An efficient one-point extrapolation method for linear convergence

Abstract

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 6 -process has an efficiency of just -J2, 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.

Anderson-Björck for linear sequences

“…Examples. All three of the examples are taken from [7], with computations done in quadruple precision on an IBM 370.…”

“…Both Kn and mn are given in the tables. Results for the two methods reviewed in Section 1-the 52-process and the scheme of [7]-are also included for comparison. Extrapolation substeps have a P (for prime) following the step number n in the first column.…”

“…But we can attain a higher convergence rate merely by retaining, at each step, information held over from the previous step. This idea is incorporated (with a change of notation) into the one-point extrapolation method with memory described in [7], to wit: define <pn = <¡>ix"), and get started from x0 by taking x, = <f>(). For each and every subsequent step, apply a secant-method step to the related function g = x -<p. Thus the method is (1)(2)(3)(4)(5) xn + 2 = xn+t-ix"-xn+x) _"+x , 6/1 ¥>n t I starting at g0 = gix()) = x0 -<p0 and g, = g(x,) = jc, -<£,.…”

“…For each and every subsequent step, apply a secant-method step to the related function g = x -<p. Thus the method is (1)(2)(3)(4)(5) xn + 2 = xn+t-ix"-xn+x) _"+x , 6/1 ¥>n t I starting at g0 = gix()) = x0 -<p0 and g, = g(x,) = jc, -<£,. Since ultimately |e"+, |<|e"| , the error equation for the one-point method of [7] turns out to be the following (from (1.5) and the error expression (1.2) for <f>):…”

“….499813 -.499178Í-21) Since Newton's method requires the calculation of both / and /' to get <p, the effective efficiency for the ô2-process is really only 2'/4 -1.189, and for Method [7] is (1.618)1/2 -1.272. Similarly the proposed scheme with Newton's method has an effective efficiency of (1.839)1/2 = 1.356.…”