Rational Interpolation Methods for Symmetric Sylvester Equations

IF 0.8 4区数学 Q3 MATHEMATICS, APPLIED

Electronic Transactions on Numerical Analysis Pub Date : 2014-01-01 DOI:10.17617/2.2060858

P. Benner, T. Breiten

引用次数: 3

Abstract

We discuss low-rank approximation methods for large-scale symmetric Sylvester equations. Follow- ing similar discussions for the Lyapunov case, we introduce an energy norm by the symmetric Sylvester operator. Given a rank nr, we derive necessary conditions for an approximation being optimal with respect to this norm. We show that the norm minimization problem is related to an objective function based on the H2-inner product for sym- metric state space systems. This objective function leads to first-order optimality conditions that are equivalent to the ones for the norm minimization problem. We further propose an iterative procedure and demonstrate its efficiency by means of some numerical examples.

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对称Sylvester方程的有理插值方法

讨论了大规模对称Sylvester方程的低秩逼近方法。在对Lyapunov情况进行类似讨论之后，我们引入了对称Sylvester算子的能量范数。给定一个秩n，我们得到了关于这个范数的最优逼近的必要条件。我们证明了范数最小化问题与一个基于h2 -内积的对称状态空间系统的目标函数有关。这个目标函数导致一阶最优性条件，等同于范数最小化问题的条件。我们进一步提出了一种迭代方法，并通过一些数值算例证明了它的有效性。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

Electronic Transactions on Numerical Analysis 数学-应用数学

CiteScore

2.10

自引率

7.70%

发文量

审稿时长

6 months

期刊介绍： Electronic Transactions on Numerical Analysis (ETNA) is an electronic journal for the publication of significant new developments in numerical analysis and scientific computing. Papers of the highest quality that deal with the analysis of algorithms for the solution of continuous models and numerical linear algebra are appropriate for ETNA, as are papers of similar quality that discuss implementation and performance of such algorithms. New algorithms for current or new computer architectures are appropriate provided that they are numerically sound. However, the focus of the publication should be on the algorithm rather than on the architecture. The journal is published by the Kent State University Library in conjunction with the Institute of Computational Mathematics at Kent State University, and in cooperation with the Johann Radon Institute for Computational and Applied Mathematics of the Austrian Academy of Sciences (RICAM).