牛頓法

牛頓法者，導數之法也，以解方程之近似值。

方程${\displaystyle f(x)}$，導數${\displaystyle f'(x)}$且

${\displaystyle f'(x)=\lim _{\Delta x\to 0}{\frac {f(x+\Delta x)-f(x)}{\Delta x}$

則有

${\displaystyle \Delta x=\lim _{\Delta x\to 0}{\frac {f(x+\Delta x)-f(x)}{f'(x)}$

又方程一側值爲〇，則

${\displaystyle x_{1}=x_{0}+\Delta x=x_{0}+{\frac {0-f(x_{0})}{f'(x_{0})}=x_{0}-{\frac {f(x_{0})}{f'(x_{0})}$

得

${\displaystyle x_{n}=x_{n-1}-{\frac {f(x_{n-1})}{f'(x_{n-1})}$

求值${\displaystyle {\sqrt {2}$

設${\displaystyle y=x^{2}-2}$，則${\displaystyle y'=2x}$

${\displaystyle x_{0}=1.4}$

${\displaystyle x_{1}=1.4-(-0.04\div 2.8)=1.4142857\cdots \cdots \thickapprox 1.4142}$

${\displaystyle x_{2}\approx 1.41421356}$

${\displaystyle \cdots \cdots }$

故${\displaystyle {\sqrt {2}\approx 1.4142135623}$