A general approach for improving the Padé iterations for the matrix sign function

Abstract

The Padé family of iterations is a well-known set of methods used to compute the matrix sign function, which includes classical methods such as Newton’s method, the Newton–Schultz iteration, and Halley’s method. In this paper, we present a general approach to enhance the Padé iterations by choosing an arbitrary three-point family of methods based on weight functions. We determine the weight functions in a way that, for a complex quadratic with distinct roots, the three-point methods are conformally conjugate to a complex polynomial with as many parameters as desired. This approach leads to a multi-parameter family of iterations for the matrix sign function, which allows us to discover many new methods, including the Padé family of iterations as a special case. We provide a convergence and stability analysis of the multi-parameter family and conduct numerical experiments to confirm the improved performance of the new methods. Although the three-point family of methods is arbitrarily chosen, our approach can be easily extended to any other multipoint methods.