{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,24]],"date-time":"2026-03-24T16:23:17Z","timestamp":1774369397128,"version":"3.50.1"},"reference-count":17,"publisher":"Springer Science and Business Media LLC","issue":"1-3","license":[{"start":{"date-parts":[[2002,3,1]],"date-time":"2002-03-01T00:00:00Z","timestamp":1014940800000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"},{"start":{"date-parts":[[2002,3,1]],"date-time":"2002-03-01T00:00:00Z","timestamp":1014940800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Numerical Algorithms"],"published-print":{"date-parts":[[2002,3]]},"DOI":"10.1023\/a:1014864007293","type":"journal-article","created":{"date-parts":[[2002,12,29]],"date-time":"2002-12-29T19:13:07Z","timestamp":1041189187000},"page":"97-105","source":"Crossref","is-referenced-by-count":7,"title":["The Breakdowns of BiCGStab"],"prefix":"10.1007","volume":"29","author":[{"given":"P.R.","family":"Graves-Morris","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","reference":[{"key":"395852_CR1","doi-asserted-by":"crossref","first-page":"49","DOI":"10.1007\/978-94-011-0970-3_5","volume-title":"Nonlinear Numerical Methods and Rational Approximation II","author":"C. Brezinski","year":"1994","unstructured":"C. Brezinski and M Redivo-Zaglia, Breakdowns in the computation of orthogonal polynomials, in: Nonlinear Numerical Methods and Rational Approximation II, ed. A. Cuyt (Kluwer, Dordrecht, 1994) pp. 49\u201359."},{"key":"395852_CR2","doi-asserted-by":"crossref","first-page":"169","DOI":"10.1007\/BF01737161","volume":"35","author":"C. Brezinski","year":"1995","unstructured":"C. Brezinski and M Redivo-Zaglia, Look-ahead in BiCGStab and other product methods for linear systems, BIT 35 (1995) 169\u2013201.","journal-title":"BIT"},{"key":"395852_CR3","series-title":"Lecture Notes in Mathematics","doi-asserted-by":"crossref","first-page":"73","DOI":"10.1007\/BFb0080116","volume-title":"Numer. Analysis, Dundee, 1975","author":"R. Fletcher","year":"1976","unstructured":"R. Fletcher, Conjugate gradient methods for indefinite systems, in: Numer. Analysis, Dundee, 1975, ed. G.A. Watson, Lecture Notes in Mathematics, Vol. 506 (Springer, Berlin, 1976) pp. 73\u201389."},{"key":"395852_CR4","doi-asserted-by":"crossref","first-page":"137","DOI":"10.1137\/0914009","volume":"14","author":"R.W. Freund","year":"1993","unstructured":"R.W. Freund, M.H. Gutknecht and N. Nachtigal. An implementation of look-ahead Lanczos algorithm for non-Hermitian matrices, SIAM J. Sci. Comput. 14 (1993) 137\u2013158.","journal-title":"SIAM J. Sci. Comput."},{"key":"395852_CR5","doi-asserted-by":"crossref","first-page":"63","DOI":"10.1093\/oso\/9780198500148.003.0003","volume-title":"The State of the Art in Numerical Analysis","author":"G.H. Golub","year":"1997","unstructured":"G.H. Golub and H.A. Van der Vorst, Closer to the solution: Iterative linear solvers, in: The State of the Art in Numerical Analysis, eds. I.S. Duff and G.A. Watson (Clarendon Press, Oxford, 1997) pp. 63\u201392."},{"key":"395852_CR6","doi-asserted-by":"crossref","first-page":"247","DOI":"10.1023\/A:1019162308133","volume":"15","author":"P.R. Graves-Morris","year":"1997","unstructured":"P.R. Graves-Morris, A 'Look-around Lanczos' algorithm for solving a system of linear equations, Numer. Algorithms 15 (1997) 247\u2013274.","journal-title":"Numer. Algorithms"},{"key":"395852_CR7","unstructured":"P.R. Graves-Morris, VPAStab and its breakdowns, submitted to Numer. Algorithms (2002)."},{"key":"395852_CR8","doi-asserted-by":"crossref","first-page":"205","DOI":"10.1023\/A:1019145010317","volume":"21","author":"P.R. Graves-Morris","year":"1999","unstructured":"P.R. Graves-Morris and A. Salam, Avoiding breakdown in van der Vorst's method, Numer. Algorithms 21 (1999) 205\u2013223.","journal-title":"Numer. Algorithms"},{"key":"395852_CR9","doi-asserted-by":"crossref","first-page":"535","DOI":"10.1137\/S0895479895284944","volume":"18","author":"A. Greenbaum","year":"1997","unstructured":"A. Greenbaum, Estimating the attainable accuracy of recursively computed residual methods, SIAM J. Matrix Anal. Appl. 18 (1997) 535\u2013551.","journal-title":"SIAM J. Matrix Anal. Appl."},{"key":"395852_CR10","doi-asserted-by":"crossref","first-page":"271","DOI":"10.1017\/S0962492900002737","volume":"6","author":"M.H. Gutknecht","year":"1997","unstructured":"M.H. Gutknecht, Lanczos-type solvers for non-symmetric linear systems of equations, Acta Numerica 6 (1997) 271\u2013397.","journal-title":"Acta Numerica"},{"key":"395852_CR11","doi-asserted-by":"crossref","first-page":"1051","DOI":"10.1137\/S0895479896306744","volume":"21","author":"M.H. Gutknecht","year":"2000","unstructured":"M.H. Gutknecht and K.J. Ressel, Look-ahead procedures for Lanczos-type product methods based on three-term Lanczos recurrences, SIAM J. Matrix Anal. Appl. 21 (2000) 1051\u20131078.","journal-title":"SIAM J. Matrix Anal. Appl."},{"key":"395852_CR12","doi-asserted-by":"crossref","first-page":"213","DOI":"10.1137\/S0895479897331862","volume":"22","author":"M.H. Gutknecht","year":"2000","unstructured":"M.H. Gutknecht and Z. Strako\u0161, Accuracy of two three-term and three two-term recurrences for Krylov space solvers, SIAM J. Matrix Anal. Appl. 22 (2000) 213\u2013229.","journal-title":"SIAM J. Matrix Anal. Appl."},{"key":"395852_CR13","unstructured":"MATLAB 6.0, The MathWorks Inc., Natick, MA, USA."},{"key":"395852_CR14","doi-asserted-by":"crossref","unstructured":"J.K. Reid, The use of conjugate gradients for systems of equations possessing 'Property A', SIAM J. Numer. Anal. (1972) 325-332.","DOI":"10.1137\/0709032"},{"key":"395852_CR15","doi-asserted-by":"crossref","first-page":"203","DOI":"10.1007\/BF02140769","volume":"10","author":"G.L.G. Sleijpen","year":"1995","unstructured":"G.L.G. Sleijpen and H.A. van der Vorst, Maintaining convergence properties of BiCGStab methods in finite precision arithmetic, Numer. Algorithms 10 (1995) 203\u2013223.","journal-title":"Numer. Algorithms"},{"key":"395852_CR16","doi-asserted-by":"crossref","first-page":"141","DOI":"10.1007\/BF02309342","volume":"56","author":"G.L.G. Sleijpen","year":"1996","unstructured":"G.L.G. Sleijpen and H.A. van der Vorst, Reliable updated residuals in hybrid Bi-CG methods, Computing 56 (1996) 141\u2013163.","journal-title":"Computing"},{"key":"395852_CR17","doi-asserted-by":"crossref","first-page":"631","DOI":"10.1137\/0913035","volume":"13","author":"H.A. Van der Vorst","year":"1992","unstructured":"H.A. Van der Vorst, Bi-CGStab: A fast and smoothly convergent variant of Bi-CG for the solution of non-symmetric linear systems, SIAM J. Sci. Statist. Comput. 13 (1992) 631\u2013644.","journal-title":"SIAM J. Sci. Statist. Comput."}],"container-title":["Numerical Algorithms"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1023\/A:1014864007293.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1023\/A:1014864007293\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1023\/A:1014864007293.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,7,7]],"date-time":"2025-07-07T07:09:48Z","timestamp":1751872188000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1023\/A:1014864007293"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2002,3]]},"references-count":17,"journal-issue":{"issue":"1-3","published-print":{"date-parts":[[2002,3]]}},"alternative-id":["395852"],"URL":"https:\/\/doi.org\/10.1023\/a:1014864007293","relation":{},"ISSN":["1017-1398","1572-9265"],"issn-type":[{"value":"1017-1398","type":"print"},{"value":"1572-9265","type":"electronic"}],"subject":[],"published":{"date-parts":[[2002,3]]}}}