%0 Journal Article
%T New Approach for the Inversion of Structured Matrices via Newton¡¯s Iteration
%A Mohammad M. Tabanjeh
%J Advances in Linear Algebra & Matrix Theory
%P 1-15
%@ 2165-3348
%D 2015
%I Scientific Research Publishing
%R 10.4236/alamt.2015.51001
%X <p>
	<span>Newton</span><span>¡¯</span><span>s iteration is a fundamental tool for numerical solutions of systems of
equations. The well-known iteration <img src=\"Edit_2236544f-be48-482d-9b4d-c22fd048bebf.bmp\" alt=\"\" /></span><span></span><span>&nbsp;</span><span>rapidly refines a crude
initial approximation X&lt;sub&gt;0&lt;/sub&gt;</span><span></span><span>&nbsp;to the inverse of a general
nonsingular matrix. In this paper, we will extend and apply this me</span><span>thod to <b><i><span lang=\"EN-US\">n</span></i></b><span>¡Á </span><b><i><span lang=\"EN-US\">n&nbsp;</span></i></b></span><span>structured matrices <strong><em>M</em></strong>&nbsp;</span><span></span><span>,</span><span> in which matrix multiplication has a lower computational cost. These
matrices can be represented by their short generators which allow faster compu</span><span>tations 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 grow</span><span><span>th 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 </span><span>these approaches. Our main algorithm can be
applied to more general classes of structured </span><span>matrices.</span></span>
</p>
%K Newton Iteration
%K Structured Matrices
%K Superfast Algorithm
%K Displacement Operators
%K Matrix Inverse.
%U http://www.scirp.org/journal/PaperInformation.aspx?PaperID=53902