改进的不变多面体算法及其应用
Improved invariant polytope algorithm and applications
摘要：
在2013 - 2016年的几篇论文中，Guglielmi和Protasov在联合谱半径计算的问题上取得了突破，开发了不变多面体算法，该算法对于大多数矩阵族都可以找到联合谱半径的精确值。该算法在泛函分析、逼近论、组合数学等问题中有许多应用。在本文中，我们提出了一个修改的不变多面体算法，使其大约3倍的速度，适合更高的维度。修改后的版本适用于大多数矩阵族的维数高达25，非负矩阵的维数高达三千。此外，我们介绍了一种新的，快速算法计算良好的联合谱半径的下限。给出了相应的算例和数值结果的统计。我们的算法的几个应用。特别是，我们发现的Daubechies小波的高阶和代码的能力，避免某些差异模式的正则性指数的确切值。
英文摘要：
In several papers of 2013 - 2016, Guglielmi and Protasov made a breakthrough in the problem of the joint spectral radius computation, developing the invariant polytope algorithm which for most matrix families finds the exact value of the joint spectral radius. This algorithm found many applications in problems of functional analysis, approximation theory, combinatorics, etc.. In this paper we propose a modification of the invariant polytope algorithm making it roughly 3 times faster and suitable for higher dimensions. The modified version works for most matrix families of dimensions up to 25, for non-negative matrices the dimension is up to three thousand. Besides we introduce a new, fast algorithm for computing good lower bounds for the joint spectral radius. The corresponding examples and statistics of numerical results are provided. Several applications of our algorithms are presented. In particular, we find the exact values of the regularity exponents of Daubechies wavelets of high orders and the capacities of codes that avoid certain difference patterns.
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