On the optimal complex extrapolation of the complex Cayley transform

Abstract

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.