The semi-iterative method (SIM) is applied to the hyper-power (HP) iteration, which is defined by $R_k=P(I-AX_k)$, $X_{k+1}=X_k(I+R_k+\dots +R_k^{q-1})$, $k=0,1,2,\dots $, where $P^2=P$ and $A$ and $X_0$ are arbitrary and complex matrices. Necessary and sufficient conditions are given for the convergence of the semi-iterative-hyper-power (SIM-HP) iteration. The root-convergence rate is computed for both the HP and SIM-HP methods, and the quotient convergence rate is given for the HP iteration.
Reviewer: Jan Chleboun (Praha)