Computation of matrix gamma function

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.