7.1 INTRODUCTION

Many engineering problems can not be solved exactly by analytical techniques. The knowledge of numerical methodology is essential for determining approximate solutions. Numerical methods, in one form or another, have been studied for several centuries. We will look, for example, at Newton s method of approximating the solution of an algebraic or transcendental equation, at the Gaussian elimination method for solving linear systems of equations and at the Euler and Runge-Kutta methods for solving initial-value problems. Since the arrival of computers, the potential of these methods has been realized and the entire character of numerical methods has changed. We can now use iterative methods with much greater speed. We can also solve large systems of equations numerically. Many non-linear equations which were intractable in the past can now be solved approximately by numerical methods. As a result of these possibilities, mathematical modelling is now more realistic and a new field of numerical simulation has been opened up. Successful simulations of complex processes in science and engineering are now being achieved. In this chapter, we present different methods which will be useful to applied science students.

We first briefly examine the possible sources of error that may occur in the solutions obtained by numerical methods. Most numbers have an infinite decimal representation and for computational purposes they have to be rounded to a finite number of decimal places. This type of error is known as round off error and is unavoidable. During the process of calculation, we generate errors.