A Completed Theory of the Unsymmetric Lanczos Process and Related Algorithms. Part II
A Completed Theory of the Unsymmetric Lanczos Process and Related Algorithms. Part II
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DOI：
10.1137/s0895479890188803
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发表时间：
1994
期刊：
影响因子：
--
通讯作者：
M. Gutknecht
中科院分区：
文献类型：
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作者：
M. Gutknecht
本文是第一部分[M·H·古特克内希特，SIAM J·矩阵分析]的续篇。Appl.，13(1992)，pp.594--639]，其中非对称Lanczos双正交化(BO)算法的理论和相应的迭代方法BIORES被扩展到非一般情况。本文对双共轭梯度法(或Biomin法)和相关的BIODIR法进行了类似的推广。在这里，这些方法的缺陷也是可以治愈的。作为准备，给出了非正规Pade表中两个相邻对角线的形式正交多项式序列的混合递推公式，并给出了这些递推公式的矩阵解释。这种矩阵解释直接导致渐进QD算法的完整公式，在非正常Pade表的情况下也是有效的。最后，我们展示了精确击穿的曲线如何扩展到接近击穿，使得(在精确算术中)条件良好的形式正交多项式和相应的Krylov空间向量不依赖于指定接近击穿的阈值。
This paper is a continuation of Part I [M. H. Gutknecht, SIAM J. Matrix Anal. Appl., 13 (1992), pp. 594--639], where the theory of the "unsymmetric" Lanczos biorthogonalization (BO) algorithm and the corresponding iterative method BIORES for non-Hermitian linear systems was extended to the nongeneric case. The analogous extension is obtained here for the biconjugate gradient (or BIOMIN) method and for the related BIODIR method. Here, too, the breakdowns of these methods can be cured. As a preparation, mixed recurrence formulas are derived for a pair of sequences of formal orthogonal polynomials belonging to two adjacent diagonals in a nonnormal Pade table, and a matrix interpretation of these recurrences is developed. This matrix interpretation leads directly to a completed formulation of the progressive qd algorithm, valid also in the case of a nonnormal Pade table. Finally, it is shown how the cure for exact breakdown can be extended to near-breakdown in such a way that (in exact arithmetic) the well-conditioned formal orthogonal polynomials and the corresponding Krylov space vectors do not depend on the threshold specifying the near-breakdown.