Computing the square roots of matrices with central symmetry

摘要

For computing square roots of a nonsingular matrix A, which are functions of A, two well known fast and stable algorithms, which are based on the Schur decomposition of A, were proposed by Björk and Hammarling [A. Björk, S. Hammarling, A Schur method for the square root of a matrix, Linear Alg. Appl. 52/53 (1983) 127–140], for square roots of general complex matrices, and by Higham [N.J. Higham, Computing real square roots of a real matrix, Linear Alg. Appl. 88/89 (1987) 405–430], for real square roots of real matrices. In this paper we further consider (the computation of) the square roots of matrices with central symmetry. We first investigate the structure of the square roots of these matrices and then develop several algorithms for computing the square roots. We show that our algorithms ensure significant savings in computational costs as compared to the use of standard algorithms for arbitrary matrices.