For several years the mathematics department at Lancaster University has organised a summer school in numerical analysis. These events last for two weeks, and are intended to cater for post-graduate research students, and yet to be sufficiently advanced to challenge and extend the knowledge of established researchers.

This book is a record of the proceedings of the fifth of the series, held in July 1992; like each of the series, the two weeks of this school each had a main topic, the two topics being closely related.

The first week was on the subject of "Large-scale matrix problems", while the second dealt with "Numerical solution of partial differential equations". As numerical methods for the solution of partial differential equations inevitably lead to large matrix problems, the link between the two weeks is very close.

There were six invited speakers, each of whom gave five lectures. In the first week the three speakers all came from the United States.

ÁñÁ«ÊÓƵ

Jesse Barlow discussed recent advances in the application of modern parallel computers to the symmetric eigenvalue problem.

Jack Dongarra reviewed the progress of the Lapack project, that has the aim of providing efficient software for a range of matrix problems which will be portable over a wide range of contemporary computers.

ÁñÁ«ÊÓƵ

Howard Elman surveyed some of the important iterative techniques used to solve large sparse systems of linear equations.

In the second week there were two more American speakers, and one from Italy.

Randolph Bank surveyed a class of iterative methods for elliptic problems, and Joseph Jerome discussed the challenging problems which arise in mathematical models of semi conductors.

Finally Maurizio Pandolfi spoke about hyperbolic systems, with particular applications in fluid flow problems. The texts of these contributions appear very much as the authors provided them, with the minimum of editing.

This means that the six sections are completely self-contained, with no cross references at all. It is a pity that the last contribution is only an extended abstract, which will only be useful to those who are already familiar with the field. It is also a pity that there is no index.

The contributors evidently prepared their text with great care, and there is a welcome lack of typographical errors; there is however an unfortunate error on the cover, where Dongarra is incorrectly credited as being at the Oak Ridge National Library.

ÁñÁ«ÊÓƵ

As always in a book of this kind there is some variation in the style of the contributions.

Sections 1 and 2 are largely descriptive, while 3, 4 and 5 are more formally mathematical, with a good many lemmas and theorems, though the reader is referred elsewhere for most of the detailed proofs.

ÁñÁ«ÊÓƵ

The first two sections both deal in some detail with the use of parallel computers, and are concerned with the details of data structures and the optimisation of the inner loops of an algorithm to obtain the maximum efficiency from such machines.

The summer school was a stimulating but rather exhausting experience, and in some ways the same is true of this book.

It contains a great deal of information which is still about as up to date as we can reasonably expect from a book in hard covers, but it is not the sort of book which could easily be read through from beginning to end.

For those with an interest in unusual and challenging numerical problems, Jerome's study of the modelling of semi conductors brings to bear a range of mathematical and computational tools and presents the state of the art in masterly fashion.

All the sections of this book are useful, and it is worth having just for Jerome's section alone.

D. F. Mayers is university lecturer in numerical analysis, Computing Laboratory, University of Oxford.

ÁñÁ«ÊÓƵ