On High-Order Iterative Schemes for the Matrix <i>p</i>th Root Avoiding the Use of Inverses

Abstract

This paper is devoted to the approximation of matrix pth roots. We present and analyze a family of algorithms free of inverses. The method is a combination of two families of iterative methods. The first one gives an approximation of the matrix inverse. The second family computes, using the first method, an approximation of the matrix pth root. We analyze the computational cost and the convergence of this family of methods. Finally, we introduce several numerical examples in order to check the performance of this combination of schemes. We conclude that the method without inverse emerges as a good alternative since a similar numerical behavior with smaller computational cost is obtained.

Keywords

• matrix <i>p</i>th root
• inverse operator
• iterative method
• order of convergence
• stability
• semilocal convergence