计算矩阵李雅普诺夫指数的凸优化方法
Convex Optimization methods for computing the Lyapunov Exponent of matrices
AI总结：
本文提出一种基于正齐次泛函的凸优化方法，用于计算非负矩阵族的最大李雅普诺夫指数，并推广到一般矩阵，给出新的通用上界。
摘要：
我们引入了一种新方法来评估非负矩阵族的最大李雅普诺夫指数。该方法基于在$R^{d}_+$上使用特殊的正齐次泛函，从而给出李雅普诺夫指数的迭代下界和上界。它们改进了先前已知的界，并收敛到真实值。估计了收敛速度，并通过来自应用（泛函分析、组合学和语言理论）的几个问题以及随机生成矩阵的数值示例证明了该算法的效率。该方法在相对较高的维度（高达60）下以指定精度计算李雅普诺夫指数。我们将此方法推广到所有矩阵（不一定是非负矩阵），推导出李雅普诺夫指数的一个新的通用上界，并表明这样的下界通常不存在。
英文摘要：
We introduce a new approach to evaluate the largest Lyapunov exponent of a family of nonnegative matrices. The method is based on using special positive homogeneous functionals on $R^{d}_+,$ which gives iterative lower and upper bounds for the Lyapunov exponent. They improve previously known bounds and converge to the real value. The rate of convergence is estimated and the efficiency of the algorithm is demonstrated on several problems from applications (in functional analysis, combinatorics, and lan- guage theory) and on numerical examples with randomly generated matrices. The method computes the Lyapunov exponent with a prescribed accuracy in relatively high dimensions (up to 60). We generalize this approach to all matrices, not necessar- ily nonnegative, derive a new universal upper bound for the Lyapunov exponent, and show that such a lower bound, in general, does not exist.
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