How can a two term Taylor Series be used to derive Newton-Raphson root finding formula?

Question

Can someone please show how to derive the Newton root finding formula from a term taylor series. My main issue is I am not sure what mathematically the Newton Root finding formula actually is as I have only learned about it through my numerical methods class through MATLAB

Two Term Taylor Series: $f(x_i) + f '(x-x_i) + f ''(x_i)(1/2)(x-x_i)^2$

Thanks for the help!

Answer 1

We have that the 2 term Taylor polynomial is given by

$$f(x)\approx f(x_0)+f'(x_0)(x-x_0)$$

Setting $f(x)=0$, we end up with

$$ \begin{align} 0&=f(x_0)+f'(x_0)(x-x_0)\\ f'(x_0)(x-x_0)&=-f(x_0)\\ x-x_0&=-\frac{f(x_0)}{f'(x_0)}\\ x&=x_0-\frac{f(x_0)}{f'(x_0)} \end{align} $$

Answer 2

Suppose $f(a)=0$, then Taylor's formula says:

$$0=f(a)=f(x)+(a-x)f'(x)+\frac{1}{2}(a-x)^2f''(\xi)$$

for some $\xi$ between $a$ and $x$. Solving for $a$ (sort of) gives:

$$a=x-\frac{f(x)+\frac{1}{2}(a-x)^2f''(\xi)}{f'(x)}$$

which, for $x\approx a$, we can (under certain conditions) expect to say that:

$$a\approx x-\frac{f(x)}{f'(x)}$$

which is, in essence, the Newton-Raphson method.

One can be a bit more careful with this argument to provide a more formal proof of the convergence of the method to $a$ under certain conditions.