na1 swMATH ID: 11516 Software Authors: Brezinski, C.; Redivo Zaglia, M.; Sadok, H. Description: Avoiding breakdown and near-breakdown in Lanczos type algorithms. The paper deals with methods which the authors have developed to avoid breakdown and near-breakdown due to division by a scalar product whose value is zero or is different from zero but small in Lanczos type algorithms for solving linear systems.par In particular, the bulk of the paper concentrates on a method called by the authors the method of recursive zoom (MRZ) and its variants: SMRZ, BMRZ, GMRZ and BSMRZ, where S, B, G stand for symmetric, balancing and general, respectively. It is shown that a breakdown can be avoided by considering only the existing orthogonal polynomials in the Lanczos type algorithms. The methods described in the paper are able to detect if such a polynomial does not exist in order to jump over it.par Pseudo-codes for MRZ and BSMRZ are given and some numerical results are presented. (netlib numeralg na1) Homepage: http://www.netlib.org/numeralgo/index.html Keywords: Lanczos method; biconjugate gradient; projection; orthogonal polynomials; methods of moments; method of recursive zoom Related Software: CGS; na5; BiCGstab; Harwell-Boeing sparse matrix collection; CMRH; Regularization tools; GpBiCg; QMRPACK; LSQR; LAPACK; mftoolbox; OPQ; blgaussexp; ABLE; Eigtool; mctoolbox; JDQZ; JDQR; eigs; BLAS Cited in: 57 Documents Standard Articles 1 Publication describing the Software, including 1 Publication in zbMATH Year Avoiding breakdown and near-breakdown in Lanczos type algorithms. Zbl 0748.65033Brezinski, C.; Redivo Zaglia, M.; Sadok, H. 1991 all top 5 Cited by 64 Authors 17 Brezinski, Claude 13 Redivo-Zaglia, Michela 11 Sadok, Hassane 5 Chan, Tony Fan-Cheong 4 Salhi, Abdellah 4 Van der Vorst, Henk Albertus 3 Farooq, Muhammad Umar 3 Ye, Qiang 2 Cao, Zhihao 2 Demmel, James Weldon 2 Dongarra, Jack J. 2 Freund, Roland W. 2 Gutknecht, Martin H. 2 Pozza, Stefano 2 Szeto, Tedd 2 Weiss, Rüdiger 1 Alqahtani, Hessah 1 Bai, Zhaojun 1 Bank, Randolph E. 1 Barrett, Richard K. 1 Berry, Michael W. 1 Bouyghf, F. 1 Broyden, Charles George 1 Bruaset, Are Magnus 1 Bultheel, Adhemar François 1 Carpentieri, Bruno 1 Chehab, Jean-Paul 1 Chesneaux, Jean-Marie 1 Cipolla, Stefano 1 Depillis, Lisette 1 Donato, June M. 1 Draux, André 1 Duan, Yong 1 Eijkhout, Victor L. 1 El Guennouni, Ahmed 1 Gemignani, Luca 1 Golub, Gene Howard 1 Graves-Morris, Peter R. 1 Greenbaum, Anne 1 Heyouni, Mohammed 1 Huang, Ting-Zhu 1 Jia, Zhongxiao 1 Jing, Yanfei 1 Larasati, Niken 1 Maharani, Maharani 1 Mashwani, Wali Khan 1 Matos, Ana Cristina 1 Meurant, Gérard A. 1 Nachtigal, Noël M. 1 Pozo, Roldan 1 Pranic, Miroslav S. 1 Reichel, Lothar 1 Romine, Charles H. 1 Ruhe, Axel 1 Saad, Yousef 1 Schönauer, Willi 1 Simoncini, Valeria 1 Szyld, Daniel B. 1 Tong, Charles H. 1 Triyani, Triyani 1 Ullah, Zakir 1 Van Barel, Marc 1 van Buggenhout, Niel 1 Yeniay, Ozgur all top 5 Cited in 18 Serials 17 Numerical Algorithms 5 Applied Numerical Mathematics 4 Journal of Computational and Applied Mathematics 3 Numerische Mathematik 3 SIAM Journal on Scientific Computing 3 The Journal of Prime Research in Mathematics 2 Computers & Mathematics with Applications 2 Linear Algebra and its Applications 1 Mathematics of Computation 1 BIT 1 Calcolo 1 Applicationes Mathematicae 1 Numerical Linear Algebra with Applications 1 Reliable Computing 1 Journal of Applied Mathematics 1 Hacettepe Journal of Mathematics and Statistics 1 Software - Environments - Tools 1 Princeton Series in Applied Mathematics all top 5 Cited in 10 Fields 56 Numerical analysis (65-XX) 4 Harmonic analysis on Euclidean spaces (42-XX) 3 Linear and multilinear algebra; matrix theory (15-XX) 3 Approximations and expansions (41-XX) 2 History and biography (01-XX) 1 General and overarching topics; collections (00-XX) 1 Partial differential equations (35-XX) 1 Operator theory (47-XX) 1 Computer science (68-XX) 1 Information and communication theory, circuits (94-XX) Citations by Year