Discuss the bisection method for finding roots of nonlinear equations. Provide an example.

The bisection method is a numerical technique used to find the roots of nonlinear equations. It is a simple and robust method that relies on the intermediate value theorem. The basic idea behind the bisection method is to repeatedly divide the interval containing the root in half and determine which subinterval the root lies in. This process is continued until the desired level of accuracy is achieved.

The steps involved in the bisection method are as follows:

1. Start with an interval [a, b] such that f(a) and f(b) have opposite signs, indicating that a root lies within the interval.

2. Calculate the midpoint c = (a + b) / 2.

3. Evaluate f(c) and check if it is close enough to zero. If f(c) is sufficiently small, then c is considered as the root.

4. If f(c) is not close enough to zero, determine which subinterval [a, c] or [c, b] contains the root. This can be done by checking the signs of f(a) and f(c) or f(c) and f(b).

5. Repeat steps 2-4 until the desired level of accuracy is achieved.

Here is an example to illustrate the bisection method:

Consider the equation f(x) = x^3 - 2x - 5. We want to find a root of this equation using the bisection method.

Let's start with the interval [2, 3]. Evaluating f(2) and f(3), we find that f(2) = -1 and f(3) = 16. Since f(2) and f(3) have opposite signs, we can conclude that a root lies within the interval [2, 3].

The midpoint of the interval is c = (2 + 3) / 2 = 2.5. Evaluating f(2.5), we find that f(2.5) = 2.375. Since f(2.5) is positive, the root must lie within the subinterval [2, 2.5].

We repeat the process by calculating the midpoint of the subinterval [2, 2.5]. The new midpoint is c = (2 + 2.5) / 2 = 2.25. Evaluating f(2.25), we find that f(2.25) = 0.609375. Since f(2.25) is positive, the root must lie within the subinterval [2, 2.25].

We continue this process until we reach the desired level of accuracy. After several iterations, we find that the root of the equation f(x) = x^3 - 2x - 5 is approximately x = 2.094.

The bisection method is a reliable and straightforward technique for finding roots of nonlinear equations. However, it may require a large number of iterations to achieve high accuracy, especially for functions with complex behavior. Other numerical methods, such as Newton's method or the secant method, may be more efficient in such cases.