Second derivative of an iterative solver boosts its acceleration by Koçak’s method

摘要

Many mathematical applications involve the solution of a nonlinear equation x = g(x). The target fixed-point z of g may be a zero of a related function f(x), or equivalently, a root of the equation f(x) = 0. It is an important and challenging task to develop practical, efficient, and robust solvers. Koçak’s method achieves acceleration by generating a superior secondary solver gK via a transformation gK = (g − mx)/(1 − m) where m is a linear combination of the first derivative of g at two different x values, namely the current iterate and the target z. gK is the result of a simple rearrangement of the equation x = g after subtracting the product mx from both sides. Therefore gK(z) = g(z) = z. The process is piecewise linearization where m is the slope of a straight line approximating g. Ideally, m = (g − z)/(x − z). Let n denote the convergence order of g. The initial version employs w = 1/2 irrespective of n. Its gK is of third order when n is 1 or 2; when n exceeds 2 the order remains the same. The second version harnesses w = 1/n when n exceeds 2 and obtains a gK of order n + 1. This manuscript reports a third version involving the second derivative of g.