The convergence of this rearrangement can be demonstrated graphically. Part of the graph of the curves y=x and y=g(x) is shown. Using x1=1, the iteration is represented by the red line: The approximations get better as the iteration progresses. This is because the iteration converges to the intersection, which has the same x-value as the root. If g(x)=(x3+1)/3 then, g/(x)=d/dx (x3/3 – 1/3)=3×2/3= x2 The root we have already found is x=0. 3472964 (to seven decimal places). If we apply function g(x) to this root, (i. e. square it,) we obtain 0. 1206148 (to seven decimal places).
The absolute value of g/(x)<1, therefore convergence to the root, x=0. 3472964, will occur provided that a suitable initial value is chosen. It converges rather than diverges at the point shown because from one approximation to the next, the change in y is less than the change in x, i. e. the gradient is less than 1. For example, in the graph above, AC is smaller than AB. In terms of speed of convergence, the rearrangement method took 11 steps to converge to the root to 7 decimal places, the Newton-Raphson method took 3 steps to do this and the decimal search method took 8 sets to achieve this.
It would seem that the Newton-Raphson method is the fastest, followed by the decimal search method and then followed by the rearrangement method. This may not always be the case since the speed of convergence for both Newton-Raphson and the rearrangement method is subject to the initial starting value chosen. The decimal search method always takes one more table than the number of decimal places required (as explained before). In many cases, the speed of convergence for the rearrangement method has been faster than the decimal search method, but the Newton-Raphson method is always quickest to converge (given that it does converge).
I used Microsoft Excel to do all of these tables and graphs. In terms of ease of use, the rearrangement method required the least steps as all I had to do was fill the formula in one cell, using cell above as the xn value, and ‘drag down’. This was the quickest and easiest method to produce a table for. The Newton-Raphson method was much the same, however it required the input of a much longer formula, and therefore was slightly more tedious. The decimal search method required the input of a lot of data, even after entering the formula into one of the cells.
It requires n+1 tables if I want to find a root to n decimal places. Therefore this was by far the slowest method to use. In terms of failure, the rearrangement method only gives one root for one rearrangement. In this respect the Newton-Raphson method is better, since it converges to all roots given a suitable starting value. However it has the potential to diverge which is unhelpful. In this respect, the decimal search method is the best, given small enough intervals, as it hardly ever fails once a root is identified in a particular interval.