## BiCGstab

 swMATH ID: 4022 Software Authors: Sleijpen, G.L.G.; van der Vorst, H.A.; Fokkema, D.R. Description: BiCGstab($$l$$) and other hybrid Bi-CG methods. Various implementations of the modification BiCGstab(l) of the bi-conjugate gradient method for the solution of a system of linear algebraic equations with a non-singular n×n matrix are studied. The basis of these method forms various investigations of the BiCG part and the polynomial part. The second one is used for further reduction of the BiCG residue and for improving the numerical stability. Both parts are studied in detail. The theory is illustrated by numerical examples. Homepage: http://www.math.uu.nl/people/sleijpen Keywords: hybrid Bi-CG methods; non-symmetric linear systems; bi-conjugate gradient method; numerical stability; numerical examples Related Software: CGS; GpBiCg; mctoolbox; SparseMatrix; Algorithm 913; MatrixMarket; JDQZ; QMRPACK; na1; Matlab; JDQR; IDRStab; GMRES; PETSc; na5; INMOST; LAPACK; CUDA; Harwell-Boeing sparse matrix collection; KELLEY Cited in: 131 Publications

### Standard Articles

1 Publication describing the Software, including 1 Publication in zbMATH Year
BiCGstab($$l$$) and other hybrid Bi-CG methods. Zbl 0810.65027
Sleijpen, G. L. G.; van der Vorst, H. A.; Fokkema, D. R.
1994
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### Cited by 231 Authors

 16 Sleijpen, Gerard L. G. 15 Van der Vorst, Henk Albertus 7 Abe, Kuniyoshi 6 Gutknecht, Martin H. 6 Van Gijzen, Martin Bastiaan 5 Carson, Erin Claire 5 Strakoš, Zdeněk 5 van den Eshof, Jasper 4 Aihara, Kensuke 4 Huang, Ting-Zhu 4 Terekhov, Kirill M. 3 Chan, Tony Fan-Cheong 3 de Sturler, Eric 3 Graves-Morris, Peter R. 3 Gu, Xian-Ming 3 Sonneveld, Peter 3 Tichý, Petr 3 Zemke, Jens-Peter Max 2 Ahuja, Kapil 2 Akhunov, R. R. 2 Astudillo, Reinaldo 2 Carpentieri, Bruno 2 Chen, Zhangxin 2 Demmel, James Weldon 2 Fokkema, Diederik R. 2 Gazizov, T. R. 2 Imakura, Akira 2 Ishiwata, Emiko 2 Klawonn, Axel 2 Kuksenko, S. P. 2 Li, Liang 2 Liang, Yu 2 Liu, Hui 2 Lowengrub, John Samuel 2 Macklin, Paul 2 Meerbergen, Karl 2 Rozložník, Miroslav 2 Simoncini, Valeria 2 Szyld, Daniel B. 2 Turkel, Eli L. 2 Weiss, Rüdiger 2 Wittum, Gabriel 2 Ye, Qiang 2 Zhang, Shao-Liang 1 Agullo, Emmanuel 1 Alavani, Chitra 1 Alexiou, Terpsichori S. 1 Amritkar, Amit 1 Ang, E. Y. M. 1 Babich, R. 1 Bai, Zhaojun 1 Bai, Zhongzhi 1 Ballard, Grey M. 1 Banjai, Lehel 1 Barros, Kipton 1 Baumann, Manuel 1 Benner, Peter 1 Bermel, Peter 1 Berrill, Mark A. 1 Bertaccini, Daniele 1 Bertolazzi, Enrico 1 Bilotta, Giuseppe 1 Bogachëv, K. Yu. 1 Borges, António 1 Botta, Eugen F. F. 1 Brändle de Motta, Jorge César 1 Brezinski, Claude 1 Brower, Richard C. 1 Bruyelle, Jérémie 1 Cai, Wenjun 1 Centorrino, Veronica 1 Chang, Hyuk-Jae 1 Chen, Tyler 1 Chertovskih, Roman A. 1 Chung, Peter W. 1 Clark, Michael A. 1 Clarno, Kevin T. 1 Clemens, Markus 1 Climent, Éric 1 Cools, Siegfried 1 Cullum, Jane K. 1 Dalrymple, Robert A. 1 Darmofal, David L. 1 De Rossi, Alessandra 1 del Negro, Ciro 1 Deng, GanBo 1 Depillis, Lisette 1 Didier, Eric 1 Djodom, Landry Fokoua 1 Dongarra, Jack J. 1 Engwer, Christian 1 Erlangga, Yogi A. 1 Estivalezes, Jean-Luc 1 Evans, Thomas M. 1 Feng, Lihong 1 Filelis-Papadopoulos, Christos K. 1 Fischer, Bernd 1 Frego, Marco 1 Frommer, Andreas 1 Fujino, Seiji ...and 131 more Authors
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### Cited in 50 Serials

 12 Journal of Computational and Applied Mathematics 11 SIAM Journal on Scientific Computing 10 SIAM Journal on Matrix Analysis and Applications 9 Applied Numerical Mathematics 7 Journal of Computational Physics 6 Numerical Algorithms 4 Computers and Fluids 4 Computer Physics Communications 3 Applied Mathematics and Computation 3 BIT 3 Numerical Linear Algebra with Applications 2 Computers & Mathematics with Applications 2 Computer Methods in Applied Mechanics and Engineering 2 Mathematical and Computer Modelling 2 Journal of Scientific Computing 2 Japan Journal of Industrial and Applied Mathematics 2 Journal of Mathematical Sciences (New York) 2 ETNA. Electronic Transactions on Numerical Analysis 2 Computational Geosciences 2 Acta Numerica 1 IMA Journal of Numerical Analysis 1 International Journal of Engineering Science 1 Journal of Engineering Mathematics 1 Journal of Fluid Mechanics 1 Moscow University Mathematics Bulletin 1 Computing 1 Mathematics and Computers in Simulation 1 SIAM Journal on Scientific and Statistical Computing 1 Physica D 1 Applications of Mathematics 1 M$$^3$$AS. Mathematical Models & Methods in Applied Sciences 1 International Journal of Computer Mathematics 1 Linear Algebra and its Applications 1 Computational Optimization and Applications 1 Advances in Computational Mathematics 1 Mathematical Problems in Engineering 1 The Asian Journal of Mathematics 1 Engineering Computations 1 International Journal of Applied Mathematics 1 Journal of Computational and Applied Mechanics 1 Journal of Numerical Mathematics 1 International Journal of Computational Methods 1 European Series in Applied and Industrial Mathematics (ESAIM): Mathematical Modelling and Numerical Analysis 1 Classics in Applied Mathematics 1 Studies in Computational Mathematics 1 Cambridge Monographs on Applied and Computational Mathematics 1 Software - Environments - Tools 1 Communications in Computational Physics 1 Journal of Theoretical Biology 1 JSIAM Letters
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### Cited in 19 Fields

 114 Numerical analysis (65-XX) 19 Fluid mechanics (76-XX) 17 Partial differential equations (35-XX) 7 Linear and multilinear algebra; matrix theory (15-XX) 5 Mechanics of deformable solids (74-XX) 4 Biology and other natural sciences (92-XX) 3 Optics, electromagnetic theory (78-XX) 3 Quantum theory (81-XX) 2 Ordinary differential equations (34-XX) 2 Computer science (68-XX) 2 Statistical mechanics, structure of matter (82-XX) 2 Geophysics (86-XX) 2 Operations research, mathematical programming (90-XX) 2 Systems theory; control (93-XX) 1 General and overarching topics; collections (00-XX) 1 Approximations and expansions (41-XX) 1 Calculus of variations and optimal control; optimization (49-XX) 1 Classical thermodynamics, heat transfer (80-XX) 1 Information and communication theory, circuits (94-XX)