This volume pays special attention to the combined issues of speed and numerical reliability in algorithm development. These two requirements have often been regarded as competitive, so much so that the design of fast and numerically reliable algorithms for large-scale structured systems of linear equations, in many cases, remains a significant open issue. This work seeks to bridge this gap by providing the reader with contributions written by several experts in the field. The authors deal with both the theory and the practice of fast numerical algorithms for large-scale structured linear systems. Each chapter covers in detail different aspects of the most recent trends in the theory of fast algorithms, with emphasis on implementation and application issues. Both direct and iterative methods are covered. The book is not merely a collection of articles. The editors seek to blend the individual papers into a consistent presentation. Each chapter exposes the reader to research whilst providing enough background material to put the work into proper context.