Optimization of extrapolated Cayley transform with non-Hermitian positive definite matrix

For the extrapolated Cayley transform, we give necessary and sufficient conditions for guaranteeing its convergence and contraction (in the Euclidean norm). We derive upper bounds for the convergence and the contraction factors, and compute the optimal parameters minimizing these upper bounds and the corresponding optimal values of these upper bounds. Numerical computations show that these upper bounds are reasonably sharp compared with the exact convergence and the exact contraction factors of the extrapolated Cayley transform, respectively.