牛顿迭代法

　

牛顿迭代法

设 f(x)" role="presentation" style="position: relative;">f(x)$f(x)$ 在 R" role="presentation" style="position: relative;">ℝ$\mathbb{R}$ 有二阶连续导数，且 f′(x)≠0" role="presentation" style="position: relative;">f′(x)≠0$f^{\prime }(x)\ne 0$ 则

∀x0,x∈R," role="presentation" style="position: relative;">∀x0,x∈ℝ,$\mathrm{\forall }{x}_0,x\in \mathbb{R},$ 若 f(x0)=0" role="presentation" style="position: relative;">f(x0)=0$f(x_0)=0$ 则

f(x0)=f(x)+f′(x)Δx+12f″(ε)Δx2=0" role="presentation" style="position: relative;">f(x0)=f(x)+f′(x)Δx+12f″(ε)Δx2=0$f(x_0)=f(x)+f^{\prime }(x)\mathrm{\Delta }x+{\displaystyle \frac{1}{2}}{f}^{″}(\epsilon )\mathrm{\Delta }{x}^2=0$

⇒" role="presentation" style="position: relative;">⇒$\Rightarrow$

x0−x=Δx=−1f′(x)[f(x)+12f″(ε)Δx2]" role="presentation" style="position: relative;">x0−x=Δx=−1f′(x)[f(x)+12f″(ε)Δx2]$x_0-x=\mathrm{\Delta }x=-{\displaystyle \frac{1}{f^{\prime }(x)}}\left[f(x)+{\displaystyle \frac{1}{2}}{f}^{″}(\epsilon )\mathrm{\Delta }{x}^2\right]$

=−f(x)f′(x)−12f″(ε)f′(x)Δx2" role="presentation" style="position: relative;">=−f(x)f′(x)−12f″(ε)f′(x)Δx2$=-{\displaystyle \frac{f(x)}{f^{\prime }(x)}}-{\displaystyle \frac{1}{2}}{\displaystyle \frac{f^{″}(\epsilon )}{f^{\prime }(x)}}\mathrm{\Delta }{x}^2$

⇒" role="presentation" style="position: relative;">⇒$\Rightarrow$

x0=x−f(x)f′(x)−12f″(ε)f′(x)Δx2" role="presentation" style="position: relative;">x0=x−f(x)f′(x)−12f″(ε)f′(x)Δx2$x_0=x-{\displaystyle \frac{f(x)}{f^{\prime }(x)}}-{\displaystyle \frac{1}{2}}{\displaystyle \frac{f^{″}(\epsilon )}{f^{\prime }(x)}}\mathrm{\Delta }{x}^2$

由于 limx→x0[−12f″(ε)f′(x)Δx2]=0" role="presentation" style="position: relative;">limx→x0[−12f″(ε)f′(x)Δx2]=0$\underset{x\to x_0}{lim}\left[-{\displaystyle \frac{1}{2}}{\displaystyle \frac{f^{″}(\epsilon )}{f^{\prime }(x)}}\mathrm{\Delta }{x}^2\right]=0$

因此 limx→x0[x−f(x)f′(x)]=x0" role="presentation" style="position: relative;">limx→x0[x−f(x)f′(x)]=x0$\underset{x\to x_0}{lim}\left[x-{\displaystyle \frac{f(x)}{f^{\prime }(x)}}\right]=x_0$

令 F(x)=x−f(x)f′(x)" role="presentation" style="position: relative;">F(x)=x−f(x)f′(x)$F(x)=x-{\displaystyle \frac{f(x)}{f^{\prime }(x)}}$，则 limx→x0F(x)=x0" role="presentation" style="position: relative;">limx→x0F(x)=x0$\underset{x\to x_0}{lim}F(x)=x_0$

由此得到迭代公式：

xk+1=F(xk)=xk−f(xk)f′(xk)" role="presentation" style="position: relative;">xk+1=F(xk)=xk−f(xk)f′(xk)$x_{k+1}=F(x_k)=x_k-{\displaystyle \frac{f(x_k)}{f^{\prime }(x_k)}}$

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牛顿迭代法

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