Other Rearrangements

• Another rearrangement of $f(x)$; Let us start the iterations again with $x_0=4$. Successive values then are:

$$x=g_2(x)=\frac{3}{(x-2)}$$

$x_0=4$	$\rightarrow$	$x_1=1.5$	$\rightarrow$
$x_2=-6$	$\rightarrow$	$x_3=-0.375$	$\rightarrow$
$x_4= - 1.263158$	$\rightarrow$	$x_5 = -0.919355$	$\rightarrow$
$x_5 = -0.919355$	$\rightarrow$	$x_6= - 1.02762$	$\rightarrow$
$x_7=-0.990876$	$\rightarrow$	$x_8= - 1.00305$

• It seems that we now converge to the other root, at $x = -1$.
• Consider a third rearrangement; starting again with $x_0=4$, we get

$$x=g_3(x)=\frac{(x^2-3)}{2}$$

$x_0=4$	$\rightarrow$	$x_1=6.5$	$\rightarrow$
$x_2 = 19.625$	$\rightarrow$	$x_3 = 191.070$

• The iterations are obviously diverging.

• The fixed point of $x=g(x)$ is the intersection of the line $y=x$ and the curve $y = g(x)$ plotted against $x$.

Figure 3.11: The fixed point of $x=g(x)$ is the intersection of the line $y=x$ and the curve $y = g(x)$ plotted against $x$. Where A: $x = g_1(x) =\sqrt {2x + 3}$. B: $x=g_2(x)=\frac {3}{(x-2)}$. C: $x=g_3(x)=\frac {(x^2-3)}{2}$.

Figure 3.11 shows the three cases.

• Start on the x-axis at the initial $x_0$, go vertically to the curve, then horizontally to the line $y=x$, then vertically to the curve, and again horizontally to the line.
• Repeat this process until the points on the curve converge to a fixed point or else diverge.
• The method may converge to a root different from
  the expected one, or it may diverge.
• Different rearrangements will converge at different rates.
• Iteration algorithm with the form $x=g(x)$