Computation of matrix gamma function

Abstract

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.Since the reciprocal gamma function, here denoted by ∆(z) := 1 Γ(z) , is an entire function, for any matrix A ∈ C n×n , the n × n matrix ∆(A) = (Γ(A)) −1 is a well defined matrix. Furthermore, if A does not have any non-positive integer eigenvalue, then A + mI is invertible, for all integer m ≥ 0, and one gets the following formula [28], which uses the Pochhammer notation: (A) m = A(A + I) . . . (A + (m − 1)I) = Γ(A + mI)∆(A), m ≥ 1 and (A) 0 = I. An alternative definition of the matrix gamma function, as a limit of a sequence of matrices, is provided in [28]: Γ(A) = lim m→∞ (m − 1)! [(A) m ] −1 m A .We shall note that many definitions of the scalar gamma function (see, for instance, [1, Ch. 6]) may be easily extended to the matrix case. The matrix gamma function has connections with other special functions, which in turn play an important role to solving certain matrix differential equations; see [28] and the references therein. Two of those special functions are the matrix beta and Bessel functions. If the matrices A, B ∈ C n×n satisfy the spectral conditions [16] Re(z) > −1, ∀z ∈ σ(A), Re(z) < 1, ∀z ∈ σ(B), then 1 −1 (1 + t) A (1 − t) B dt = 2 A+I B(A + I, B + I) 2 B , where B(A, B) is the Beta matrix function [28], defined by

Verified computation of matrix gamma function

“…Here, the word "computable" means that we can numerically obtain a rigorous upper bound which takes rounding and truncation errors into account. We can find a perturbation bound for Γ(A) also in [2]. On the other hand, the bound in [2] is not a computable one.…”

“…where t A−In := e (A−In) log(t) and I n denotes the n × n identity matrix. The function Γ(A) has connections with other special functions, which play an important role in solving certain matrix differential equations [2]. Two of these special functions are the matrix beta and Bessel functions.…”

“…Two of these special functions are the matrix beta and Bessel functions. In [2], mathematical properties of Γ(A) are elegantly clarified, and fast and accurate algorithms for computing Γ(A) are proposed.…”

A Technique for Improving the Computation of Functions of Triangular Matrices

“…To illustrate the gain that preconditioning a matrix may bring, let us consider 1). We finish this section by noticing that preconditioning the original matrix may also bring benefits if combined with the MATLAB function funm, which implements the Schur-Parlett method of [6] for computing several matrix functions, and with methods for computing special functions [5,9].…”

The gamma function via interpolation

“…This sum is given in terms of the Bernoulli numbers 4 B 2k = (−1) k+1 2ζ(2k)(2k)!/ (2π) 2k , which grow quickly with increasing k and render the series divergent for finite z. From the standpoint of numerical computation, many techniques have been considered [9,13,19,21,22,24,31,33,34,[36][37][38]40] (we aim to provide merely a representative, but by no means comprehensive view of the literature). The approaches can be categorically sorted into (i) asymptotic expansions, (ii) rational functions, and (iii) numerical quadrature.…”