整方根函数

整方根函数${\displaystyle \lfloor {\sqrt {a}}\rfloor }$用原始递归函数可定义为：[1]

${\displaystyle {\begin{cases}\lfloor {\sqrt {0}}\rfloor =0\\\lfloor {\sqrt {Sa}}\rfloor =\lfloor {\sqrt {a}}\rfloor +N(Sa{\dot {-}}(S\lfloor {\sqrt {a}}\rfloor )^{2}\end{cases}}}$

由牛顿法迭代公式${\displaystyle x_{n+1}=x_{n}-{\frac {f(x_{n})}{f'(x_{n+1})}}}$，欲计算${\displaystyle \lfloor {\sqrt {a}}\rfloor }$，可令

${\displaystyle f(x)=x^{2}-a}$，由${\displaystyle x^{2}-a=0}$，得

${\displaystyle f(x)}$与${\displaystyle x}$轴相交于${\displaystyle x={\sqrt {a}}}$，可计算平方根，于是

${\displaystyle f'(x)=2x}$，代入迭代公式可得

${\displaystyle x_{n+1}=x_{n}-{\frac {x_{n}^{2}-a}{2x_{n}}}}$，整理得

${\displaystyle x_{n+1}={\frac {x_{n}}{2}}+{\frac {a}{2x_{n}}}}$。

算法结束条件为${\displaystyle \Delta _{n}=\left\vert x_{n+1}-x_{n}\right\vert =0}$，即${\displaystyle x_{n+1}=x_{n}}$。[2]