The well-known first-order nonlinear difference equation

$$ y_{n+1}=2y_{n}-xy_{n}^{2}, \quad n\in {\mathbb {N}}_{0}, $$

naturally appeared in the problem of computing the reciprocal value of a given nonzero real number x. One of the interesting features of the difference equation is that it is solvable in closed form. We show that there is a class of theoretically solvable higher-order nonlinear difference equations that include the equation. We also show that some of these equations are also practically solvable.

中文翻译：

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关于一类可解决的差分方程，概括了用于计算倒数的迭代过程

著名的一阶非线性差分方程

$$ y_ {n + 1} = 2y_ {n} -xy_ {n} ^ {2}，\ quad n \ in {\ mathbb {N}} _ {0}，$$

很自然地出现在计算给定非零实数x的倒数的问题中。差分方程的有趣特征之一是它可以闭合形式求解。我们表明，有一类理论上可解的高阶非线性差分方程包括该方程。我们还表明，其中的一些方程式实际上也是可解的。