A description of numerical analysis

The subject of numerical analysis is concerned with devising methods for approximating, in an efficient way, the solutions to mathematically expressed problems (Trefethen gives a more elaborate definition for numerical analysis). The efficiency of the method depends both upon the accuracy required and the ease with which the method can be implemented. In a practical situation, the mathematical problem is derived from a physical (chemical, biological, economical) phenomenon where some simplifying assumptions have been made to allow the mathematical representation to develop. Generally a relaxation on the physical assumptions leads to a more appropriate mathematical model, but at the same time one that is more difficult or impossible to solve explicitly. Since the mathematical problem ordinarily does not solve the physical problem exactly in any case, it is often more appropriate to find an approximate solution to a more complicated mathematical model of a physical problem than to find an exact solution of a simplified model. To obtain such an approximation a method called an algorithm is devised. The algorithm consists of a sequence of algebraic and logical operations that produces the approximation to the mathematical problem, and, it is hoped, to the physical problem as well, within a prescribed tolerance or accuracy.

Since the efficiency of a method depends upon its of implementation, the choice of the appropriate method for approximation the solution to a problem is influenced significantly by changes in calculator and computer technology. Twenty-five years ago, before the widespread use of digital computing equipment, methods requiring a large amount of computational effort could not be reasonably applied. Since that time, however, the advances in computing equipment have made some of these methods increasingly attractive. At present, the limiting factor generally involves the amount of computer storage requirements of the method, although the cost factor associated with a large amount of computation time is, of course, also important. The availability of personal computers and low cost programmable calculators is also an influencing factor in the choice of an approximation method, since these can be used to solve many relatively simple problems.

The basic ideas that underlie most current numerical techniques have been known for some time, as, have the methods used in predicting bounds for the maximum error that can be produced in an application of the methods. It is of primary interest, then, to determine the way in which these methods have developed and how their error can be estimated, since variations of these techniques will undoubtedly be used to develop and apply numerical procedures in the future, irrespective of the technology.

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