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 that 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, and so on. In this article, we propose a modification of the invariant polytope algorithm making it roughly 3 times faster (single threaded), suitable for higher dimensions, and parallelise it. The modified version works for most matrix families of dimensions up to 25, for non-negative matrices up to 3,000. In addition, we introduce a new, fast algorithm, called modified Gripenberg 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 up to order 42 and the capacities of codes that avoid certain difference patterns.

中文翻译：

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算法 1011

在 2013-2016 年的几篇论文中，Guglielmi 和 Protasov 在联合谱半径计算问题上取得了突破，开发了不变多面体算法，可以为大多数矩阵族找到联合谱半径的精确值。该算法在泛函分析、逼近论、组合学等问题中有很多应用。在本文中，我们提出了对不变多面体算法的修改，使其大约快 3 倍（单线程），适用于更高的维度，并将其并行化。修改后的版本适用于维数高达 25 的大多数矩阵族，以及高达 3,000 的非负矩阵。此外，我们引入了一种新的快速算法，称为改进的 Gripenberg 算法，用于计算联合光谱半径的良好下界。提供了相应的例子和数值结果的统计数据。介绍了我们算法的几种应用。特别是，我们找到了高达 42 阶的 Daubechies 小波的正则指数的精确值以及避免某些差异模式的代码的容量。