In this piece of coursework, I will use three different methods (Change of Sign, Newton-Raphson, and the Rearrangement Method) to find the roots of a series of different equations. The number of roots found differs from method to method.

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Colin Tebbett

Pure Mathematics 2: Coursework Assessment

In this piece of coursework, I will use three different methods (Change of Sign, Newton-Raphson, and the Rearrangement Method) to find the roots of a series of different equations.  The number of roots found differs from method to method.  A comparison of the three methods will be made at the end of the report.

Change of Sign Method

Equation:        x3 + x2 - 2x + 3 = 0

Using the above equation, the function f(x) = x3 + x2 - 2x + 3 may be generated.

This may be expressed in graphical form (see page 2)

This may be solved using the method of Interval Bisection (see page 3)

The Change of Sign Method can, however, fail when certain equations must be solved.  Below is an example:

Equation:        (x-3.1)2 (x + 4.2) = 0

Function:        f(x) = (x-3.1)2 (x + 4.2)

This fails because it has a repeated root ‘(x-3.1)2’, therefore there is no change of sign at this point.  A change of sign is required for this method to work, as it is the basis from which a root is found.

The graph of the function is displayed on page 5.

An attempt to solve the equation using Interval Bisection is shown on page 6.

This can be seen to have failed because, when looking at the graph on page 5, it can clearly be seen that the root is not located at x = 3.5.

Newton-Raphson Method

Equation:        x3 - 5x + 1 = 0

Using the above equation, the function f(x) = x3 – 5x + 1 may be generated.

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This may be expressed in a graphical form (see page 8)

The process of Newton-Raphson Iteration is displayed on the graph with regards to root ‘c’.  The window in the corner shows the steps taken to reach the value for the root.

The equation may be solved to find roots a, b, and c (see page 9)

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