牛顿法

求方程${\displaystyle \cos(x)-x^{3}=0}$的根。令${\displaystyle f(x)=\cos(x)-x^{3}}$，两边求导，得${\displaystyle f'(x)=-\sin(x)-3x^{2}}$。由于${\displaystyle -1\leq \cos(x)\leq 1(\forall x)}$，则${\displaystyle -1\leq x^{3}\leq 1}$，即${\displaystyle -1\leq x\leq 1}$，可知方程的根位于${\displaystyle 0}$和${\displaystyle 1}$之间。我们从${\displaystyle x_{0}=0.5}$开始。

${\displaystyle {\begin{matrix}x_{1}&=&x_{0}-{\frac {f(x_{0})}{f'(x_{0})}}&=&0.5-{\frac {\cos(0.5)-0.5^{3}}{-\sin(0.5)-3\times 0.5^{2}}}&=&1.112141637097\\x_{2}&=&x_{1}-{\frac {f(x_{1})}{f'(x_{1})}}&=&\vdots &=&{\underline {0.}}909672693736\\x_{3}&=&\vdots &=&\vdots &=&{\underline {0.86}}7263818209\\x_{4}&=&\vdots &=&\vdots &=&{\underline {0.86547}}7135298\\x_{5}&=&\vdots &=&\vdots &=&{\underline {0.8654740331}}11\\x_{6}&=&\vdots &=&\vdots &=&{\underline {0.865474033102}}\end{matrix}}}$

牛顿法亦可发挥与泰勒展开式，对于函式展开的功能。

求${\displaystyle a}$的${\displaystyle m}$次方根。

${\displaystyle x^{m}-a=0}$

设${\displaystyle f(x)=x^{m}-a}$，${\displaystyle f'(x)=mx^{m-1}}$

而a的m次方根，亦是x的解，

以牛顿法来迭代：

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

${\displaystyle x_{n+1}=x_{n}-{\frac {x_{n}^{m}-a}{mx_{n}^{m-1}}}}$

${\displaystyle x_{n+1}=x_{n}-{\frac {x_{n}}{m}}(1-ax_{n}^{-m})}$

（或 ${\displaystyle x_{n+1}=x_{n}-{\frac {1}{m}}\left(x_{n}-a{\frac {x_{n}}{x_{n}^{m}}}\right)}$）

牛頓法也被用於求函數的極值。由於函數取極值的點處的導數值為零，故可用牛頓法求導函數的零點，其疊代式為

${\displaystyle x_{n+1}=x_{n}-{\frac {f^{\prime }(x_{n})}{f^{\prime \prime }(x_{n})}}.}$

求拐点的公式以此类推