2.2: Newton's Method

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May 31, 2022
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- 2.1: Bisection Method
- 2.3: Secant Method

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This is the fastest method, but requires analytical computation of the derivative of $f(x)$ . Also, the method may not always converge to the desired root.

We can derive Newton’s Method graphically, or by a Taylor series. We again want to construct a sequence $x_{0}, x_{1}, x_{2}, \ldots$ that converges to the root $x=r$ . Consider the $x_{n+1}$ member of this sequence, and Taylor series expand $f\left(x_{n+1}\right)$ about the point $x_{n}$ . We have

$f\left(x_{n+1}\right)=f\left(x_{n}\right)+\left(x_{n+1}-x_{n}\right) f^{\prime}\left(x_{n}\right)+\ldots . \nonumber$

To determine $x_{n+1}$ , we drop the higher-order terms in the Taylor series, and assume $f\left(x_{n+1}\right)=0 .$ Solving for $x_{n+1}$ , we have

$x_{n+1}=x_{n}-\frac{f\left(x_{n}\right)}{f^{\prime}\left(x_{n}\right)} \nonumber$

Starting Newton’s Method requires a guess for $x_{0}$ , hopefully close to the root $x=r .$

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