arXivDaily arXiv每日学术速递 周一至周五更新
arXiv 0906.2242math.NAcs.NA
收藏

一种改进的调和Lanczos双对角化方法和用于计算大型矩阵最小奇异三元组的隐式重启算法

A Refined Harmonic Lanczos Bidiagonalization Method and an Implicitly Restarted Algorithm for Computing the Smallest Singular Triplets of Large Matrices

Tsinghua University（清华大学）；Dalian University of Technology（大连理工大学）

Zhongxiao Jia, Datian Niu

备注：
31 pages, 14 figures, SIAM Journal on Scientific Computing, to appear
发表信息：
SIAM Journal on Scientific Computing, 32 (2) (2010): 714-744
阅读 PDF
arXiv 页面 ↗HTML 原文 ↗

AI总结：

针对调和Lanczos双对角化方法在计算大型矩阵最小奇异三元组时可能遗漏奇异值或收敛不稳定的问题，提出了一种改进的调和Lanczos双对角化方法，并开发了隐式重启算法，通过残差最小化提取最佳近似奇异向量，理论分析和数值实验表明新算法具有竞争力且更高效。

摘要：

调和Lanczos双对角化方法可用于计算大型矩阵$A$的最小奇异三元组。我们证明，对于足够好的投影子空间，如果$A$的列是强线性无关的，调和Ritz值收敛。另一方面，当$A$的列几乎线性相关时，调和Ritz值可能遗漏一些期望的奇异值。此外，调和Ritz向量可能不规则收敛甚至无法收敛。基于第一作者提出的大型矩阵特征问题的改进投影原理，我们提出了一种改进的调和Lanczos双对角化方法，该方法将调和Ritz向量的Rayleigh商作为近似奇异值，并从给定子空间中提取最佳近似奇异向量（称为改进的调和Ritz近似），其意义是残差最小化。我们证明，一旦子空间足够好且Rayleigh商收敛，改进的近似将收敛到期望的奇异向量。我们开发了一种隐式重启的改进调和Lanczos双对角化算法（IRRHLB）。我们研究了如何选择最佳可能的位移，并提出了改进的调和位移，理论上优于隐式重启Lanczos双对角化算法（IRHLB）中使用的调和位移。我们提出了一种新颖的过程，可以高效且准确地数值计算改进的调和位移。我们报告了数值实验，将IRRHLB与其他五种基于Lanczos双对角化过程的算法进行比较。结果表明，IRRHLB至少与它们相当，并且在计算最小奇异三元组时可以显著更高效。

英文摘要：

The harmonic Lanczos bidiagonalization method can be used to compute the smallest singular triplets of a large matrix $A$. We prove that for good enough projection subspaces harmonic Ritz values converge if the columns of $A$ are strongly linearly independent. On the other hand, harmonic Ritz values may miss some desired singular values when the columns of $A$ almost linearly dependent. Furthermore, harmonic Ritz vectors may converge irregularly and even may fail to converge. Based on the refined projection principle for large matrix eigenproblems due to the first author, we propose a refined harmonic Lanczos bidiagonalization method that takes the Rayleigh quotients of the harmonic Ritz vectors as approximate singular values and extracts the best approximate singular vectors, called the refined harmonic Ritz approximations, from the given subspaces in the sense of residual minimizations. The refined approximations are shown to converge to the desired singular vectors once the subspaces are sufficiently good and the Rayleigh quotients converge. An implicitly restarted refined harmonic Lanczos bidiagonalization algorithm (IRRHLB) is developed. We study how to select the best possible shifts, and suggest refined harmonic shifts that are theoretically better than the harmonic shifts used within the implicitly restarted Lanczos bidiagonalization algorithm (IRHLB). We propose a novel procedure that can numerically compute the refined harmonic shifts efficiently and accurately. Numerical experiments are reported that compare IRRHLB with five other algorithms based on the Lanczos bidiagonalization process. It appears that IRRHLB is at least competitive with them and can be considerably more efficient when computing the smallest singular triplets.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

↑