New Approach for the Inversion of Structured Matrices via Newton’s Iteration

Abstract

Newton’s iteration is a fundamental tool for numerical solutions of systems of equations. The well-known iteration ( ) 1 2 , 0 ii i X X I MX i + = − ≥ rapidly refines a crude initial approximation 0 X to the inverse of a general nonsingular matrix. In this paper, we will extend and apply this method to nn × structured matrices M , in which matrix multiplication has a lower computational cost. These matrices can be represented by their short generators which allow faster computations based on the displacement operators tool. However, the length of the generators is tend to grow and the iterations do not preserve matrix structure. So, the main goal is to control the growth of the length of the short displacement generators so that we can operate with matrices of low rank and carry out the computations much faster. In order to achieve our goal, we will compress the computed approximations to the inverse to yield a superfast algorithm. We will describe two different compression techniques based on the SVD and substitution and we will analyze these approaches. Our main algorithm can be applied to more general classes of structured matrices.