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Reviewer: Donald G. M. Anderson

Together with its companion volume [1], this book constitutes the most comprehensive and definitive treatise on the numerical solution of ordinary differential equation initial value problems currently available. The main competitor is Butcher [2], for which a second volume is projected. The two volumes of the set are characterized by a scholarly and historical approach, and by the inclusion of challenging examples and exercises (often of scientific or historical significance) and of extensive comparisons of numerical results for different methods applied to difficult problems. The thrust, however, is the theory underlying the principal classes of methods, a unifying theme being the use where appropriate of the rooted tree formalism introduced by Butcher, who provides an analogous treatment [2]. The sheer volume of material precludes the use of these books as a text other than in an advanced, specialized course. This volume is a second revised edition of a book originally published in 1986 [3]. The work is organized as three long chapters with many major sections. The first chapter reviews historical and mathematical background. The second chapter deals with Runge-Kutta and extrapolation methods, not only for the standard first-order problem but also for various extensions thereof, such as second-order and delay equations. The third chapter discusses multistep and general linear methods. These classes of methods are revisited in the context of stiff and differential-algebraic problems in the second volume. An appendix lists FORTRAN implementations of some of the methods tested. The physical presentation has been upgraded to the standards of the second volume. Sections on topics that have become fashionable since the first edition was prepared, such as parallel integrators, symplectic methods, and dense output (and related issues such as discontinuities), have been added or extended. Some sections have been substantially rewritten, and some material has been deleted (with the first edition unfortunately being cited as the only alternate reference in some cases). The codes in the appendix have been revised, and the numerical results updated. Basically, however, the bulk of the material in the first edition is unchanged in the second. The book is generally well written, though obscure in places. Some of the more algebraically complicated arguments are just sketched or the results merely quoted and discussed, with details left to cited references. Though a lengthy bibliography is included, in keeping with the scholarly and historical character of the presentation, the authors explicitly acknowledge that it is not comprehensive, and indeed, important references that one could reasonably expect to find there (including volume 2 of this work [1]) are inexplicably omitted, a missed opportunity in my opinion.