given":"G\u00e9rard","family":"Meurant","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-6008-4056","authenticated-orcid":false,"given":"Petr","family":"Tich\u00fd","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2018,11,20]]},"reference":[{"issue":"1","key":"634_CR1","doi-asserted-by":"publisher","first-page":"138","DOI":"10.1137\/0613012","volume":"13","author":"M Arioli","year":"1992","unstructured":"Arioli, M., Duff, I.S., Ruiz, D.: Stopping criteria for iterative solvers. SIAM J. Matrix Anal. Appl. 13(1), 138\u2013144 (1992)","journal-title":"SIAM J. Matrix Anal. Appl."},{"key":"634_CR2","doi-asserted-by":"publisher","DOI":"10.1137\/1.9781611971538","volume-title":"Templates for the Solution of Linear Systems: Building Blocks for Iterative Methods","author":"R Barrett","year":"1994","unstructured":"Barrett, R., Berry, M., Chan, T.F., et al.: Templates for the Solution of Linear Systems: Building Blocks for Iterative Methods. Society for Industrial and Applied Mathematics (SIAM), Philadelphia (1994)"},{"issue":"2","key":"634_CR3","doi-asserted-by":"publisher","first-page":"312","DOI":"10.1137\/0611021","volume":"11","author":"CH Bischof","year":"1990","unstructured":"Bischof, C.H.: Incremental condition estimation. SIAM J. Matrix Anal. Appl. 11(2), 312\u2013322 (1990)","journal-title":"SIAM J. Matrix Anal. Appl."},{"key":"634_CR4","doi-asserted-by":"publisher","first-page":"151","DOI":"10.1016\/0022-247X(72)90264-8","volume":"37","author":"G Dahlquist","year":"1972","unstructured":"Dahlquist, G., Eisenstat, S.C., Golub, G.H.: Bounds for the error of linear systems of equations using the theory of moments. J. Math. Anal. Appl. 37, 151\u2013166 (1972)","journal-title":"J. Math. Anal. Appl."},{"key":"634_CR5","doi-asserted-by":"crossref","unstructured":"Dahlquist, G., Golub, G.H., Nash, S.G.: Bounds for the error in linear systems. In: Semi-Infinite Programming (Proc. Workshop, Bad Honnef, 1978), Lecture Notes in Control and Information Sci., vol. 15, pp. 154\u2013172. Springer, Berlin (1979)","DOI":"10.1007\/BFb0003890"},{"issue":"2","key":"634_CR6","doi-asserted-by":"publisher","first-page":"300","DOI":"10.1023\/A:1021946924741","volume":"42","author":"IS Duff","year":"2002","unstructured":"Duff, I.S., V\u00f6mel, C.: Incremental norm estimation for dense and sparse matrices. BIT 42(2), 300\u2013322 (2002)","journal-title":"BIT"},{"issue":"1","key":"634_CR7","doi-asserted-by":"publisher","first-page":"174","DOI":"10.1137\/130922872","volume":"35","author":"J Duintjer Tebbens","year":"2014","unstructured":"Duintjer Tebbens, J., Tuma, M.: On incremental condition estimators in the 2-norm. SIAM J. Matrix Anal. Appl. 35(1), 174\u2013197 (2014)","journal-title":"SIAM J. Matrix Anal. Appl."},{"key":"634_CR8","doi-asserted-by":"publisher","first-page":"251","DOI":"10.1017\/S0962492901000046","volume":"10","author":"M Eiermann","year":"2001","unstructured":"Eiermann, M., Ernst, O.G.: Geometric aspects of the theory of Krylov subspace methods. Acta Numer. 10, 251\u2013312 (2001)","journal-title":"Acta Numer."},{"key":"634_CR9","doi-asserted-by":"crossref","unstructured":"Fischer, B., Golub, G.H.: On the error computation for polynomial based iteration methods. In: Recent Advances in Iterative Methods, IMA Vol. Math. Appl., Vol. 60, pp. 59\u201367. Springer, New York (1994)","DOI":"10.1007\/978-1-4613-9353-5_5"},{"key":"634_CR10","unstructured":"Golub, G.H., Meurant, G.: Matrices, moments and quadrature. In: Numerical Analysis 1993 (Dundee, 1993), Pitman Res. Notes Math. Ser., Vol. 303, pp. 105\u2013156. Longman Sci. Tech., Harlow (1994)"},{"issue":"3","key":"634_CR11","doi-asserted-by":"publisher","first-page":"687","DOI":"10.1007\/BF02510247","volume":"37","author":"GH Golub","year":"1997","unstructured":"Golub, G.H., Meurant, G.: Matrices, moments and quadrature. II. How to compute the norm of the error in iterative methods. BIT 37(3), 687\u2013705 (1997)","journal-title":"BIT"},{"key":"634_CR12","volume-title":"Matrices, Moments and Quadrature with Applications","author":"GH Golub","year":"2010","unstructured":"Golub, G.H., Meurant, G.: Matrices, Moments and Quadrature with Applications. Princeton University Press, Princeton (2010)"},{"issue":"2-4","key":"634_CR13","doi-asserted-by":"publisher","first-page":"241","DOI":"10.1007\/BF02142693","volume":"8","author":"GH Golub","year":"1994","unstructured":"Golub, G.H., Strako\u0161, Z.: Estimates in quadratic formulas. Numer. Algorithms 8(2-4), 241\u2013268 (1994)","journal-title":"Numer. Algorithms"},{"key":"634_CR14","volume-title":"Matrix Computations. Johns Hopkins Studies in the Mathematical Sciences","author":"GH Golub","year":"2013","unstructured":"Golub, G.H., Van Loan, C.F.: Matrix Computations. Johns Hopkins Studies in the Mathematical Sciences, 4th edn. Johns Hopkins University Press, Baltimore (2013)","edition":"4th edn."},{"issue":"3","key":"634_CR15","doi-asserted-by":"publisher","first-page":"535","DOI":"10.1137\/S0895479895284944","volume":"18","author":"A Greenbaum","year":"1997","unstructured":"Greenbaum, A.: Estimating the attainable accuracy of recursively computed residual methods. SIAM J. Matrix Anal. Appl. 18(3), 535\u2013551 (1997)","journal-title":"SIAM J. Matrix Anal. Appl."},{"issue":"2","key":"634_CR16","doi-asserted-by":"publisher","first-page":"189","DOI":"10.1023\/A:1011889705659","volume":"27","author":"MH Gutknecht","year":"2001","unstructured":"Gutknecht, M.H., Rozlo\u017en\u00edk, M.: By how much can residual minimization accelerate the convergence of orthogonal residual methods? Numer. Algorithms 27(2), 189\u2013213 (2001)","journal-title":"Numer. Algorithms"},{"issue":"1","key":"634_CR17","doi-asserted-by":"publisher","first-page":"86","DOI":"10.1023\/A:1021917801600","volume":"41","author":"MH Gutknecht","year":"2001","unstructured":"Gutknecht, M.H., Rozlo\u017enik, M.: Residual smoothing techniques: do they improve the limiting accuracy of iterative solvers? BIT 41(1), 86\u2013114 (2001)","journal-title":"BIT"},{"key":"634_CR18","doi-asserted-by":"publisher","first-page":"409","DOI":"10.6028\/jres.049.044","volume":"49","author":"MR Hestenes","year":"1952","unstructured":"Hestenes, M.R., Stiefel, E.: Methods of conjugate gradients for solving linear systems. J. Res. Nat. Bur. Stand. 49, 409\u2013436 (1952)","journal-title":"J. Res. Nat. Bur. Stand."},{"key":"634_CR19","volume-title":"Accuracy and stability of numerical algorithms","author":"NJ Higham","year":"1996","unstructured":"Higham, N.J.: Accuracy and stability of numerical algorithms. Society for Industrial and Applied Mathematics (SIAM), Philadelphia (1996)"},{"key":"634_CR20","volume-title":"Description of the CYLSHELL set","author":"R Kouhia","year":"1998","unstructured":"Kouhia, R.: Description of the CYLSHELL set. Laboratory of Structural Mechanics, Finland (1998)"},{"issue":"1","key":"634_CR21","doi-asserted-by":"publisher","first-page":"77","DOI":"10.1023\/A:1019178811767","volume":"16","author":"G Meurant","year":"1998","unstructured":"Meurant, G.: The computation of bounds for the norm of the error in the conjugate gradient algorithm. Numer. Algorithms 16(1), 77\u201387 (1998)","journal-title":"Numer. Algorithms"},{"key":"634_CR22","volume-title":"The Lanczos and Conjugate Gradient Algorithms, from Theory to Finite Precision Computations, Software, Environments and Tools, vol. 19","author":"G Meurant","year":"2006","unstructured":"Meurant, G.: The Lanczos and Conjugate Gradient Algorithms, from Theory to Finite Precision Computations, Software, Environments and Tools, vol. 19. Society for Industrial and Applied Mathematics (SIAM), Philadelphia (2006)"},{"issue":"2","key":"634_CR23","doi-asserted-by":"publisher","first-page":"163","DOI":"10.1007\/s11075-012-9591-9","volume":"62","author":"G Meurant","year":"2013","unstructured":"Meurant, G., Tich\u00fd, P.: On computing quadrature-based bounds for the A-norm of the error in conjugate gradients. Numer. Algo. 62(2), 163\u2013191 (2013)","journal-title":"Numer. Algo."},{"issue":"3","key":"634_CR24","doi-asserted-by":"publisher","first-page":"679","DOI":"10.1007\/s11075-014-9868-2","volume":"66","author":"G Meurant","year":"2014","unstructured":"Meurant, G., Tich\u00fd, P.: Erratum to: On computing quadrature-based bounds for the A-norm of the error in conjugate gradients [mr3011386]. Numer. Algorithms 66(3), 679\u2013680 (2014)","journal-title":"Numer. Algorithms"},{"issue":"1","key":"634_CR25","doi-asserted-by":"publisher","first-page":"405","DOI":"10.1007\/BF01386090","volume":"6","author":"W Oettli","year":"1964","unstructured":"Oettli, W., Prager, W.: Compatibility of approximate solution of linear equations with given error bounds for coefficients and right-hand sides. Numer. Math. 6(1), 405\u2013409 (1964)","journal-title":"Numer. Math."},{"issue":"1","key":"634_CR26","doi-asserted-by":"publisher","first-page":"43","DOI":"10.1145\/355984.355989","volume":"8","author":"CC Paige","year":"1982","unstructured":"Paige, C.C., Saunders, M.A.: LSQR: an algorithm for sparse linear equations and sparse least squares. ACM Trans. Math. Software 8(1), 43\u201371 (1982)","journal-title":"ACM Trans. Math. Software"},{"key":"634_CR27","doi-asserted-by":"crossref","unstructured":"Parlett, B.N., Dhillon, I.S.: Relatively robust representations of symmetric tridiagonals. In: Proceedings of the International Workshop on Accurate Solution of Eigenvalue Problems (University Park, PA, 1998), vol. 309, pp. 121\u2013151 (2000)","DOI":"10.1016\/S0024-3795(99)00262-1"},{"issue":"3","key":"634_CR28","doi-asserted-by":"publisher","first-page":"543","DOI":"10.1145\/321406.321416","volume":"14","author":"JL Rigal","year":"1967","unstructured":"Rigal, J.L., Gaches, J.: On the compatibility of a given solution with the data of a linear system. Journal of the ACM (JACM) 14(3), 543\u2013548 (1967)","journal-title":"Journal of the ACM (JACM)"},{"key":"634_CR29","first-page":"56","volume":"13","author":"Z Strako\u0161","year":"2002","unstructured":"Strako\u0161, Z., Tich\u00fd, P.: On error estimation in the conjugate gradient method and why it works in finite precision computations. Electron. Trans. Numer. Anal. 13, 56\u201380 (2002)","journal-title":"Electron. Trans. Numer. Anal."},{"issue":"4","key":"634_CR30","doi-asserted-by":"publisher","first-page":"789","DOI":"10.1007\/s10543-005-0032-1","volume":"45","author":"Z Strako\u0161","year":"2005","unstructured":"Strako\u0161, Z., Tich\u00fd, P.: Error estimation in preconditioned conjugate gradients. BIT 45(4), 789\u2013817 (2005)","journal-title":"BIT"}],"container-title":["Numerical Algorithms"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s11075-018-0634-8.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s11075-018-0634-8","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s11075-018-0634-8.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,5,11]],"date-time":"2026-05-11T09:38:42Z","timestamp":1778492322000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s11075-018-0634-8"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2018,11,20]]},"references-count":30,"journal-issue":{"issue":"3","published-print":{"date-parts":[[2019,11]]}},"alternative-id":["634"],"URL":"https:\/\/doi.org\/10.1007\/s11075-018-0634-8","relation":{},"ISSN":["1017-1398","1572-9265"],"issn-type":[{"value":"1017-1398","type":"print"},{"value":"1572-9265","type":"electronic"}],"subject":[],"published":{"date-parts":[[2018,11,20]]},"assertion":[{"value":"4 October 2018","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"12 November 2018","order":2,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"20 November 2018","order":3,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}}]}}