On a Newton-type family of high-order iterative methods for some matrix functions

Abstract

The main goal of this paper is to approximate some matrix functions by using a family of high-order Newton-type iterative methods. We analyse the semilocal convergence and the speed of convergence of these methods. Concerning stability, it is well known that even the simplified Newton method can be unstable. Despite it, we present stable versions of our family of algorithms for several matrix functions. We test numerically the methods: we check the numerical robustness and stability by considering matrices that are close to be singular and badly conditioned. We find algorithms of the family with better numerical behavior than the Newton and the Halley methods. These two algorithms are basically the iterative methods proposed in the literature to solve many of this type of problems.