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

Kazuki Maeda

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arXiv · 2309.13582

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

Kazuki Maeda

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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.

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Kazuki Maeda. 2023-09-24. Solvable difference equations similar to the Newton-Raphson iteration for algebraic equations. https://arxiv.org/abs/2309.13582

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