{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,24]],"date-time":"2026-08-24T09:33:50Z","timestamp":1787564030983,"version":"build-2736575974"},"reference-count":18,"publisher":"Society for Industrial & Applied Mathematics (SIAM)","issue":"5","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["SIAM J. Sci. Comput."],"published-print":{"date-parts":[[2010,1]]},"abstract":"<jats:p>IDR(s) [P. Sonneveld and M. B. van Gijzen, SIAM J. Sci. Comput., 31 (2008), pp. 1035\u20131062] and BiCGstab($\\ell$) [G. L. G. Sleijpen and D. R. Fokkema, Electron. Trans. Numer. Anal., 1 (1993), pp. 11\u201332] are two of the most efficient short-recurrence iterative methods for solving large nonsymmetric linear systems of equations. Which of the two is best depends on the specific problem class. In this paper we describe IDRstab, a new method that combines the strengths of IDR(s) and BiCGstab($\\ell$). To derive IDRstab we extend the results that we reported on in [G. L. G. Sleijpen, P. Sonneveld, and M. B. van Gijzen, Appl. Numer. Math., (2009), DOI: 10.1016\/j.apnum.2009.07.001], where we considered Bi-CGSTAB as an induced dimension reduction (IDR) method. We will analyze the relation between hybrid Bi-CG methods and IDR and introduce the new concept of the Sonneveld subspace as a common framework. Through numerical experiments we will show that IDRstab can outperform both IDR(s) and BiCGstab($\\ell$).<\/jats:p>","DOI":"10.1137\/090752341","type":"journal-article","created":{"date-parts":[[2010,8,31]],"date-time":"2010-08-31T18:25:08Z","timestamp":1283279108000},"page":"2687-2709","source":"Crossref","is-referenced-by-count":37,"title":["Exploiting BiCGstab($\\ell$) Strategies to Induce Dimension Reduction"],"prefix":"10.1137","volume":"32","author":[{"given":"Gerard L. G.","family":"Sleijpen","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Martin B.","family":"van Gijzen","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2010,8,31]]},"reference":[{"key":"R1","doi-asserted-by":"publisher","DOI":"10.1016\/0377-0427(95)00227-8"},{"key":"R2","doi-asserted-by":"publisher","DOI":"10.1137\/0914062"},{"key":"R3","first-page":"126","volume":"36","author":"Gutknecht M. H.","year":"2010","journal-title":"Electron. Trans. Numer. Anal.","ISSN":"https:\/\/id.crossref.org\/issn\/1097-4067","issn-type":"print"},{"key":"R4","doi-asserted-by":"publisher","DOI":"10.1137\/S0895479896306744"},{"key":"R5","doi-asserted-by":"publisher","DOI":"10.1137\/S1064827596304812"},{"key":"R6","first-page":"11","volume":"1","author":"Sleijpen G. L. G.","year":"1993","journal-title":"Electron. Trans. Numer. Anal.","ISSN":"https:\/\/id.crossref.org\/issn\/1097-4067","issn-type":"print"},{"key":"R7","doi-asserted-by":"crossref","unstructured":"G. L. G. Sleijpen, P. Sonneveld, and M. B. van Gijzen,\n                      Bi-CGSTAB as an induced dimension reduction method\n                      , Appl. Numer. Math. (2009), DOI: 10.1016\/j.apnum.2009.07.001.","DOI":"10.1016\/j.apnum.2009.07.001"},{"key":"R8","doi-asserted-by":"publisher","DOI":"10.1007\/BF02140769"},{"key":"R9","doi-asserted-by":"publisher","DOI":"10.1007\/BF02309342"},{"key":"R10","unstructured":"G. L. G. Sleijpen and M. B. van Gijzen,\n                      BiCGstab($\\ell$) Strategies to Induce Dimension Reduction\n                      , Report ISSN 1389-6520, Department of Applied Mathematical Analysis, Delft University of Technology, The Netherlands, 2009."},{"key":"R11","doi-asserted-by":"publisher","DOI":"10.1137\/0910004"},{"key":"R12","doi-asserted-by":"publisher","DOI":"10.1137\/070685804"},{"key":"R13","unstructured":"M. Tanio and M. Sugihara,\n                      \n                        GBi-CGSTAB(s,L): IDR(\n                        s\n                        ) with Higher-Order Stabilization Polynomials\n                      \n                      , Report METR 2009-16, Department of Mathematical Informatics, University of Tokyo, Japan, 2009."},{"key":"R14","doi-asserted-by":"publisher","DOI":"10.1137\/0913035"},{"key":"R15","unstructured":"M. van Gijzen and P. Sonneveld,\n                      \n                        An Elegant IDR(\n                        s\n                        ) Variant That Efficiently Exploits Bi-orthogonality Properties\n                      \n                      , Preprint 08-21, Delft Institute of Applied Mathematics, Delft University of Technology, Delft, The Netherlands, 2008."},{"key":"R16","doi-asserted-by":"crossref","unstructured":"P. Wesseling and P. Sonneveld,\n                      Numerical experiments with a multiple grid and a preconditioned Lanczos type method\n                      , in Approximation Methods for Navier-Stokes Problems, Lecture Notes in Math. 771, Springer Verlag, Heidelberg, 1980, pp. 543\u2013562.","DOI":"10.1007\/BFb0086930"},{"key":"R17","doi-asserted-by":"publisher","DOI":"10.1137\/S1064827597321581"},{"key":"R18","doi-asserted-by":"publisher","DOI":"10.1137\/S1064827592236313"}],"container-title":["SIAM Journal on Scientific Computing"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/epubs.siam.org\/doi\/pdf\/10.1137\/090752341","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T16:43:18Z","timestamp":1787330598000},"score":1,"resource":{"primary":{"URL":"https:\/\/epubs.siam.org\/doi\/10.1137\/090752341"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2010,1]]},"references-count":18,"journal-issue":{"issue":"5","published-print":{"date-parts":[[2010,1]]}},"alternative-id":["10.1137\/090752341"],"URL":"https:\/\/doi.org\/10.1137\/090752341","relation":{},"ISSN":["1064-8275","1095-7197"],"issn-type":[{"value":"1064-8275","type":"print"},{"value":"1095-7197","type":"electronic"}],"subject":[],"published":{"date-parts":[[2010,1]]}}}