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## Approximate Numerical Methods

Most numerical methods for solving equations in one variable are best illustrated graphically, indeed it is always a good idea to sketch the graph of the function before looking for the roots. Consider the roots of satisfying (1.8)

where is a continuous function of . Setting we can sketch the graph of . This is shown schematically in Figure 1.1. The object is to obtain the value of the root in the interval , where or equivalently where the graph crosses the -axis.

Before using the method described below, it is a good idea to identify the number of roots in the given interval. Consider Figure 1.2.  has an infinite number of roots. If we consider the interval , there is clearly only one root. Note that Thus, there is a change in the sign of the function values at the two ends of the interval. However, if we consider the interval , there is again a sign change since but this time there are three roots in the interval. So a change in sign means that there are an odd number of roots. In the first interval the function does not have a turning point but in the second interval the function has two turning points, one lying above the -axis and one lying below. These ideas are illustrated in the following example.

Example 1. .4

1. Show that has only one root in the interval . (This is a short hand notation for .) The function is continuous and Hence, there is a sign change and there must at least one root in . Now consider the derivative The derivative is positive and so there are no turning points at all. Hence, there can only be one root.

2. Show that has only one root in . Note Again there is a sign change and so there must be at least one root in the interval. To check the number of roots we look at the derivative There are two turning points where In this example, the turning points are outside the interval and so there can only be one root in the interval . In fact, it turns out for this function that there is only one root since at the turning points and Both turning points lie above the -axis and so there is only one root.   Next: Newton-Raphson Method Up: Approximate Numerical Methods for Previous: Forms of Numbers and
Prof. Alan Hood
2000-02-01