Failure
The decimal search will generally fail when two roots are close together or when there is a double root present.
The function has a repeated root; therefore the decimal search will fail to find the root.
The two diagrams above show that there is a root present, however, as there is a double root, decimal search has failed.
The table clearly shows that decimal search has failed to find the root as there is no sign change and f(x) is not close to 0.
Newton – Raphson Method
I will find the root of the following function using the Newton-Raphson method.
The following diagram shows the Newton-Raphson method being used in order to find the three roots of the equation.
I had to change the function into an iterative formula:
This formula allowed me to work out the roots
The first root was calculated like the following, using the iterative formula:
and are the same to 6 decimal places, indicating that an approximation of the root has been found.
The other two roots were calculated in the same manner.
For the last approximation, the error bounds are -0.43265 ± 0.000005
In order to confirm the approximation as the root, I will check for sign change;
As there is a sign change, when the boundaries of the approximation are put into the equation, a root exists.
Failure
In order to show the failure of this method, I will use following equation:
y = 3.5x⁴+2.8x³+5.4x+3
The above graph and the table shows that the Newton-Raphson method has failed even thought the starting root was close to the actual root.
Rearrangement equation
In the rearrangement method, an equation is rearranged to find.
The graph below shows the equation which I will rearrange to find one root.
In order to find the root, I rearranged my equation to
The graphs above are and. The two functions intersect at the roots of.
The graph below shows the success of the rearrangement method. Here the root had been found to 5 significant figures.
The rearrangement method is calculated as follows:
The function above is the basis of the iterative formula of the rearrangement method:
As the root was found using the negative graph, my equation will also be negative.
Using this method, I will show how the root was found in my first graph:
The root is approximately 0.32150 to 5 significant figures.
As the gradient of was between, the method was successful.
Failure
The following graphs show the failure of the rearrangement method.
The diagram and table above show that the method has failed as it diverges from the actual root.
As the gradient of the rearranged equation is not between, therefore the method has failed.
Comparison
In order to compare the three methods with each other, I will use the equation, which I used for my rearrangement method.
Decimal Search
I used decimal search to find the root in between 0 and 1.
[0,1] [0.3,0.4] [0.32,0.33]
[0.321,0.322] [0.3215,0.3216]
From the tables, it is clear that there is a root between 0.32150 and 0.32151.
Therefore the root is 0.3215 to four significant figures.
Newton Raphson Method
The Newton-Raphson Method clearly found the root within 3 iterations.
From this method, we can see that the root is 0.3215 to 4 significant figures.
Rearrangement Method
The rearrangement method found the root within 6 iterations. The root is 0.3215 to 4 significant figures.
The table above shows the numbers of iterations each method took in order to come to the same degree of accuracy. The Newton-Raphson method was the quickest, finding the root within a certain degree of accuracy in only three iterations. Second was the Decimal Search, which took five iterations and last was the Rearrangement Method, which took the most number of iterations, 6.
Newton-Raphson method is clearly the fastest and the most efficient method to use as the number of iterations needed to find a root to a degree of accuracy is small. However, this method is very tiresome to calculate by hand and the tiniest mistake can result in a wrong answer.
The Decimal Search takes more iteration; but, this method is the easiest and easily understood. However, this method is best done on a spreadsheet, where you would be able to spot the sign change easily.
The Rearrangement Method takes slightly more iteration but it provides the root to any degree of accuracy. Also, the formula is iterative, therefore, it is not very time consuming. However, finding can be tricky.
Software
In terms of the software used, Decimal Search was the easiest as it only required spreadsheet which is not difficult to use. Although making the tables can be repetitive, any faults can easily be rectified.
Both the Newton-Raphson Method and the Rearrangement Method used a calculator to work out the iterative steps. This was often very time-consuming and frustrating as simple mistakes could let to the wrong route.
Autograph was used to draw all the graphs and show the methods at work. It was not hard to use but tricky, due to the different options available.
