A framework for analyzing nonlinear eigenproblems and parametrized linear systems
A framework for analyzing nonlinear eigenproblems and parametrized linear systems
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DOI：
10.1016/j.laa.2009.12.038
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发表时间：
2011-08
影响因子：
1.1
通讯作者：
L. Grammont;N. Higham;F. Tisseur
中科院分区：
文献类型：
--
作者：
L. Grammont;N. Higham;F. Tisseur
与n×n阶矩阵多项式P（λ）=∑j=0 <$λjAj相关联的是特征值问题P（λ）x=0和线性系统问题P（ω）x=B，其中在后一种情况下，x是针对参数ω的许多值计算的。这两个问题都可以通过转换为等价的问题L（λ）z=0或L（ω）z=c来解决，该问题在参数λ或ω中是线性的。这种线性化过程近年来在特征值问题中受到了很大的关注，但在线性系统问题中却没有得到很好的理解。我们开发了一个框架，在其中可以分析这两个问题的更一般版本，基于将一般非线性矩阵函数N（λ）连接到更简单的函数M（λ）（通常是1次或2次多项式）的单侧因子分解。我们的分析将原始问题和低次问题的解联系起来，并在线性系统的情况下指出如何选择右侧c并从z恢复解x。对于特征值问题，这个框架包括许多文献中研究的特殊情况，包括最近由Mackey，Mackey，Mehl和Mehrmann引入的铅笔L1（P）和L2（P）的向量空间以及一类理性问题。我们使用这个框架来研究参数化线性系统P（ω）x=B的条件性和稳定性，从而研究原始多项式和束L的尺度效应.我们的研究结果确定的情况下，缩放可以大大提高空调和稳定性和我们的数值结果表明，在实践中可以实现显着的改善。
Associated with an n×n matrix polynomial of degree ℓ,P(λ)=∑j=0ℓλjAj, are the eigenvalue problem P(λ)x=0 and the linear system problem P(ω)x=b, where in the latter case x is to be computed for many values of the parameter ω. Both problems can be solved by conversion to an equivalent problem L(λ)z=0 or L(ω)z=c that is linear in the parameter λ or ω. This linearization process has received much attention in recent years for the eigenvalue problem, but it is less well understood for the linear system problem. We develop a framework in which more general versions of both problems can be analyzed, based on one-sided factorizations connecting a general nonlinear matrix function N(λ) to a simpler function M(λ), typically a polynomial of degree 1 or 2. Our analysis relates the solutions of the original and lower degree problems and in the linear system case indicates how to choose the right-hand side c and recover the solution x from z. For the eigenvalue problem this framework includes many special cases studied in the literature, including the vector spaces of pencils L1(P) and L2(P) recently introduced by Mackey, Mackey, Mehl, and Mehrmann and a class of rational problems. We use the framework to investigate the conditioning and stability of the parametrized linear system P(ω)x=b and thereby study the effect of scaling, both of the original polynomial and of the pencil L. Our results identify situations in which scaling can potentially greatly improve the conditioning and stability and our numerical results show that dramatic improvements can be achieved in practice.