# Solution of Equations by Numerical Methods

Damian Chandler 13MP        66555 0474        Maths Coursework

Solution of Equations by Numerical Methods

Change of Sign Method – Numerical Search

By using the computer program ‘Omnigraph’, I have produced this graph of the equation y = x^3-9x+7.

The roots of this equation are shown where the line crosses the X-axis. I can now see that the three roots of this equation lie between (-4,-3), (0,1) and (2,3). Due to the fact that I have only declared the roots to be within wide margins, I will need to use a different method to acquire a more accurate answer. Instead of simply saying “the root is about -3.3”, I can use a method called decimal search to find an answer to any degree of accuracy that I decide.

Decimal search works by finding two points on the x-axis of a graph, where between, the sign of Y changes, i.e. where the line crosses the x-axis. Without the graph I would be able to work out that the sign changes three times, meaning there are three roots.

This table shows the values of Y when X is between -5 and 4:

From this I can now work out a more accurate answer to where the roots lie. I now know that a root lies between (-4,-3). To investigate further, I need to find just where between these points the root is. I can do this by the same method as before accept I will use numbers between the points I already know the root lies.

This table shows the values of Y when X is between -4 and -3:

This shows that the root lies between (-3.4,-3.3). As the sign changed at -3.3, it was needless to continue to 3. To find an even more accurate answer, I simply repeat the method again for all the numbers between (-3.4,-3.3).

This table shows the values of Y when X is between -3.4 and -3.3:

This shows that the root lies between (-3.34,-3.33).

This table shows the values of Y when X is between -3.34 and -3.33:

This shows that the root lies between (-3.332,-3.331). From these tables I now know that X = -3.3315 ± 0.0005.

Failure of Decimal Search

Decimal search is only effective in certain equations. When using decimal search it is usual to begin by using whole numbers. If an equation ...