×

A COCR method for solving complex symmetric linear systems. (English) Zbl 1108.65028

Summary: The conjugate orthogonal conjugate gradient (COCG) method has been recognized as an attractive Lanczos-type Krylov subspace method for solving complex symmetric linear systems; however, it sometimes shows irregular convergence behavior in practical applications. In the present paper, we propose a conjugate A-orthogonal conjugate residual (COCR) method, which can be regarded as an extension of the conjugate residual (CR) method. Numerical examples show that COCR often gives smoother convergence behavior than COCG.

MSC:

65F10 Iterative numerical methods for linear systems
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Z. Bai, D. Day, J. Demmel, J. Dongarra, A test matrix collection for non-Hermitian eigenvalue problems, Technical Report CS-97-355, University of Tennessee, Knoxville, TN, 1997.; Z. Bai, D. Day, J. Demmel, J. Dongarra, A test matrix collection for non-Hermitian eigenvalue problems, Technical Report CS-97-355, University of Tennessee, Knoxville, TN, 1997.
[2] Bayliss, A.; Goldstein, C. I.; Turkel, E., An iterative method for the Helmholtz equation, J. Comput. Phys., 49, 443-457 (1983) · Zbl 0524.65068
[3] Bunse-Gerstner, A.; Stöver, R., On a conjugate gradient-type method for solving complex symmetric linear systems, Lin. Alg. Appl., 287, 105-123 (1999) · Zbl 0941.65031
[4] Clemens, M.; Weiland, T., Comparison of Krylov-type methods for complex linear systems applied to high-voltage problems, IEEE Trans. Mag., 34, 5, 3335-3338 (1998)
[5] R. Fletcher, Conjugate gradient methods for indefinite systems, Lecture Notes in Mathematics, vol. 506, Springer, Berlin, 1976, pp. 73-89.; R. Fletcher, Conjugate gradient methods for indefinite systems, Lecture Notes in Mathematics, vol. 506, Springer, Berlin, 1976, pp. 73-89. · Zbl 0326.65033
[6] Freund, R. W., Conjugate gradient-type methods for linear systems with complex symmetric coefficient matrices, SIAM J. Sci. Statist. Comput., 13, 425-448 (1992) · Zbl 0761.65018
[7] Golub, G. H.; Van Loan, C. F., Matrix Computations (1996), The Johns Hopkins University Press: The Johns Hopkins University Press Baltimore · Zbl 0865.65009
[8] Hestenes, M. R.; Stiefel, E., Methods of conjugate gradients for solving linear systems, J. Res. Nat. Bur. Standards, 49, 409-436 (1952) · Zbl 0048.09901
[9] Meijerink, J. A.; van der Vorst, H. A., An iterative solution method for linear systems of which the coefficient matrix is a symmetric M-matrix, Math. Comput., 31, 148-162 (1977) · Zbl 0349.65020
[10] Saad, Y., Iterative Methods for Sparse Linear Systems (2003), SIAM: SIAM Philadelphia, PA · Zbl 1002.65042
[11] Sogabe, T.; Zhang, S.-L., Extended conjugate residual methods for solving nonsymmetric linear systems, (Yuan, Y., Numerical Linear Algebra and Optimization (2004), Science Press: Science Press Beijing), 88-99
[12] van der Vorst, H. A.; Melissen, J. B.M., A Petrov-Galerkin type method for solving \(A x = b\), where \(A\) is symmetric complex, IEEE Trans. Mag., 26, 2, 706-708 (1990)
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.