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arXiv 1905.00749math.DScs.NAmath.NA
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快速计算p-半径、矩阵压力或广义李雅普诺夫指数用于正矩阵和主导矩阵

Fast approximation of the $p$-radius, matrix pressure or generalised Lyapunov exponent for positive and dominated matrices

Ian D. Morris

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AI总结：

本文提出了一种新的算法，用于计算p-半径，假设矩阵均为正矩阵或更一般地满足主导条件，该算法基于将p-半径视为希尔伯特空间上迹类算子的主特征值，并通过近似算子的弗雷德霍姆行列式来估计该特征值。

摘要：

如果A_1,...,A_N是实方阵，则p-半径、广义李雅普诺夫指数或矩阵压力被定义为$\sum_{i_1,\ldots,i_n=1}^N \|A_{i_n}\cdots A_{i_1}\|^p$的渐近指数增长速率，其中p是一个实参数。在各种名称下，这个量已被研究用于波let正则性和细化方程、分形几何和随机矩阵乘积的大偏差理论等应用。在本文中，我们提出了一种新的算法来计算p-半径，假设矩阵均为正矩阵，或更一般地假设它们满足一个称为主导的较弱条件。该算法基于将p-半径视为希尔伯特空间上迹类算子的主特征值，并通过近似算子的弗雷德霍姆行列式来估计该特征值。在这方面，我们的方法与Z.-Q. Bai和M. Pollicott关于计算随机矩阵乘积的顶部李雅普诺夫指数的工作密切相关。对于低维正矩阵的对，我们的方法在现有方法上取得了显著改进。

英文摘要：

If A_1,...,A_N are real square matrices then the p-radius, generalised Lyapunov exponent or matrix pressure is defined to be the asymptotic exponential growth rate of the sum $\sum_{i_1,\ldots,i_n=1}^N \|A_{i_n}\cdots A_{i_1}\|^p$, where p is a real parameter. Under its various names this quantity has been investigated for its applications to topics including wavelet regularity and refinement equations, fractal geometry and the large deviations theory of random matrix products. In this article we present a new algorithm for computing the p-radius under the hypothesis that the matrices are all positive, or more generally under the hypothesis that they satisfy a weaker condition called domination. This algorithm is based on interpreting the p-radius as the leading eigenvalue of a trace-class operator on a Hilbert space and estimating that eigenvalue via approximations to the Fredholm determinant of the operator. In this respect our method is closely related to the work of Z.-Q. Bai and M. Pollicott on computing the top Lyapunov exponent of a random matrix product. For pairs of positive matrices of low dimension our method yields substantial improvements over existing methods.

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