A Quasi-Minimal Residual Variant of the Bi-CGSTAB Algorithm for Nonsymmetric Systems
@article{Chan1994AQR, title={A Quasi-Minimal Residual Variant of the Bi-CGSTAB Algorithm for Nonsymmetric Systems}, author={Tony F. Chan and Efstratios Gallopoulos and Valeria Simoncini and Tedd Szeto and Charles H. Tong}, journal={SIAM J. Sci. Comput.}, year={1994}, volume={15}, pages={338-347} }
Motivated by a recent method of Freund [SIAM J. Sci. Comput., 14 (1993), pp. 470–482], who introduced a quasi-minimal residual (QMR) version of the conjugate gradients squared (CGS) algorithm, a QMR variant of the biconjugate gradient stabilized (Bi-CGSTAB) algorithm of van der Vorst that is called QMRCGSTAB, is proposed for solving nonsymmetric linear systems. The motivation for both QMR variants is to obtain smoother convergence behavior of the underlying method. The authors illustrate this…
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References
SHOWING 1-10 OF 24 REFERENCES
QMR: a quasi-minimal residual method for non-Hermitian linear systems
- Computer Science
- 1991
A novel BCG-like approach, the quasi-minimal residual (QMR) method, which overcomes the problems of BCG is presented and how BCG iterates can be recovered stably from the QMR process is shown.
Bi-CGSTAB: A Fast and Smoothly Converging Variant of Bi-CG for the Solution of Nonsymmetric Linear Systems
- MathematicsSIAM J. Sci. Comput.
- 1992
Numerical experiments indicate that the new variant of Bi-CG, named Bi- CGSTAB, is often much more efficient than CG-S, so that in some cases rounding errors can even result in severe cancellation effects in the solution.
Variants of BICGSTAB for Matrices with Complex Spectrum
- Computer ScienceSIAM J. Sci. Comput.
- 1993
The author presents for real nonsymmetric matrices a method BICGSTAB2 in which the second factor may have complex conjugate zeros, and versions suitable for complex matrices are given for both methods.
How Fast are Nonsymmetric Matrix Iterations?
- Computer ScienceSIAM J. Matrix Anal. Appl.
- 1992
It is shown that the convergence of CGN is governed by singular values and that of GMRES and CGS by eigenvalues or pseudo-eigenvalues, and the three methods are found to be fundamentally different.
CGS, A Fast Lanczos-Type Solver for Nonsymmetric Linear systems
- Computer Science
- 1989
A Lanczos-type method is presented for nonsymmetric sparse linear systems as arising from discretisations of elliptic partial differential equations. The method is based on a polynomial variant of…
Quasi-Kernel Polynomials and Convergence Results for Quasi-Minimal Residual Iterations
- Mathematics, Computer Science
- 1992
This paper derives bounds for the norms of quasi-kernel polynomials and applies these results to obtain convergence theorems both for the QMR method, and for a transpose-free variant of QMR, the TFQMR algorithm.
The convergence behaviour of preconditioned CG and CG-S in the presence of rounding errors
- Biology
- 1990
Incomplete Choleski decompositions and modified versions thereof are quite effective preconditioners for the conjugate gradients method and it is shown why the convergence behaviour of Conjugate Gradients-Squared can be, sometimes unacceptable, irregular.
A Transpose-Free Quasi-Minimal Residual Algorithm for Non-Hermitian Linear Systems
- Computer ScienceSIAM J. Sci. Comput.
- 1993
The biconjugate gradient method (BCG) for solving general non-Hermitian linear systems $Ax = b$ and its transpose-free variant, the conjugate gradients squared algorithm (CGS), both typically exhib...
Solution of Systems of Linear Equations by Minimized Iterations1
- Mathematics
- 1952
In an earlier publication [14] a method was described which generated the eigenvalues and eigenvectors of a matrix by a successive algorithm based on minimizations by least squares. The advantage of…
An Implementation of the Look-Ahead Lanczos Algorithm for Non-Hermitian Matrices
- Computer ScienceSIAM J. Sci. Comput.
- 1993
An implementation of a look-ahead version of the Lanczos algorithm that overcomes problems by skipping over those steps in which a breakdown or near-breakdown would occur in the standard process.