TITLE:
New Approach for the Inversion of Structured Matrices via Newton’s Iteration
AUTHORS:
Mohammad M. Tabanjeh
KEYWORDS:
Newton Iteration, Structured Matrices, Superfast Algorithm, Displacement Operators, Matrix Inverse.
JOURNAL NAME:
Advances in Linear Algebra & Matrix Theory,
Vol.5 No.1,
February
10,
2015
ABSTRACT:
Newton’s iteration is a fundamental tool for numerical solutions of systems of
equations. The well-known iteration rapidly refines a crude
initial approximation X0to the inverse of a general
nonsingular matrix. In this paper, we will extend and apply this method to n× nstructured 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.