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Zbl 0802.65038
Bank, Randolph E.; Chan, Tony F.
An analysis of the composite step biconjugate gradient method. (English)
[J] Numer. Math. 66, No.3, 295-319 (1993). ISSN 0029-599X; ISSN 0945-3245/e

The paper shows that $2\times 2$ composite steps can cure breakdowns in the biconjugate gradient matrix caused by (near) singularity of principal submatrices of the tridiagonal matrix generated by the underlying Lanczos process. In section 2, the factorization of general nonsingular tridiagonal matrices is considered. In section 3 and 4, the composite step biconjugate gradient method is derived, an analysis of its convergence is presented and a ``best approximation'' result is proved. In section 5, three of many possible implementations of the method are presented. Some illustrations showing the effect of roundoff error are given in the last section.
[Xie Shenquan (Xiangtan)]

MSC 2000:: *65F10 Iterative methods for linear systems

Keywords: iterative method; numerical examples; Lanczos process; tridiagonal matrices; composite step biconjugate gradient method; convergence; roundoff error

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