Pure Mathematics 2 Katie Ruck

Solution of Equations by Numerical Methods

## Change of Sign Method

If we let f(x)=3x³-0.5x²-0.5x-1, in order to solve this equation and determine its roots, it is necessary for it to be written in the form 3x³-0.5x²-0.5x-1=0.

The root of the equation f(x)=0 is indicated where y=f(x) crosses the x-axis.

Roots of the equation 3x³-0.5x²-0.5x-1=0, will be found to a three decimal place accuracy.

Having illustrated the equation graphically using Autograph, it is evident that the equation has only root that lies between the interval [0,1].

Before proceeding, it is necessary to check that there is a sign change in the above interval:

f(0) = 0-0-0-1= -1

f(1) = 3-0.5-0.5-1 =1

The method that will be used for the numerical solution of the equation is the decimal search method.

I will first take the increments in x of size 0.1 within the interval [0,1], calculating the value of the function 3x³-0.5x²-0.5x-1 for each one, until a change of sign is found.

The above table illustrates that there is a sign change. It can therefore be understood that the root lies in the interval [0.8,0.9].

Having narrowed down the interval, I will continue with the decimal search, but now using increments of 0.01 within the interval [0.8,0.9].

This indicates that the root lies in the new interval [0.83,0.84].

I will continue the process using the increments of 0.001 within the interval [0.83,0.84].

The root therefore lies in the interval [0.838,0.839]. I shall now use increments of 0.0001 in the interval [0.838,0.839].

Therefore the root lies between 0.8389 and 0.8390. To three decimal places, the root = 0.839.

## Error Bounds

A change of sign method such as the one used, provides bounds within which a root lies so that the maximum possible error in a result is known.

When x = 0.8385, f(0.8385) = -0.00218772512

When x = 0.8395, f(0.8395) = 0.00280856463

The error bounds of the root 0.839 are 0.839 ± 0.0005.

However, I am able to say that I have a more accurate solution, as I know that the root lies in the interval [0.8389,0.8390].

## Failure of the Change of Sign Method

There are a number of situations that can cause problems for change of sign methods. For example, let y=f(x)=x³-2x²-x+2.63.

This curve is shown graphically below.

With this example I shall use the decimal search method to find a change of sign and so investigate the roots of the previous equation. Integers will be used as the x-values.

From the graph it is evident that the equation x³-2x²-x+2.63=0 has more than one root, i.e. roots that lie in the intervals [-2,-1] and [1,2]. However, the above table illustrates that the method used only detects one such root that lies in the interval [-2,-1].