na5 swMATH ID: 11489 Software Authors: Brezinski, C.; Redivo-Zaglia, M. Description: Treatment of near-breakdown in the CGS algorithm. Lanczos’ method for solving the system of linear equationsAx=b consists in constructing a sequence of vectors (x k ) such thatr k =b−Ax k =P k (A)r 0 wherer 0=b−Ax 0.P k is an orthogonal polynomial which is computed recursively. The conjugate gradient squared algorithm (CGS) consists in takingr k =P k 2 (A)r0. In the recurrence relation forP k , the coefficients are given as ratios of scalar products. When a scalar product in a denominator is zero, then a breakdown occurs in the algorithm. When such a scalar product is close to zero, then rounding errors can seriously affect the algorithm, a situation known as near-breakdown. In this paper it is shown how to avoid near-breakdown in the CGS algorithm in order to obtain a more stable method. Homepage: http://www.netlib.org/numeralgo/index.html Keywords: orthogonal polynomials; numerical examples; near-breakdown situation; conjugate gradient squared algorithm; CGS algorithm; Lanczos’ method; recurrence relations; method of recursive zoom algorithm Related Software: na1; CGS; BiCGstab; GpBiCg; ARPACK; JDQR; DSUBSP; LSQR; MA32; MA42; MA47; DRIC; ScaLAPACK; LAPACK; testmatrix; QMRPACK; JDQZ; eigs; SRRIT; RODAS Cited in: 24 Publications Standard Articles 1 Publication describing the Software, including 1 Publication in zbMATH Year Treatment of near-breakdown in the CGS algorithm. Zbl 0810.65028Brezinski, C.; Redivo-Zaglia, M. 1994 all top 5 Cited by 20 Authors 12 Brezinski, Claude 9 Redivo-Zaglia, Michela 5 Van der Vorst, Henk Albertus 4 Sadok, Hassane 2 Chan, Tony Fan-Cheong 2 Gutknecht, Martin H. 1 Chesneaux, Jean-Marie 1 Dongarra, Jack J. 1 Duff, Iain S. 1 Golub, Gene Howard 1 Matos, Ana Cristina 1 Nyman, Gunnar 1 Ressel, Klaus J. 1 Salam, Ahmed 1 Sleijpen, Gerard L. G. 1 Sorensen, Danny C. 1 Szeto, Tedd 1 Tichý, Petr 1 Yu, Hua-Gen 1 Zítko, Jan all top 5 Cited in 11 Serials 5 Numerical Algorithms 2 Numerische Mathematik 2 Applied Numerical Mathematics 2 Linear Algebra and its Applications 1 Computers & Mathematics with Applications 1 BIT 1 Journal of Computational and Applied Mathematics 1 SIAM Journal on Matrix Analysis and Applications 1 Applications of Mathematics 1 Journal of Computational Methods in Sciences and Engineering 1 Software - Environments - Tools all top 5 Cited in 9 Fields 24 Numerical analysis (65-XX) 2 Harmonic analysis on Euclidean spaces (42-XX) 1 History and biography (01-XX) 1 Linear and multilinear algebra; matrix theory (15-XX) 1 Approximations and expansions (41-XX) 1 Operator theory (47-XX) 1 Fluid mechanics (76-XX) 1 Quantum theory (81-XX) 1 Biology and other natural sciences (92-XX) Citations by Year