Barabanov范数的迭代构建与矩阵集合联合谱半径的计算
Iterative building of Barabanov norms and computation of the joint spectral radius for matrix sets
AI总结：
本文提出两种迭代方法，用于数值构建不可约矩阵集合的Barabanov范数并同时计算其联合谱半径，填补了Barabanov范数单位球几何性质分析的空白。
摘要：
关于构建Barabanov范数以分析矩阵集合的联合（广义）谱半径性质的问题已在多篇文献中讨论。Barabanov范数方法是反驳Lagarias-Wang有限性猜想的关键工具。相关构造主要基于对某些特定Barabanov范数单位球的几何性质的研究。在此背景下，当前文献中未能找到对Barabanov范数单位球几何性质的详细分析，这看起来有些矛盾。部分原因在于Barabanov范数是通过隐式程序非构造性地定义的。因此，即使在最简单的情况下，也很难可视化其单位球的形状。本文可视为弥补这一缺陷的第一步。文中考虑了两种迭代过程，允许数值构建不可约矩阵集合的Barabanov范数，并同时计算这些集合的联合谱半径。
英文摘要：
The problem of construction of Barabanov norms for analysis of properties of the joint (generalized) spectral radius of matrix sets has been discussed in a number of publications. The method of Barabanov norms was the key instrument in disproving the Lagarias-Wang Finiteness Conjecture. The related constructions were essentially based on the study of the geometrical properties of the unit balls of some specific Barabanov norms. In this context the situation when one fails to find among current publications any detailed analysis of the geometrical properties of the unit balls of Barabanov norms looks a bit paradoxical. Partially this is explained by the fact that Barabanov norms are defined nonconstructively, by an implicit procedure. So, even in simplest cases it is very difficult to visualize the shape of their unit balls. The present work may be treated as the first step to make up this deficiency. In the paper two iteration procedure are considered that allow to build numerically Barabanov norms for the irreducible matrix sets and simultaneously to compute the joint spectral radius of these sets.
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