Computing

As we have already mentioned in the preface to this book, we consider numerical analysis to be the mathematics of constructive methods which can be realized numerically. Thus, one of the problems of numerical analysis is to design computer algorithms for either exactly or approximately solving problems in mathematics itself, or in its applications in the natural sciences, technology, economics, and so forth. Our aim is to design algorithms which can be programmed and run on a calculator or digital computer. The key to this approach is to have an appropriate way of representing numbers which is compatible with the physical properties of the memory of the computer. In a practical computer, each number can only be stored to a finite number of digits, and thus some way of rounding off numbers is needed. This in turn implies that for complicated algorithms, there may be an accumulation of errors, and hence it is essential to have a way of performing an error analysis of our methods. In this connection there are several different kinds of error types, which in addition to the roundoff error mentioned above, include data error and method error.