Numerical methods

This chapter explores numerical methods. It covers using graphical methods to find the number of roots of the equation f(x)=0, proving whether a root lies within a given interval, using iteration within a given interval, using iteration to find an approximation to the root of the equation f(x)=0.
We can use graphs to find approximations for the roots of the equation f(x)=0. The following example explores this;

Example

Suppose we had to use a graph to show that an equation x³-3x²+3x-4=0 has a root between x=2 and x=3.
First we would draw the equation graph.
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You must know that where y=0 is also the x-axis so y=0 when the curve crosses the x-axis. We can find this point on the plotted graph, so we find the x-values when y=0. On the graph it is clear that the curve crosses the x-axis between x=2 and x=3, that must mean there is a root of the equation between x=2 and x=3

Example

In this example we’re going to prove the change of sign of the y values when the curve crosses the x-axis. Suppose we had to show that the values of y for points on the graph of y=4+2x-x³ change sign as the graph crosses the x-axis. First we would draw the curve as shown below;
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From the graph we can see that the graph curve lies above the x-axis for all points below x=2 therefore the y values are positive. And also the curve lies below the x-axis for all values above x=2. Therefore there is a change of sign as the graph crosses the x-axis.

Example

In this example we shall prove that an equation has roots in a certain interval without using graphics. Suppose we had to show that e^x + 2x – 3 = 0 has a root between x=0.5 and x=0.6. First let;
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We shall substitute x=0.5 and 0.6 into the equation to see whether f(x) changes sign. We shall start with x=0.5.
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…now we can try x=0.6.
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We can conclude that the root lies between x=0.5 and 0.6 because f(x) changes sign between these values.
It has become evident from the above examples that if you find an interval in which f(x) changes sign then the interval must contain a root of the equation f(x)=0 or crosses the x-axis between this interval. However this is not true for when f(x) has a discontinuity in the interval for instance the following equation has a discontinuity at x=0.
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This is shown in the graph below;
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In the above graph we can see that to the left of x=0, f(x)>0. The graph above does not change sign at x=0 but does change sign for any interval which contains x=0. We say there is a discontinuity at x=0.

Iteration