On the optimal complex extrapolation of the complex Cayley transform

The Cayley transform, F := F(A) = (I + A)(-1) (I - A), with A epsilon C(n.n) and -1 is not an element of sigma (A), where sigma(.) denotes spectrum, and its extrapolated counterpart F (omega A), omega epsilon C\{0} and -1 is not an element of sigma (omega A), are of significant theoretical and practical importance (see, e.g. [A. Hadjidimos, M. Tzoumas, On the principle of extrapolation and the Cayley transform, Linear Algebra Appl., in press]). In this work, we extend the theory in [8] to cover the complex case. Specifically, we determine the optimal extrapolation parameter omega epsilon C\{0} for which the spectral radius of the extrapolated Cayley transform rho(F(omega A)) is minimized assuming that sigma(A) subset of H, where H is the smallest closed convex polygon, and satisfies O(0) is not an element of H. As an application, we show how a complex linear system, with coefficient a certain class of indefinite matrices, which the ADI-type method of Hermitian/Skew-Hermitian splitting fails to solve, can be solved in a "best" way by the aforementioned method. (C) 2008 Elsevier Inc. All rights reserved

Nonstationary Extrapolated Modulus Algorithms for the solution of the Linear Complementarity Problem

The Linear Complementarity Problem (LCP) has many applications as, e.g., in the solution of Linear and Convex Quadratic Programming, in Free Boundary Value problems of Fluid Mechanics, etc. In the present work we assume that the matrix coefficient M is an element of R(n,n) of the LCP is symmetric positive definite and we introduce the (optimal) nonstationary extrapolation to improve the convergence rates of the well-known Modulus Algorithm and Block Modulus Algorithm for its solution. Two illustrative numerical examples show that the (Optimal) Nonstationary Extrapolated Block Modulus Algorithm is far better than all the previous similar Algorithms. (C) 2009 Elsevier Inc. All rights reserved

The principle of extrapolation and the Cayley Transform

The Cayley Transform, F := (I + A)(-1)(I - A), with A is an element of C-n,C-n and -1 is not an element of sigma (A), where sigma(.) denotes spectrum, is of significant theoretical importance and interest and has many practical applications. E.g., in the solution of the Linear Complementarity Problem (LCP), in the solution of linear systems arising from the discretization of model problems elliptic PDEs by Alternating Direction Implicit (ADI) iterative methods, in the solution of complex linear systems by ADI-type methods of Hermitian/Skew Hermitian or Normal/Skew Hermitian Splittings, etc. In the present work we apply the principle of Extrapolation to generalize the Cayley Transform and determine in an optimal sense the Extrapolation parameter involved so that problems in many practical applications are solved much more efficiently. (C) 2007 Elsevier Inc. All rights reserved