线性算子联合谱特征的精确计算
Exact computation of joint spectral characteristics of linear operators
摘要：
我们解决的问题的精确计算的两个联合谱特征的一个家庭的线性算子，联合谱半径（简称JSR）和下谱半径（简称LSR），这是众所周知的不同的推广到一组运营商的通常谱半径的线性算子。在这篇文章中，我们开发了一种方法，它-在适当的假设下-允许计算的JSR和LSR的一个有限的家庭的矩阵。我们注意到，到目前为止，在文献中没有算法可以精确地计算LSR。本文提出了必要的理论结果极值范数（和极值反范数）的线性算子，这构成了我们的程序的基本工具，并详细描述了相应的算法计算的JSR和LSR（最后一个限制家庭共享不变锥）。算法实现简单，描述简短.如果算法在有限时间内终止，则它们构造极值范数（在JSR情况下）或逆范数（在LSR情况下）并找到它们的精确值;否则它们提供都收敛到精确值的上界和下界。在有限时间内终止的理论准则也被导出。根据数值实验，JSR的算法发现，在尺寸小于20的矩阵族的绝大多数的确切值。对于非负矩阵，它工作得更快，并且在几次迭代内找到100阶的JSR;对于计算LSR的算法也是如此。为了说明新方法的效率，我们能够应用它，以回答最近在组合学，数论和形式语言理论中提出的几个问题。
英文摘要：
We address the problem of the exact computation of two joint spectral characteristics of a family of linear operators, the joint spectral radius (in short JSR) and the lower spectral radius (in short LSR), which are well-known different generalizations to a set of operators of the usual spectral radius of a linear operator. In this article we develop a method which - under suitable assumptions - allows to compute the JSR and the LSR of a finite family of matrices exactly. We remark that so far no algorithm was available in the literature to compute the LSR exactly. The paper presents necessary theoretical results on extremal norms (and on extremal antinorms) of linear operators, which constitute the basic tools of our procedures, and a detailed description of the corresponding algorithms for the computation of the JSR and LSR (the last one restricted to families sharing an invariant cone). The algorithms are easily implemented and their descriptions are short. If the algorithms terminate in finite time, then they construct an extremal norm (in the JSR case) or antinorm (in the LSR case) and find their exact values; otherwise they provide upper and lower bounds that both converge to the exact values. A theoretical criterion for termination in finite time is also derived. According to numerical experiments, the algorithm for the JSR finds the exact value for the vast majority of matrix families in dimensions less than 20. For nonnegative matrices it works faster and finds JSR in dimensions of order 100 within a few iterations; the same is observed for the algorithm computing the LSR. To illustrate the efficiency of the new method we are able to apply it in order to give answers to several conjectures which have been recently stated in combinatorics, number theory, and the theory of formal languages.
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