The 1996 conference on the State of the Art in Numerical Analysis was organized to provide the numerical analysis community, and users of numerical methods, with a forum where an account of the important recent developments in the subject could be presented in a coherent and concentrated way in a manner accessible to the non-specialist in the sub-area. This volume extends to a much wider audience the opportunity given to those who attended the meeting. It contains full versions of all the papers presented, with one exception on Neural Nets. In addition to recent developments in mainstream topics, linear algebra, ordinary differential equations, approximation and optimization, it contains papers devoted to two important application areas, numerical tomography and image processing. Compare with the corresponding 1986 volume, some topics show a continuous and natural development, while others show significant departure from conventional trends. A recurring theme is the solution of large problems and exploitation of structure. Underlying many of the developments is the fact that increasingly complicated and sophisticated problems are now amenable to an increasingly powerful range of numerical techniques. This is greatly helped by the advent of excellent computer languages, like MATLAB, and state of the art Fortran software such as LAPACK, so that the fundamental building blocks for much of the armoury of a numerical analysis are now readily accessible.

The subject of sparse matrices has its root in such diverse fields as management science, power systems analysis, surveying, circuit theory, and structural analysis. Efficient use of sparsity is a key to solving large problems in many fields. This second edition is a complete rewrite of the first edition published 30 years ago. Much has changed since that time. Problems have grown greatly in size and complexity; nearly all examples in the first edition were of order less than 5,000 in the first edition, and are often more than a million in the second edition. Computer architectures are now much more complex, requiring new ways of adapting algorithms to parallel environments with memory hierarchies. Because the area is such an important one to all of computational science and engineering, a huge amount of research has been done in the last 30 years, some of it by the authors themselves. This new research is integrated into the text with a clear explanation of the underlying mathematics and algorithms. New research that is described includes new techniques for scaling and error control, new orderings, new combinatorial techniques for partitioning both symmetric and unsymmetric problems, and a detailed description of the multifrontal approach to solving systems that was pioneered by the research of the authors and colleagues. This includes a discussion of techniques for exploiting parallel architectures and new work for indefinite and unsymmetric systems.