I calculated each of the f(x) values in order to see where the change in sign was, as shown in the table on follow page 3 : There is a root in the interval of [1, 2] which I have chosen to investigate. Now I use excel to calculate the change values of f(x) by taking increments in x of size 0. 1 for the equation that we are using. As the graph and the table show, there is a change of sign between x=1. 5 and x=1. 6. and thus the new interval in [1. 5,1. 6]. I then take decimal search to 2 decimal places.
I will carry on using this method to find the root to 6 decimal places for accuration. And the result tables are placed on the next page.. By increasing in x of size 0. 001 By increasing in x of size 0. 0001 By increasing in x of size 0. 00001 So, using these tables we can say that the root is 1. 525685 with maximum error bounds of 0. 000005 Where this method fails The change of sign method was succesfull in the equation above. However it can fail in many cases when the curve touches the x axis, but does not cross it.
It is repeated root, so it does not have the change of sign from positve to negetive or vice versa. Therefore the change of sign can not be carried on. For this case I’ll survey this equation: . So in speeds of convergence, the Change of Sign method takes 20 steps, x=g(x) method takes 10 steps and the Newton-Raphson method takes 5 for this equation. This order of speed is a good representation of most equations that I have investigated. The three methods (Change of Sign, Newton-Raphson, and the Rearrangement Method) differ in their ease-of-use and speed.
The Change of Sign method takes the simple initial work; for the Newton-Raphson method and rearranging, f'(x) and g'(x) needs to be calculated. For many equations, differentiating them is very complicated, more calculations are required. And the rearranging method requires rearranging the formula to start with. There for spend more time to build the formula. For g(x) functions with more than one root, the rearranging method will usually fail to find at least one of them. The Hardware available such as computer is used a large part in the ease of each method, it helps greatly, as complex calculations can be done easily.
This is helpful in all three methods, as may be seen above. Graphical calculators can also make a comprehensive view of three methods. The Software available also can make different methods a lot easier. Spreadsheets (using Microsoft Excel) and Graphs (entirely drawn using Autograph) can be link into text documents (Microsoft Word) quickly and clearly, and thus reference can be made to them throughout a report. Therefore, the software available is used a large part in the ease of which each method can be completed.