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.