# Solution of equations by Numerical Methods.

## Pure 2 Coursework

### Solution of equations by Numerical Methods

#### Method 1: The Change of Sign method

The simplest method for solving an f(x) function is to use a change of sign method; these include the methods of bisection, decimal search and linear interpolations.

Unfortunately, as well as being the simplest methods, they are also relatively cumbersome. The bisection was employed below for the function of f(x)=x5+3x2+x+2. This was also displayed graphically in the graph below.

The actual calculations for this method are summarised below in this table. I placed these with relevant diagrams

The error bounds for this result are conveniently provided by the use of this method, and are -1.485<x<-1.475. The maximum error is therefore ± 0.005.

However, this method is clearly not particularly easy to work with. In addition, as with many methods there are some functions for x, which do not work. An example is the function f(x)=x3-2x2-x. This is shown below.

The problem in this situation is the fact that there is in fact two roots in a very short space. (Labelled (a.) and (b.) on the graph).

Using the bisection method within the interval of [-1, 0], this is bisected into 0.5. The root of 0 is the one to which the calculations tend. However, the other root, of –0.4142 (to 5 s.f.) is missed. This is illustrated in more detail in the diagram below.

#### Method 2: The Newton-Raphson Method

The next method, which can be used to solve functions of x numerically, is the Newton-Raphson method. An example of how this method works is shown below.

The graph of f(x)=x5-6x+1 was plotted using a graph-plotting program, the result is shown below.

Using an estimate for (a.) as 0, the Newton-Raphson iterative formula was then used.

This next diagram gives some indication as to how the Newton- ...