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Bisection

Success

The success of the bisection method can be proved by using the following equation:

        

f(x) = 0.64x4+0.28x2-2x+0.28

For this method, I have chosen to find the root between 0 and 1.  Below is the process taken to find this root, each table represents zooming in once (halving the window e.g. first window is 1 unit, second window is 0.5 units).

As to achieve the desired accuracy (of 3 decimal places) this process had to be repeated many times, the last of which is shown below:

Shown in the lightly shaded area; are the X values for which the first 3 decimal places are the same.

From the image above it can be seen that the process has succeeded in finding the root, although on the diagram it appears that the line is cutting at 0.143, it does indeed cut just before, however due to the limitations of the software, it cannot zoom in anymore to show properly labelled numbers.

Failure

The theory behind bisection is by splitting up two boundaries in two (by halves) so to defeat this method, there will need to be two roots within 1 unit (e.g. between 0 and 1) this way only one of the two roots can be found, and thus resulting in a failure.

The chosen formula is shown below:

f(x) = -12.4x4-0.3x3+9.9x2-0.6x-0.6

From this it can be seen that the bisection will fail (from theory) however the proof and calculations are below:

As seen above, by using the bisection method, only one of the two roots is being zoomed into, meaning that it has failed in finding the remaining root contained within the specified region.

Accuracy

The accuracy of each f(x) value within the tables is to 6 decimal places, which means that all figures have an accuracy of ±0.0000005.  This I feel is very accurate and due to the nature of zooming in, accuracy can be obtained to an infinite degree (depending on the function) however I feel that 6 decimal places are enough.


Fixed point iteration – Newton Raphson

This method is based on the theory of utilising gradients of the curve to eventually zoom in close enough to the root.  The stated formula (shown below) will result in the finding the gradient of the specified point and its root will be used as a point for another gradient, and if the graph satisfies the criteria, then with each gradient taken, the root of the gradient will move increasingly closer to the root of the original curve.

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Xn+1 =   Xn–    f(Xn)
                                       f '(Xn)

From this equation it can be seen that when the Y value of f(x) is divided by the deferential of that equation, the change in X is the result.  Therefore, if the change in X is taken away from the original X value (X0 will always be an integer value) it will result in a value of X which is closer to the root.  So by repeating this iterative process, the root can be found ...

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