Download:
|
by Charles H. Tong, Qiang Ye
Math. Comp
http://www-sccm.stanford.edu/pub/sccm/sccm95-11.ps.gz
Add To MetaCart
Abstract:
Abstract. In this paper we analyze the BiCG algorithm in finite precision arithmetic and suggest reasons for its often observed robustness. By using a tridiagonal structure, which is preserved by the finite precision BiCG iteration, we are able to bound its residual norm by a minimum polynomial of a perturbed matrix (i.e. the exact GMRES residual norm applied to a perturbed matrix) multiplied by some amplification factors. Furthermore, the same analysis can be applied to the CG algorithm and we are able to relate the slowing down of convergence to loss of orthogonality in finite precision arithmetic. Finally, numerical examples are given to gain insights into these bounds. Key words. Bi-conjugate gradient algorithm, error analysis, convergence analysis, nonsymmetric linear systems AMS subject classifications. 65F10, 65N20 1. Introduction. Since its introduction by Lanczos [14] and later re-discovery by Fletcher [5] in its present form, the bi-conjugate gradient (BiCG) algorithm has evolved many variations (e.g. CGS, BiCGSTAB, QMR, CSBCG [19, 21, 6, 2]), each of which was specially designed to overcome some of its inherent difficulties (the need for adjoint matrix vector product, potential breakdowns, erratic convergence behavior,
Citations
|
284
|
QMR: a quasi-minimal residual method for non-Hermitian linear systems
– Freund, Nachtigal
- 1991
|
|
174
|
der Vorst, Bi-CGSTAB: A fast and smoothly converging variant of Bi-CG for the solution of nonsymmetric linear systems
– van
- 1992
|
|
141
|
CGS, a fast Lanczos-type solver for nonsymmetric linear systems
– Sonneveld
- 1989
|
|
140
|
Solution of systems of linear equations by minimized iterations
– Lanczos
- 1952
|
|
123
|
Conjugate gradient methods for indefinite systems
– Fletcher
- 1975
|
|
69
|
Analysis and Implementation o/ an Implicitly Restarted Arnoldi Iteration
– Lehoucq
- 1995
|
|
38
|
Approximation theory and numerical linear algebra, in Algorithms for Approximation 11
– TREFETHEN
- 1990
|
|
34
|
Error analysis of the Lanczos algorithm for tridiagonalizing a symmetric matrix
– Paige
- 1976
|
|
33
|
Analysis of some Krylov subspace approximations to the matrix exponential operator
– Saad
- 1992
|
|
31
|
The convergence of inexact Chebyshev and Richardson iterative methods for solving linear systems
– Golub, Overton
- 1988
|
|
31
|
Behavior of slightly perturbed Lanczos and conjugate-gradient recurrences
– Greenbaum
- 1989
|
|
29
|
An analysis of the composite step biconjugate gradient method
– Bank, Chan
|
|
22
|
Accuracy and effectiveness of the Lanczos algorithm for symmetric eigenproblems
– Paige
- 1980
|
|
21
|
Error analysis of the Lanczos algorithm for the nonsymmetric eigenvalues problem
– Bai
- 1994
|
|
20
|
Predicting the behavior of finite precision Lanczos and conjugate gradient computations
– Greenbaum, Strako˘s
- 1992
|
|
15
|
A comparative study of preconditioned Lanczos methods for nonsymmetric linear systems
– TONG
- 1992
|
|
13
|
Variable metric conjugate gradient methods
– Barth, Manteuffel
- 1994
|
|
8
|
der Vorst, The convergence behaviour of preconditioned CG
– van
- 1990
|
|
8
|
A convergence analysis for nonsymmetric Lanczos algorithms
– Ye
- 1991
|
|
7
|
Convergence of a two-stage Richardson iterative procedure for solving systems of linear equations
– Golub, Overton
- 1982
|
|
7
|
On the convergence rate of the conjugate gradients in presence of rounding errors
– Notay
- 1993
|
|
5
|
Accuracy of Computed Solutions from Conjugate-Gradient-Like Methods
– Greenbaum
- 1994
|
|
2
|
Semi-duality in the two-sided Lanczos algorithm
– Day
- 1993
|
|
2
|
Private communication
– Greenbaum, Druskin, et al.
- 1973
|
|
2
|
Predicting the behaviour of finite precision Lanczos and Conjugate Gradient computations
– Greenbaum, Strakos
- 1992
|
|
1
|
Relation between Galerkian and norm-minimizing iterative methods for solving linear systems
– Cullum, Greenbaum
- 1996
|
|
1
|
A Talk given at the Linear and Nonlinear
– Manteuffel, Barth
- 1995
|
