×

A variable preconditioned GCR\((m)\) method using the GSOR method for singular and rectangular linear systems. (English) Zbl 1188.65051

Summary: The generalized conjugate residual (GCR) method with a variable preconditioning is an efficient method for solving a large sparse linear system \(Ax=b\). It has been clarified by some numerical experiments that the successive over relaxation (SOR) method is more effective than Krylov subspace methods such as GCR and ILU(0) preconditioned GCR for performing the variable preconditioning. However, SOR cannot be applied for performing the variable preconditioning when solving such linear systems that the coefficient matrix has diagonal entries of zero or is not square. Therefore, we propose a type of the generalized SOR (GSOR) method. By numerical experiments on the singular linear systems, we demonstrate that the variable preconditioned GCR using GSOR is effective.

MSC:

65F20 Numerical solutions to overdetermined systems, pseudoinverses
65F08 Preconditioners for iterative methods

Software:

MIQR
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Bruaset, A. M., (A survey of Preconditioned Iterative Methods. A survey of Preconditioned Iterative Methods, Frontiers in Applied Mathematics, vol. 17 (1995), Longman Scientific and Technical: Longman Scientific and Technical London) · Zbl 0834.65014
[2] Meijerink, J. A.; van der Vorst, H. A., Iterative solution method for linear systems of which the coefficient matrix is a symmetric \(M\)-matrix, Math. Comp., 31, 148-162 (1977) · Zbl 0349.65020
[3] Saad, Y., A flexible inner-outer preconditioned GMRES algorithm, SIAM J. Sci. Stat. Comput., 14, 461-469 (1993) · Zbl 0780.65022
[4] van der Vorst, H. A.; Vuik, C., GMRESR: A family of nested GMRES methods, Numer. Linear Algebra Appl., 1, 369-386 (1994) · Zbl 0839.65040
[5] Saad, Y.; Schultz, M. H., GMRES: A generalized minimal residual algorithm for solving nonsymmetric linear systems, SIAM J. Sci. Stat. Comput., 7, 856-869 (1986) · Zbl 0599.65018
[6] Eisenstat, S. C.; Elman, H. C.; Schultz, M. H., Variational iterative methods for nonsymmetric systems of linear equations, SIAM J. Numer. Anal., 20, 345-357 (1983) · Zbl 0524.65019
[7] Abe, K.; Zhang, S.-L., A variable preconditioning using the SOR Method for GCR-like methods, Int. J. Numer. Anal. Model., 2, 147-161 (2005) · Zbl 1131.65030
[8] Buoni, J. J.; Varga, R. S., Theorems of Stein-Rosenberg type, (Ansorge, R.; Glanshoff, K.; Werner, B., Numerical Mathematics (1979), Birkhäuser Verlag: Birkhäuser Verlag Basel), 65-75 · Zbl 0412.65016
[9] Buoni, J. J.; Varga, R. S., Theorems of Stein-Rosenberg type II, Optimum paths of relaxation in the complex domain, (Schultz, M. H., Elliptic Problem Solvers (1981), Academic Press: Academic Press New York), 231-240
[10] Hadjidimos, A., On the optimization of the classical iterative schemes for the solution of complex singular linear systems, SIAM J. Alg. Disc. Meth., 6, 555-566 (1985) · Zbl 0582.65017
[11] Varga, R. S., Matrix Iterative Analysis (2000), Springer-Verlag: Springer-Verlag Berlin · Zbl 0133.08602
[12] Young, D. M., Iterative Solution of Large Linear Systems (1971), Academic Press: Academic Press NewYork and London · Zbl 0204.48102
[15] Reichel, L.; Ye, Q., Breakdown-free GMRES for singular systems, SIAM J. Matrix Anal. Appl., 26, 1001-1021 (2005) · Zbl 1086.65030
[16] Li, N.; Saad, Y., MIQR: A multilevel incomplete QR preconditioner for large sparse least-square problems, SIAM J. Matrix Anal. Appl., 28, 524-550 (2006) · Zbl 1113.65036
[17] Zhang, S.-L.; Oyanagi, Y., Orthomin(k) method for linear least squares problem, J. Inform. Process., 14, 121-125 (1991) · Zbl 0767.65036
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.