Numerically stable iterative methods for computing matrix sign function

Abstract

The main objective of this study is to develop two novel iterative schemes for computing the sign of a matrix with no pure imaginary eigenvalues. Detailed convergence analysis and asymptotical stability of the proposed methods are discussed. It is shown that the new schemes converge globally by drawing attraction basins. In addition, we extend the obtained results to compute nontrivial solutions of the Yang-Baxter-like equation, provided the given matrix has no eigenvalues on the imaginary axis. To illustrate the effectiveness of theoretical developments, a class of numerical examples for matrices of various dimensions, including eigenvalue clustering, are worked out.