Gene H. Golub

Iterative methods are an important and fundamental class of solution algorithms that are used by scientists and engineers. Their applications can be found in diverse fields of science, engineering and technology, for example, computational fluid dynamics, queuing and communication problems, and image and signal processing. Because of the rapid evolution of the development of this field, as well as the fact that iterative methods are not often developed in a generic form for general applications, there is a lack of published materials that treat the topic properly and fully. These lectures from the Winter School on Iterative Methods in Scientific Computing and their Applications aims to bridge such a gap in the literature. This book provides an excellent overview of the state-of-the-art of the field as well as being a general intoduction for beginners. It will prove useful to researchers, practitioners and engineers interested in practising scientific computing.

The solution of very large sparse or structured linear algebra problems is an integral part of many scientific computations. Direct methods for solving such problems are often infeasible because of computation time and memory requirements, and so iterative techniques are used instead. In recent years much research has focussed on the efficient solution of large systems of linear equations, least squares problems, and eigenvalue problems using iterative methods. This volume on iterative methods for sparse and structured problems brings together researchers from all over the world to discuss topics of current research. Areas addressed included the development of efficient iterative techniques for solving nonsymmetric linear systems and eigenvalue problems, estimating the convergence rate of such algorithms, and constructing efficient preconditioners for special classes of matrices such as Toeplitz and Hankel matrices. Iteration strategies and preconditioners that could exploit parallelism were of special interest. This volume represents the latest results of mathematical and computational research into the development and analysis of robust iterative methods for numerical linear algebra problems. This volume will be useful for both mathematicians and for those involved in applications using iterative methods.

This computationally oriented work describes and explains the mathematical relationships among matrices, moments, orthogonal polynomials, quadrature rules, and the Lanczos and conjugate gradient algorithms.