{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T17:28:02Z","timestamp":1787333282632,"version":"build-2736575974"},"reference-count":29,"publisher":"Society for Industrial & Applied Mathematics (SIAM)","issue":"4","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["SIAM J. Sci. Comput."],"published-print":{"date-parts":[[2010,1]]},"abstract":"<jats:p>The induced dimension reduction (IDR) method of Sonneveld and van Gijzen [SIAM J. Sci. Comput., 31 (2008), pp. 1035\u20131062] is shown to be a Petrov\u2013Galerkin (projection) method with a particular choice of left Krylov subspaces; these left subspaces are rational Krylov spaces. Consequently, other methods, such as BiCGStab and ML(s)BiCGStab, which are mathematically equivalent to some versions of IDR, can also be interpreted as Petrov\u2013Galerkin methods. The connection with rational Krylov spaces inspired a new version of IDR, called Ritz-IDR, where the poles of the rational function are chosen as certain Ritz values. Experiments are presented illustrating the effectiveness of this new version.<\/jats:p>","DOI":"10.1137\/090774756","type":"journal-article","created":{"date-parts":[[2010,6,24]],"date-time":"2010-06-24T18:03:11Z","timestamp":1277402591000},"page":"1898-1912","source":"Crossref","is-referenced-by-count":27,"title":["Interpreting IDR as a Petrov\u2013Galerkin Method"],"prefix":"10.1137","volume":"32","author":[{"given":"Valeria","family":"Simoncini","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Daniel B.","family":"Szyld","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2010,6,24]]},"reference":[{"key":"R1","doi-asserted-by":"crossref","unstructured":"Z. Bai, J. Demmel, J. Dongarra, A. Ruhe, and H. A. van der Vorst, eds.\n                      Templates for the Solution of Algebraic Eigenvalue Problems: A Practical Guide\n                      , SIAM, Philadelphia, 2000.","DOI":"10.1137\/1.9780898719581"},{"key":"R2","doi-asserted-by":"publisher","DOI":"10.1137\/080741744"},{"key":"R3","doi-asserted-by":"publisher","DOI":"10.1007\/BF02141740"},{"key":"R4","doi-asserted-by":"publisher","DOI":"10.1137\/0914062"},{"key":"R5","unstructured":"M. H. Gutknecht,\n                      Block Krylov space methods for linear systems with multiple right-hand sides: An introduction\n                      , in Modern Mathematical Models, Methods and Algorithms for Real World Systems, A. H. Siddiqi, I. S. Duff, and O. Christensen, eds., Anamaya Publishers, New Delhi, 2007, pp. 420\u2013447."},{"key":"R6","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":"R7","unstructured":"N. J. Higham,\n                      The Matrix Computation Toolbox\n                      , http:\/\/www.ma.man.ac.uk\/~higham\/mctoolbox (2008)."},{"key":"R8","unstructured":"M. E. Hochstenbach, D. A. Singer, and P. F. Zachlin,\n                      On the Field of Values and Pseudospectra of the Inverse of a Large Matrix\n                      , CASA report 07-06, Department of Mathematics, TU Eindhoven, Eindhoven, The Netherlands, 2007."},{"key":"R9","doi-asserted-by":"publisher","DOI":"10.1137\/080715159"},{"key":"R10","unstructured":"Matrix Market,\n                      A Repository of Test Data for Numerical Linear Algebra\n                      , http:\/\/math.nist. gov\/MatrixMarket\/ (2007)."},{"key":"R11","doi-asserted-by":"publisher","DOI":"10.1016\/0024-3795(91)90386-B"},{"key":"R12","doi-asserted-by":"publisher","DOI":"10.1016\/0024-3795(84)90221-0"},{"key":"R13","doi-asserted-by":"publisher","DOI":"10.1007\/BF01935024"},{"key":"R14","doi-asserted-by":"crossref","unstructured":"Y. Saad,\n                      Iterative Methods for Sparse Linear systems\n                      , PWS, Boston, 1996; second edition, SIAM, Philadelphia, 2003.","DOI":"10.1137\/1.9780898718003"},{"key":"R15","doi-asserted-by":"publisher","DOI":"10.1137\/S0895479894264673"},{"key":"R16","doi-asserted-by":"publisher","DOI":"10.1002\/nla.499"},{"key":"R17","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":"R18","doi-asserted-by":"publisher","DOI":"10.1007\/BF02140769"},{"key":"R19","doi-asserted-by":"publisher","DOI":"10.1016\/0168-9274(95)00085-2"},{"key":"R20","doi-asserted-by":"publisher","DOI":"10.1007\/BF02141261"},{"key":"R21","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., to appear."},{"key":"R22","unstructured":"G. L. G. Sleijpen and M. B. van Gijzen,\n                      Exploiting bi-CGSTAB($\\ell$) Strategies to Induce Dimension Reduction\n                      , Technical report 09-02, Department of Applied Mathematical Analysis, Delft University of Technology, Delft, The Netherlands, 2009."},{"key":"R23","doi-asserted-by":"publisher","DOI":"10.1137\/070685804"},{"key":"R24","unstructured":"M. Tanio and M. Sugihara,\n                      \n                        GBi-CGSTAB($s,L$): IDR(\n                        s\n                        ) with higher-order stabilization polynomials\n                      \n                      , Technical report METR 2009\u201316, Department of Mathematical Informatics, University of Tokyo, Tokyo, Japan, 2009."},{"key":"R25","doi-asserted-by":"publisher","DOI":"10.1137\/0913035"},{"key":"R26","unstructured":"M. B. van Gijzen,\n                      IDR Website\n                      , http:\/\/ta.twi.tudelft.nl\/nw\/users\/gijzen\/IDR.html (2009)."},{"key":"R27","unstructured":"M. B. van Gijzen and P. Sonneveld,\n                      An elegant IDR(s) variant that efficiently exploits bi-orthogonality properties\n                      , Technical report 08-21, Department of Applied Mathematical Analysis, Delft University of Technology, 2008."},{"key":"R28","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, R. Rautmann, ed., Lecture Notes in Math. 771, Springer, Berlin, New York, 1980, pp. 543\u2013562.","DOI":"10.1007\/BFb0086930"},{"key":"R29","doi-asserted-by":"publisher","DOI":"10.1137\/S1064827597321581"}],"container-title":["SIAM Journal on Scientific Computing"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/epubs.siam.org\/doi\/pdf\/10.1137\/090774756","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T16:47:26Z","timestamp":1787330846000},"score":1,"resource":{"primary":{"URL":"https:\/\/epubs.siam.org\/doi\/10.1137\/090774756"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2010,1]]},"references-count":29,"journal-issue":{"issue":"4","published-print":{"date-parts":[[2010,1]]}},"alternative-id":["10.1137\/090774756"],"URL":"https:\/\/doi.org\/10.1137\/090774756","relation":{},"ISSN":["1064-8275","1095-7197"],"issn-type":[{"value":"1064-8275","type":"print"},{"value":"1095-7197","type":"electronic"}],"subject":[],"published":{"date-parts":[[2010,1]]}}}