An Enhanced Numerical Iterative Method for Expanding the Attraction Basins When Computing Matrix Signs of Invertible Matrices

The computation of the sign function of a matrix plays a crucial role in various mathematical applications. It provides a matrix-valued mapping that determines the sign of each eigenvalue of a nonsingular matrix. In this article, we present a novel iterative algorithm designed to efficiently calculate the sign of an invertible matrix, emphasizing the enlargement of attraction basins. The proposed solver exhibits convergence of order four, making it highly efficient for a wide range of matrices. Furthermore, the method demonstrates global convergence properties. We validate the theoretical outcomes through numerical experiments, which confirm the effectiveness and efficiency of our proposed algorithm.