ABSTRACT
At the end of this chapter, you should be able to:
• define iterative methods • use the method of bisection to solve equations • use an algebraic method of successive approximations to solve equations • state the Newton-Raphson formula • use Newton’s method to solve equations
Many equations can only be solved graphically or by methods of successive approximations to the roots, called iterative methods. Three methods of successive approximations are (i) bisection method, introduced in Section 25.2, (ii) an algebraic method, introduction in Section 25.3, and (iii) by using the Newton-Raphson formula, given in Section 25.4. Each successive approximation method relies on a reasonably good first estimate of the value of a root being made. One way of determining this is to sketch a graph of the function, say y= f (x), and determine the approximate values of roots from the points where the graph
cuts the x-axis. Another way is by using a functional notation method. This method uses the property that the value of the graph of f(x)=0 changes sign for values of x just before and just after the value of a root. For example, one root of the equation x2 − x−6=0 is x=3. Using functional notation:
f (x) = x2 − x − 6 f (2) = 22 − 2− 6 = −4 f (4) = 42 − 4− 6 = +6
It can be seen from these results that the value of f (x) changes from −4 at f (2) to +6 at f (4), indicating that a root lies between 2 and 4. This is shown more clearly in Figure 25.1.
