Construction of stable and globally convergent schemes for the matrix sign function

Abstract

In this paper, we propose a dynamics-based approach to develop efficient matrix iterations for computing the matrix sign function. Considering an iterative method with unknown weight functions, our approach constructs a double sequence of weight functions so that for a quadratic polynomial with distinct roots, the Julia set of the method associated with each term of the sequence is the perpendicular bisector of the line segment from one root to another. Based on the sequence-generated methods, we introduce a double sequence of matrix iterations for computing the matrix sign function, and perform precise convergence and stability analyses of these iterations. We show that our matrix iterations contain the well-known globally convergent principal Padé family and its reciprocal as special cases. Based on the result of the numerical experiments, we found several of our methods that do not belong to the principal Padé family and its reciprocal, which are more efficient in computing the matrix sign function.