Algorithm 913 swMATH ID: 12775 Software Authors: M. B. van Gijzen; P. Sonneveld Description: Algorithm 913: An elegant IDR(s) variant that efficiently exploits biorthogonality properties. The IDR(s) method that is proposed in Sonneveld and van Gijzen [2008] is a very efficient limited memory method for solving large nonsymmetric systems of linear equations. IDR(s) is based on the induced dimension reduction theorem, that provides a way to construct subsequent residuals that lie in a sequence of shrinking subspaces. The IDR(s) algorithm that is given in Sonneveld and van Gijzen [2008] is a direct translation of the theorem into an algorithm. This translation is not unique. This article derives a new IDR(s) variant, that imposes (one-sided) biorthogonalization conditions on the iteration vectors. The resulting method has lower overhead in vector operations than the original IDR(s) algorithms. In exact arithmetic, both algorithms give the same residual at every (s + 1)-st step, but the intermediate residuals and also the numerical properties differ. We show through numerical experiments that the new variant is more stable and more accurate than the original IDR(s) algorithm, and that it outperforms other state-of-the-art techniques for realistic test problems. Homepage: http://dl.acm.org/citation.cfm?id=2049667 Related Software: BiCGstab; CGS; GpBiCg; SparseMatrix; IFISS; JDQZ; MatrixMarket; mctoolbox; Matlab; IterativeSolvers.jl; CCGPAK; SciPy; ILUPACK; JDQR; SLEPc; MUMPS; BLAS; MKL; CORS; BiCOR Cited in: 24 Publications all top 5 Cited by 47 Authors 7 Van Gijzen, Martin Bastiaan 4 Astudillo, Reinaldo 4 Zemke, Jens-Peter Max 2 Il’in, Valeriĭ Pavlovich 2 Qiu, Yue 1 Abe, Kuniyoshi 1 Aihara, Kensuke 1 Ang, E. Y. M. 1 Baumann, Manuel 1 Belonosov, Mikhail 1 Benner, Peter 1 Carpentieri, Bruno 1 de Gier, J. M. 1 Demyanko, Kirill V. 1 Dmitriev, Maxim N. 1 Elman, Howard C. 1 Freitag, Melina A. 1 Grundel, Sara M. 1 Gutknecht, Martin H. 1 Huang, Ting-Zhu 1 Ishiwata, Emiko 1 Jing, Yanfei 1 Kaporin, Igor E. 1 Knibbe, H. 1 Kostin, Victor I. 1 Kürschner, Patrick 1 Meerbergen, Karl 1 Nechepurenko, Yuri M. 1 Neklyudov, Dmitry 1 Oosterlee, Cornelis Willebrordus 1 Paige, Christopher C. 1 Panayotov, Ivo 1 Plessix, René-Edouard 1 Ramage, Alison 1 Rendel, Olaf 1 Rizvanolli, Anisa 1 Sangers, Alex 1 Silvester, David J. 1 Sleijpen, Gerard L. G. 1 Sogabe, Tomohiro 1 Sonneveld, Peter 1 Stoll, Martin 1 Sun, Dong-Lin 1 Tcheverda, Vladimir A. 1 Vannieuwenhoven, Nick 1 Vuik, Cornelis 1 Yeung, Man-Chung all top 5 Cited in 16 Serials 5 Journal of Computational and Applied Mathematics 2 Linear Algebra and its Applications 2 Journal of Mathematical Sciences (New York) 2 Numerical Linear Algebra with Applications 1 Computers & Mathematics with Applications 1 Journal of Computational Physics 1 ACM Transactions on Mathematical Software 1 Applied Mathematics and Computation 1 SIAM Journal on Matrix Analysis and Applications 1 SIAM Review 1 SIAM Journal on Scientific Computing 1 ETNA. Electronic Transactions on Numerical Analysis 1 Computational Geosciences 1 Journal of Numerical Mathematics 1 Springer Series in Computational Mathematics 1 Networks and Heterogeneous Media all top 5 Cited in 7 Fields 23 Numerical analysis (65-XX) 3 Partial differential equations (35-XX) 3 Fluid mechanics (76-XX) 1 Linear and multilinear algebra; matrix theory (15-XX) 1 Dynamical systems and ergodic theory (37-XX) 1 Computer science (68-XX) 1 Operations research, mathematical programming (90-XX) Citations by Year