Журнал прикладной и вычислительной математики

The Systematic Formation of High-Order Iterative Methods

Abstract

Isaac Fried

Fixed point iteration and the Taylor-Lagrange formula are used to derive, some new, efficient, high-order, up to octic, methods to iteratively locate the root, simple or multiple, of a nonlinear equation. These methods are then systematically modified to account for root multiplicities greater than one. Also derived, are super-quadratic methods that converge contrarily, and super-linear and super-cubic methods that converge alteratingly, enabling us, not only to approach the root, but also to actually bound and bracket it.