This book contains contributions from the invited speakers to the fifth S.E.R.C Summer School in Numerical Analysis, which was held at Lancaster University from 19th to 31st July, 1992.The expositions were at a level which could be understood by post-graduate research students, yet would be advanced enough to stimulate established researchers. The book should therefore be useful to a wide class of readers.The topics which are covered include some of the most important areas of current research in numerical analysis. Part I deals with the use of parrallel computers, both for solving large sets of linear equations and calculating the eigensystems of large matrices. Such problems arise from the discretization of partial differential equations. Aspects of the solution of such equations are dealt with in Part II. These include the preconditioning of elliptic problems, the study of semi-conductors, and description of recent methods for the solution of hydrodynamic problems.The contributors are: Jesse L. Barlow, Pennsylvania State University; Professor Jack Dongarra, Oak Ridge National Library; Professor Howard C. Elman, Institute for Advanced Computer Studies, Maryland; Professor Randolph E. Bank, University of California; Professor J.W. Jerome, Northwestern University, and Professor Maurizio Pandolfi, Politecnico di Torino, Italy.