Other Rearrangements

• Another rearrangement of $f(x)$ is
  
  $$x=g_2(x)=\frac{3}{(x-2)}$$
  
  
  Let us start the iterations again with $x_0=4$. Successive values then are:
  

  $x_0=4$

  $x_1=1.5$

  $x_2=-6$

  $x_3=-0.375$

  $x_4= - 1.263158$

  $x_5 = -0.919355$

  $x_6= - 1.02762$

  $x_7=-0.990876$

  $x_8= - 1.00305$

  
  and it seems that we now converge to the other root, at $x = -1$.
• Consider a third rearrangement:
  
  $$x=g_3(x)=\frac{(x^2-3)}{2}$$
  
  
  starting again with $x_0=4$, we get
  

  $x_0=4$

  $x_1=6.5$

  $x_2 = 19.625$

  $x_3 = 191.070$

  
  and the iterates 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 shows the three cases.
• Observe that we always get the successive iterates by this construction. 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

Iteration algorithm with the form $x=g(x)$

The method may converge to a root different from the expected one, or it may diverge. Different rearrangements will converge at different rates.

Figure 3: The fixed point of $x=g(x)$ is the intersection of the line $y=x$ and the curve $y = g(x)$ plotted against x.