As can be seen, the sequence clearly converges on 1. 445281, and thus agrees with both of the other methods as to the location of this root for this function, as when rounded to 3 decimal places, this becomes 1. 445. Of the three methods, it is clear that the method that is the slowest to converge on the root is the decimal search method. The quickest to converge is the Newton-Raphson method, closely followed by the fixed point iteration.
However, although both iteration methods can find the root correct to several decimal places very quickly, there is more initial work in order to start using these methods than there is for the decimal search. Formulae must be rearranged or new formulae constructed in order to begin the search. Also, a lot of the speed of the two iteration methods is due to the ability of a computer to quickly do repeat calculations many times over.
They are well suited to the use of computers whereas the decimal search method is relatively slow even with the use of computers, as it would take a human to spot the change in sign that determines the next set of calculations. Finally, there is no reason why the decimal search method could fail when used in conjunction with an graph plotted accurately by calculator or computer to determine the rough approximations of the roots, whereas with the other two methods there are functions where it is impossible to obtain an estimate of the roots.
In order to find the root of the function f(x)=3×3-7×2-11x+17, the decimal search method was used. This is the process whereby a table of values is constructed that show whether the value of f(x) with certain values of x is positive or negative. By zooming in on the places where the value of f(x) switches from positive to negative (or vice versa), values for the root of this function can be found.