An efficient one-point extrapolation method for linear convergence

Abstract:

For iteration sequences otherwise converging linearly, the proposed one-point extrapolation method attains a convergence rate and efficiency of 1.618. This is accomplished by retaining an estimate of the linear coefficient from the previous step and using the estimate to extrapolate. For linear convergence problems, the classical Aitken-Steffensen ${\delta ^2}$-process has an efficiency of just $\sqrt 2$, while a recently proposed fourth-order method reaches an efficiency of 1.587. Not only is the method presented here more efficient, but it is also quite straightforward. Examples given are for Newton’s method in finding multiple polynomial roots and for locating a fixed point of a nonlinear function.