Solvable difference equations similar to the Newton-Raphson iteration for algebraic equations

TL;DR

This paper extends the class of solvable difference equations akin to Newton-Raphson iterations from quadratic to cubic equations, providing explicit solutions and analyzing convergence properties.

Contribution

It introduces a method to solve difference equations similar to Newton-Raphson for cubic equations using four-term recurrence relations, expanding the scope of solvable iterative methods.

Findings

Solutions can be explicitly constructed from four-term recurrence relations.

The method converges quadratically to roots of cubic equations.

The approach generalizes the known quadratic case to cubic equations.

Abstract

It is known that difference equations generated as the Newton-Raphson iteration for quadratic equations are solvable in closed form, and the solution can be constructed from linear three-term recurrence relations with constant coefficients. We show that the same construction for four-term recurrence relations gives the solution to the initial value problem of difference equations similar to the Newton-Raphson iteration for cubic equations. In many cases, the solution converges to a root of the cubic equation and the convergence rate is quadratic.