Composite-step product methods for solving nonsymmetric linear systems [electronic resource]
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| Online Access: |
Online Access (via OSTI) |
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| Corporate Author: | |
| Format: | Government Document Electronic eBook |
| Language: | English |
| Published: |
Oak Ridge, Tenn. :
distributed by the Office of Scientific and Technical Information, U.S. Department of Energy,
1994.
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| Subjects: |
| Abstract: | The Biconjugate Gradient (BCG) algorithm is the {open_quotes}natural{close_quotes} generalization of the classical Conjugate Gradient method to nonsymmetric linear systems. It is an attractive method because of its simplicity and its good convergence properties. Unfortunately, BCG suffers from two kinds of breakdowns (divisions by 0): one due to the non-existence of the residual polynomial, and the other due to a breakdown in the recurrence relationship used. There are many look-ahead techniques in existence which are designed to handle these breakdowns. Although the step size needed to overcome an exact breakdown can be computed in principle, these methods can unfortunately be quite complicated for handling near breakdowns since the sizes of the look-ahead steps are variable (indeed, the breakdowns can be incurable). Recently, Bank and Chan introduced the Composite Step Biconjugate Gradient (CSBCG) algorithm, an alternative which cures only the first of the two breakdowns mentioned by skipping over steps for which the BCG iterate is not defined. This is done with a simple modification of BCG which needs only a maximum look-ahead step size of 2 to eliminate the (near) breakdown and to smooth the sometimes erratic convergence of BCG. Thus, instead of a more complicated (but less prone to breakdown) version, CSBCG cures only one kind of breakdown, but does so with a minimal modification to the usual implementation of BCG in the hope that its empirically observed stability will be inherited. The authors note, then, that the Composite Step idea can be incorporated anywhere the BCG polynomial is used; in particular, in product methods such as CGS, Bi-CGSTAB, and TFQMR. Doing this not only cures the breakdown mentioned above, but also takes on the advantages of these product methods, namely, no multiplications by the transpose matrix and a faster convergence rate than BCG. |
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| Item Description: | Published through SciTech Connect. 12/31/1994. "conf-9404305--vol.2" "DE96005736" 17 6 ISSN 1064--8275. Colorado conference on iterative methods, Breckenridge, CO (United States), 5-9 Apr 1994. Chan, T.F.; Szeto, T. [Univ. of California, Los Angeles, CA (United States)] Front Range Scientific Computations, Inc., Boulder, CO (United States) USDOE, Washington, DC (United States) |
| Physical Description: | pp. 2, Paper 4 : digital, PDF file. |