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

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