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Definition: Newton's Method

Newton’s Method is another search method that can narrow down the minimizer $x^*$ of a unimodal function $f: \R \to \R$ over a closed interval $[a_0, b_0]$.

However, Newton’s method requires $f \in \mathcal{C}^2$.

The idea is to fit a quadratic function through $x^{(k)}, f'(x^{(k)})$ that matches its first and second derivative with that of the function $f$. The quadratic that satisfies these requirements should have the following form

$$q(x) = f(x^{(k)}) + f'(x^{(k)})(x - x^{(k)}) + \frac{ 1 }{ 2 } f^{ \prime \prime }(x^{(k)})(x - x^{(k)})^2$$

We have,

$$\begin{align*} q(x^{(k)}) &= f(x^{(k)}) \br q'(x^{(k)}) &= f'(x^{(k)}) \br q^{ \prime \prime }(x^{(k)}) &= f^{ \prime \prime }(x^{(k)}) \end{align*}$$

Then, instead of minimizing $f$, we minimize its approximation $q$.

Using FONC to restrain $q(x)$ we have,

$$x^{(k+1)} = x^{(k)} - \frac{ f'(x^{(k)})}{ f^{ \prime \prime } (x^{(k)})} $$

Notice that Newton’s method always works well if $f^{ \prime \prime (x)} > 0 $ everywhere. However, if $f^{ \prime \prime } (x) < 0$ for some $x $, it may fail to converge to the minimizer.