The principle of extrapolation and the Cayley Transform

The Cayley Transform, F:=(I+A)-1(I-A), with A∈Cn,n and -1∉σ(A), where σ(·) 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.