Fixed-point Iteration; $x=g(x)$ Method

• Rearrange $f(x)$ into an equivalent form $x=g(x)$,
• This can be done in several ways.
  • Observe that if $f(r)=0$, where $r$ is a root of $f(x)$, it follows that $r=g(r)$.
  • Whenever we have $r=g(r)$, $r$ is said to be a fixed point for the function $g$.
• The iterative form:
  
  $$x_{n+1}=g(x_n);~~n=0,1,2,3,\ldots$$
  
  
  converges to the fixed point $r$, a root of $f(x)$.
• Example: $f(x)=x^2-2x-3=0$
• Suppose we rearrange to give this equivalent form:

$$x = g_1(x) =\sqrt{2x + 3}$$

$x_0=4$	$\rightarrow$	$x_1=\sqrt{11}=3.31662$
$x_2=\sqrt{9.63325}=3.10375$	$\rightarrow$	$x_3=3.03439$
$x_4=3.01144$	$\rightarrow$	$x_5=3.00381$

• If we start with $x=4$ and iterate with the fixed-point algorithm,
• The values are converging on the root at $x = 3$.