On two families of iterative methods without memory
Anna Cima, Armengol Gasull, Víctor Mañosa, Francesc Mañosas
TL;DR
This work analyzes two memoryless families of iterative methods for solving real equations $f(x)=0$: Class I uses $x$, $f(x)$ and derivatives up to order $n-1$, and Class II uses $x$ and the iterates of $g(x)=f(x)+x$ via interpolation of $f^{-1}$. It proves that canonical Class I maps $H_f^n$ converge to a simple zero with order $n$ and extends these constructions to general Class I methods with remainder terms, providing simple, self-contained proofs; it then develops a Steffensen-type family $K_f^n$ of order $n$ in Class II by interpolation on $n$ points along the $g$-orbit. The paper further computes explicit canonical methods up to order $7$, derives simple rational variants $R^n$, and presents Halley-type generalizations, analyzing their efficiency under Kung–Traub-type cost models. Overall, the work offers a unified, rigorous treatment of two high-order, memoryless families, with practical high-order methods and a discussion of efficiency and optimality in multipoint settings.
Abstract
We study two natural families of methods of order $n\ge 2$ that are useful for solving numerically one variable equations $f(x)=0.$ The first family consists on the methods that depend on $x,f(x)$ and its successive derivatives up to $f^{(n-1)}(x)$ and the second family comprises methods that depend on $x,g(x)$ until $g^{\circ n}(x),$ where $g^{\circ m}(x)=g(g^{\circ (m-1)}(x))$ and $g(x)=f(x)+x$. The first family includes the well-known Newton, Chebyshev, and Halley methods, while the second one contains the Steffensen method. Although the results for the first type of methods are well known and classical, we provide new, simple, detailed, and self-contained proofs.