C3 Coursework: Numerical Methods

C3 Coursework:

Sapphire Mason-Brown

C3 Coursework

Numerical Methods

The place in which the graph of a line crosses the x axis is known as the root of the equation. It is not always possible to find the solution of an equation by algebraic or analytical methods such as factorising. This applies to equations such as y=3x3-11x+7. To solve equations such as these, numerical methods such as change of sign, x=g(x) and Newton-Raphson can be used to give estimates of the roots.

Change of Sign Method

The Change of sign method is a method used to look for when a sequence of numbers in the boundary of a root change from negative the positive or vice versa. This change means that the root of the equation is somewhere between the interval where there is a change of sign.

This is the graph of the equation y=3x3-11x+7

There are 3 roots to the equation y=3x3-11x+7, this is illustrated by the three intersections with the x axis. There appears to be a root between 0 and 1.

By taking increments of 0.1 between 0 and 1 it will be possible to use decimal search to look for a change in sign. This will make it possible to find an approximation for to the root between 0 and 1.

This table shows the results of the numbers in increments of 0.1 between 0 and 1. Upon looking at the results we can see that there is a change of sign between 0.7 and 0.8, this shown by the graph below.

This method is repeated again with intervals of 0.01 between 0.7 and 0.8

The table shows that there is a change if sign between 0.75 and 1.76. This means that the root of the equation lies between 0.75 and 0.76. This is shown by the graph below.

Once again this method is repeated with intervals of 0.001 between 0.75 and 0.76.

The able shows that there is change of sign from 0.752 to 0.753. This means that the root of the equation must lie between 0.752 and 0.753. Our estimate of the root is with a maximum error of ± 0.0005. This is illustrated by the graph below.

This graph shows the equation of the line y=x³+10x²+4.8x+0.576

The roots appear to be between -1 and 0. By taking increments of 0.1 between -1 and 0 it will be possible to use decimal search to attempt to look for a change in sign.

As you can see from the table there is no change of sign. As the graph of y=x³+10x²+4.8x+0.576 is a repeated root the change of sign method, and thus Change of Sign, cannot be used to find the root. Repeated roots such as y=x³+10x²+4.8x+0.576 cause the change of sign method to fail as there will be no change of sign.

Newton-Raphson Method

The Newton Raphson method is a fixed point iteration method. It uses an estimate of the root as a starting point.