This book is based on a course of introductory lectures that I have given for a number of years in Cambridge. I hope that it will be useful not only to students but also more generally to those who need to make use of a digital computer for scientific and engineering purposes. I have endeavoured to give the subject a modern slant and to confine myself to essentials.

The longest chapter is that on interpolation. This is not because of the practical importance of interpolation as such (it is, in fact, a rare operation to perform in a digital computer) but because the idea of the interpolating polynomial is fundamental to the use of finite difference methods in numerical analysis generally. The chapter on interpolation should, therefore, be regarded as laying the theoretical foundations for what is to follow.

Systematic use is made of difference operators for deriving finite difference formulae, although alternative methods are given in the more important cases. Since the view taken is that finite difference formulae are only proved when the functions concerned are polynomials, expressions containing difference operators may be regarded as convenient abbreviations for finite expressions that could be written out in full. No elaborate theoretical justification of the use of such operators is therefore called for. The finite difference formulae once derived are, of course, applied at user's risk to functions which can only approximately be represented by polynomials.

Methods for solving linear equations can be divided into direct methods, which are equivalent to elimination, and indirect or iterative methods. Direct methods are generally to be preferred, except in the following special circumstances when indirect methods are indicated:

1. (1) the number of equations is large in relation to the digital computer available,

2. (2) the equations are such that the convergence of a suitably chosen iterative method is specially rapid,

3. (3) a specially good starting approximation is available.

The number of equations that can be handled by direct methods has increased steadily with the increasing power of digital computers. With a modern computer of reasonable power (say a multiplication time of 250 μs, and at least 16,000 words of core storage) it takes between one and 2 min to solve a set of 100 equations in 100 unknowns. The time increases with the number of equations n by a factor between n3 and n4. For a given computer this rule breaks down when n becomes so large that all the coefficients cannot be accommodated at the same time in the high speed store, so that an auxiliary store with longer access time has to be used.

The above remarks refer to equations of general form. Banded equations such as arise from differential equations can be handled in much larger sets—up to several thousand in the computer mentioned above—and the time for solution increases more or less linearly with n.

In science and engineering, we are typically concerned with some particular aspect of the physical world, and this we investigate by making use of a mathematical model. The use of a model serves two purposes—it enables us to isolate the relevant aspects of a complex physical situation and it also enables us to specify with complete precision the problem to be solved. When the model has been established, the next step is to write down equations expressing the constraints and physical laws that apply. These equations may be simple algebraic equations; on the other hand they may be differential or integral equations.

The equations must now be solved and here a choice presents itself. One way is to proceed by the methods of conventional mathematical analysis, in which case we shall hope to obtain the solution in the form of a formula or a set of formulae. Inspection of this solution may then yield qualitative results of interest; for example, it may be observed that one quantity varies exponentially with respect to some other quantity* that some variable has only a second-order effect on the result, and so forth. If quantitative results are required, they may be obtained by substituting numerical values in the formulae.

The alternative procedure is to express the equations by means of numerical analysis in a form in which they can be solved by computation. This leads, of course, directly to quantitative results. However, if enough such results are obtained, then qualitative results may emerge; for example, it may appear that one quantity is proportional—to the accuracy of the computations—to another, or that changes to one variable have only a slight effect on the result.