1998 Volume 4 Issue 2 Pages 129-138
An extension of the Steffensen iteration method for solving a single nonlinear equation is considered. The point is that the iteration function is defined by using the k -th Shanks transform. The convergence rate is shown to be of order k +1. The use of ε -algorithm avoids the direct calculation of Hankel determinants, which appear in the Shanks transform, and then diminishes the computational complexity. For a special case of the Kepler equation, it is shown that the numbers of mappings are actually decreased by the use of the extended Steffensen iteration.