矩阵伽马函数的计算
Computation of matrix gamma function
AI总结：
本文研究了矩阵伽马函数的数值计算，扩展了标量伽马函数的Lanczos和Spouge方法，并提出基于递归伽马函数的新方法，通过数值实例分析了不同方法的优缺点及误差界。
摘要：
具有潜在应用的矩阵函数在科学和工程中起着重要作用。其中，由于其与某些矩阵微分方程和其他特殊矩阵函数的联系，矩阵伽马函数是基本的矩阵函数之一。本文聚焦于该函数的数值计算。已知的标量伽马函数计算技术，如Lanczos和Spouge方法，被仔细扩展到矩阵情况。这种扩展带来了许多挑战性问题，几种用于计算矩阵函数的方法，如Schur分解和块Parlett递归，需要被纳入以使方法更有效。我们还提出了一种基于递归伽马函数的第三种技术，该技术在精度上与前两种方法相当，同时具有丰富的矩阵乘法优势。所提出方法的优缺点通过一组数值示例进行了说明。还将讨论截断误差界及其他与矩阵伽马函数相关的界。
英文摘要：
Matrix functions with potential applications have a major role in science and engineering. One of the fundamental matrix functions, which is particularly important due to its connections with certain matrix differential equations and other special matrix functions, is the matrix gamma function. This research article is focused on the numerical computation of this function. Well-known techniques for the scalar gamma function, such as Lanczos and Spouge methods, are carefully extended to the matrix case. This extension raises many challenging issues and several strategies used in the computation of matrix functions, like Schur decomposition and block Parlett recurrences, need to be incorporated to turn the methods more effective. We also propose a third technique based on the reciprocal gamma function that is shown to be competitive with the other two methods in terms of accuracy, with the advantage of being rich in matrix multiplications. Strengths and weaknesses of the proposed methods are illustrated with a set of numerical examples. Bounds for truncation errors and other bounds related with the matrix gamma function will be discussed as well.
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