Verified computation of matrix gamma function

Abstract

ABSTRACT Two numerical algorithms are proposed for computing an interval matrix containing the matrix gamma function. In 2014, the author presented algorithms for enclosing all the eigenvalues and basis of invariant subspaces of . As byproducts of these algorithms, we can obtain interval matrices containing diagonal blocks whose spectrums are included in that of A. In this paper, we interpret the interval matrices containing the basis and blocks as a result of verified block diagonalization (VBD), and establish a new framework for enclosing matrix functions using the VBD. To achieve enclosure for the gamma function of the blocks, we derive computable perturbation bounds. We can apply these bounds if disks containing the spectrums of input matrices lie in the open right half plane. We incorporate matrix argument reductions (ARs) to force the input matrices to have this property, and develop theories for accelerating the ARs. The first algorithm uses the VBD based on a numerical spectral decomposition, and involves only cubic complexity if the total computational cost of the accelerated ARs is . The second algorithm adopts the VBD based on a numerical Jordan decomposition, is applicable even for defective matrices, and requires operations.