Matrix Properties and Concepts

The title of this book, Matrix Iterative Analysis, suggests that we might consider here all matrix numerical methods which are iterative in nature. However, such an ambitious goal is in fact replaced by the more practical one where we seek to consider in some detail that smaller branch of numerical analysis concerned with the efficient solution, by means of iteration, of matrix equations arising from discrete approximations to partial differential equations. These matrix equations are generally characterized by the property that the associated square matrixes are sparse, i.e., a large percentage of the entries of these matrices are zero. Furthermore, the nonzero entries of these matrices occur in some natural pattern, which, relative to a digital computer, permits even very large-order matrices to be efficiently stored. Cyclic iterative methods are ideally suited for such matrix equations, since each step requires relatively little digital computer storage or arithmetic computation. As an example of the magnitude of problems that have been successfully solved on digital computers by cyclic iterative methods, the Bettis Atomic Power Laboratory of the Westinghouse Electric Corporation had in daily use in 1960 a two-dimensional program which would treat, as a special case, Laplacian-type matrix equations of order 20,000.