, Volume 66, Issue 1, pp 295-319

An analysis of the composite step biconjugate gradient method

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Summary

The composite step biconjugate gradient method (CSBCG) is a simple modification of the standard biconjugate gradient algorithm (BCG) which smooths the sometimes erratic convergence of BCG by computing only a subset of the iterates. We show that 2×2 composite steps can cure breakdowns in the biconjugate gradient method caused by (near) singularity of principal submatrices of the tridiagonal matrix generated by the underlying Lanczos process. We also prove a “best approximation” result for the method. Some numerical illustrations showing the effect of roundoff error are given.

The work of this author was supported by the Office of Naval Research under contract N00014-89J-1440.
The work of this author was supported by the Office of Naval Research under contracts N00014-90-J-1695 and N00014-92-J-1890, the Department of Energy under, contract DE-FG03-87ER25307, the National Science Foundation under contracts ASC 90-03002 and ASC 92-01266, and the Army Research Office under contract DAAL03-91-G-0150. Part of this work was completed during a visit to the Computer Science Dept. The Chinese University of Hong Kong.