SOLUTION OF NONLINEAR EQUATIONS

Methods

1. Two Point Methods
   
2. One Point Methods
   

We'll also discuss Polynomial Root Finding later.

BASIC PROBLEM:

Find $x$ such that     $f(x)=0$      $f: R\rightarrow R\cap C^1$     $x\in R$.

In most cases not possible to solve analytically. Most methods are iterative and require some initial guess(es)

Two point: require 2 guesses
One point: require 1 guess

$\Box$

First, a reminder:

Theorem: (Intermediate Value Theorem) Suppose $f(x)\in C[a,b], k$ any number between $f(a)$ and $f(b)$.

$\Rightarrow \exists$ a number $c\in [a,b]$ such that $f(c)=k$

Use a figure to convince yourself. $\Box$

Rolle's Theorem $f(x)\in C[a,b]$ and differentiable on $(a,b)$. If $f(a)=f(b)\Rightarrow \exists$ at least one $c\in (a,b)$ such that $f'(c)=0$.

Use a figure to convince yourself. $\Box$

Two-point Methods:

Consider bracketing methods. These are inspired by the above two theorems.

General Technique in Bracketing: if $f\in C[a,b]$ and $f(a)f(b)\le 0\Rightarrow$ by the intermediate value theorem $\exists$ at least one 0 in $[a,b]$. Condition $f(a)f(b)\le 0$ means that $f(a)$ and $f(b)$ are opposite in sign. So choose $a,b$ to be bracket hopefully containing the root you want $f(\alpha_1)=0$. Search and iterate making brackets smaller till you hit root.