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.

翻译：已知作为二次方程牛顿-拉夫森迭代生成的差分方程具有闭式解，且该解可由常系数线性三项递推关系构建。我们证明，对四项递推关系采用相同构建方法，可得到与三次方程牛顿-拉夫森迭代法相似的差分方程初值问题的解。在许多情形下，该解收敛至三次方程的一个根，且收敛速度为二次收敛。