On the optimal complex extrapolation of the complex Cayley transform

The Cayley transform, F≔F(A)=(I+A)-1(I-A), with A∈Cn,n and -1∉σ(A), where σ(·) denotes spectrum, and its extrapolated counterpart F(ωA), ω∈C⧹{0} and -1∉σ(ωA), 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 ω∈C⧹{0} for which the spectral radius of the extrapolated Cayley transform ρ(F(ωA)) is minimized assuming that σ(A)⊂H, where H is the smallest closed convex polygon, and satisfies O(0)∉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.