The Convergence of Generalized Lanczos Methods for Large Unsymmetric Eigenproblems
The Convergence of Generalized Lanczos Methods for Large Unsymmetric Eigenproblems
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DOI：
10.1137/s0895479893246753
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发表时间：
1995-07
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影响因子：
--
通讯作者：
Zhongxiao Jia
中科院分区：
文献类型：
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作者：
Zhongxiao Jia
本文研究了求解大型非对称矩阵特征值问题的广义Lanczos方法的收敛性理论。建立了归一化特征向量与Krylov子空间${\calK}_m（v_1，A）$（由$v_1，Av_1，\ldots，A^{m-1}v_1$构成）之间的距离的界，并给出了当矩阵为亏损矩阵时特征元素的先验理论误差界.使用它们，我们表明，该方法仍然有利于外部的部分特征值和相关的特征向量的$A$通常，虽然他们可能会收敛得很慢的情况下，$A$是有缺陷的。同时，我们分析了收敛速度与$A$的谱之间的关系。然而，详细的分析表明，由广义Lanczos方法得到的任何非对称矩阵的近似特征向量，Ritz向量，不能保证在理论上收敛，即使近似特征值，Ritz值。因此，广义Lanczos算法，包括Arnoldi算法和修正的IOM，提供了必要的理论背景。
In this paper, we investigate the convergence theory of generalized Lanczos methods for solving the eigenproblems of large unsymmetric matrices. Bounds for the distances between normalized eigenvectors and the Krylov subspace ${\cal K}_m(v_1,A)$ spanned by $v_1, Av_1, \ldots, A^{m-1}v_1$ are established, and a priori theoretical error bounds for eigenelements are presented when matrices are defective. Using them we show that the methods will still favor the outer part eigenvalues and the associated eigenvectors of $A$ usually though they may converge quite slowly in the case of $A$ being defective. Meanwhile, we analyze the relationships between the speed of convergence and the spectrum of $A$. However, a detailed analysis exposes that the approximate eigenvectors, Ritz vectors, obtained by generalized Lanczos methods for any unsymmetric matrix cannot be guaranteed to converge in theory even if approximate eigenvalues, Ritz values, do. Therefore, generalized Lanczos algorithms including Arnoldi's algorithm and IOMs with correction are provided with necessary theoretical background.