Next: NewtonRaphson Method
Up: Approximate Numerical Methods for
Previous: Forms of Numbers and
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.
Figure 1.1:
A sketch of a continuous function, , that has a root at
.

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.
Figure:
The graph of illustrates a function that has an
infinite number of roots where the graph crosses the axis.

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
 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.
 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: NewtonRaphson Method
Up: Approximate Numerical Methods for
Previous: Forms of Numbers and
Prof. Alan Hood
20000201