{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T16:43:49Z","timestamp":1787330629223,"version":"build-2736575974"},"reference-count":23,"publisher":"Society for Industrial & Applied Mathematics (SIAM)","issue":"3","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["SIAM J. Sci. Comput."],"published-print":{"date-parts":[[2010,1]]},"abstract":"<jats:p>There is a need for flexible iterative solvers that can solve large-scale ($&gt;10^6$ unknowns) nonsymmetric sparse linear systems to a small tolerance. Among flexible solvers, flexible GMRES (FGMRES) is attractive because it minimizes the residual norm over a particular subspace. In practice, FGMRES is often restarted periodically to keep memory and work requirements reasonable; however, like restarted GMRES, restarted FGMRES can suffer from stagnation. This has led us to develop a flexible variant of the Krylov linear solver GCROT (generalized conjugate residual with inner orthogonalization and outer truncation). Unlike the original GCROT algorithm, the proposed GCROT variant uses a simplified truncation strategy similar to loose GMRES (LGMRES). This modification is motivated by numerical experiments that suggest the specific subspace retained in the outer iteration of GCROT is less important than its size. The flexible GCROT variant appears to be well suited for advection-dominated problems. In particular, when applied to an adjoint problem from computational aerodynamics, the proposed GCROT variant is robust and efficient compared with several popular truncated Krylov subspace methods. Finally, a flexible version of LGMRES is easily constructed by recognizing algorithmic similarities to GCROT.<\/jats:p>","DOI":"10.1137\/090754674","type":"journal-article","created":{"date-parts":[[2010,6,9]],"date-time":"2010-06-09T18:05:21Z","timestamp":1276106721000},"page":"1672-1694","source":"Crossref","is-referenced-by-count":63,"title":["A Simplified and Flexible Variant of GCROT for Solving Nonsymmetric Linear Systems"],"prefix":"10.1137","volume":"32","author":[{"given":"Jason E.","family":"Hicken","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"David W.","family":"Zingg","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2010,6,9]]},"reference":[{"key":"R1","doi-asserted-by":"publisher","DOI":"10.1137\/S0895479803422014"},{"key":"R2","doi-asserted-by":"crossref","unstructured":"R. F. Boisvert, R. Pozo, K. Remington, R. Barrett, and J. Dongarra,\n                      The Matrix Market: A web resource for test matrix data\n                      , in The Quality of Numerical Software, Assessment and Enhancement, R. F. Boisvert, ed., Chapman & Hall, London, 1997, pp. 125\u2013137.","DOI":"10.1007\/978-1-5041-2940-4_9"},{"key":"R3","doi-asserted-by":"publisher","DOI":"10.1016\/0377-0427(94)00123-5"},{"key":"R4","doi-asserted-by":"publisher","DOI":"10.1137\/S0036142997315950"},{"key":"R5","doi-asserted-by":"publisher","DOI":"10.1137\/0720023"},{"key":"R6","doi-asserted-by":"publisher","DOI":"10.1016\/S0167-8191(00)00075-2"},{"key":"R7","doi-asserted-by":"publisher","DOI":"10.1137\/0914062"},{"key":"R8","unstructured":"J. E. Hicken,\n                      Efficient Algorithms for Future Aircraft Design: Contributions to Aerodynamic Shape Optimization\n                      , Ph.D. thesis, University of Toronto, Toronto, 2009."},{"key":"R9","doi-asserted-by":"publisher","DOI":"10.2514\/1.34810"},{"key":"R10","doi-asserted-by":"publisher","DOI":"10.2514\/1.44033"},{"key":"R11","doi-asserted-by":"publisher","DOI":"10.1016\/j.jcp.2003.08.010"},{"key":"R12","unstructured":"Matrix Market\n                      , Technical report, National Institute of Standards and Technology, Gaithersburg, MD, 2009 (accessed 15 February 2009)."},{"key":"R13","first-page":"148","volume":"31","author":"Meijerink J. A.","year":"1977","journal-title":"Math. Comp.","ISSN":"https:\/\/id.crossref.org\/issn\/0025-5718","issn-type":"print"},{"key":"R14","doi-asserted-by":"publisher","DOI":"10.1137\/S0895479893253975"},{"key":"R15","doi-asserted-by":"publisher","DOI":"10.1137\/S0036144504443511"},{"key":"R16","doi-asserted-by":"publisher","DOI":"10.1137\/0914028"},{"key":"R17","unstructured":"Y. Saad and A. V. Malevsky,\n                      P-SPARSLIB: A Portable Library of Distributed Memory Sparse Iterative Solvers\n                      , Technical report UMSI-95-180, Minnesota Supercomputing Institute, University of Minnesota, Minneapolis, 1995."},{"key":"R18","doi-asserted-by":"publisher","DOI":"10.1137\/0907058"},{"key":"R19","doi-asserted-by":"publisher","DOI":"10.1137\/S1064827597328996"},{"key":"R20","doi-asserted-by":"publisher","DOI":"10.1002\/nla.499"},{"key":"R21","doi-asserted-by":"publisher","DOI":"10.1137\/0913035"},{"key":"R22","doi-asserted-by":"publisher","DOI":"10.1002\/nla.1680010404"},{"key":"R23","doi-asserted-by":"publisher","DOI":"10.1016\/j.amc.2006.09.116"}],"container-title":["SIAM Journal on Scientific Computing"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/epubs.siam.org\/doi\/pdf\/10.1137\/090754674","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T16:17:03Z","timestamp":1787329023000},"score":1,"resource":{"primary":{"URL":"https:\/\/epubs.siam.org\/doi\/10.1137\/090754674"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2010,1]]},"references-count":23,"journal-issue":{"issue":"3","published-print":{"date-parts":[[2010,1]]}},"alternative-id":["10.1137\/090754674"],"URL":"https:\/\/doi.org\/10.1137\/090754674","relation":{},"ISSN":["1064-8275","1095-7197"],"issn-type":[{"value":"1064-8275","type":"print"},{"value":"1095-7197","type":"electronic"}],"subject":[],"published":{"date-parts":[[2010,1]]}}}