Lambert has written a sequel to a well-received earlier book on the same general subject [1]. Comparison of the two books illustrates the dramatic evolution of the field over the past two decades: the overlap is minimal. The attractiveness of the first book derived in large part from the author’s talent for distilling the essence of complicated ideas and presenting them coherently. This remains the thrust and merit of the second book, though the attempt is less successful.

Several strategic decisions underlie the differences in approach of the two books. The first book focused on a single differential equation; the second deals primarily with systems of equations, a choice with both theoretical and practical consequences. The first surveyed the full range of existing methods; the second confines its attention to the particular methods that now provide the basis for widely available codes. The first reviewed the theoretical background of the subject in some detail; the second eschews most substantial proofs, though references to the literature are provided, in favor of motivation for and illustrations of the theory and a more thorough examination of implementation issues. Reflecting the intervening advances in the state of the art, the treatment of the stiffness phenomenon in the first book is reconsidered and elaborated upon at length.

After a chapter on background material, which might better have been relegated to an appendix, the basic concepts of convergence, consistency and stability, and their interrelations are introduced in the second chapter. The third and fourth chapters deal with linear multistep methods and their combinations as predictor-corrector methods, focusing on the Adams and backward differentiation formula (BDF) methods and their implementation in variable-step, variable-order form. The fifth chapter studies Runge-Kutta methods, organized around an excellent summary of the Butcher theory thereof. The sixth and seventh chapters cover approaches to the stiffness phenomenon through linear and nonlinear stability analysis, sorting a tangled skein of definitions and theorems but leaving most of the details to other sources. Runge-Kutta methods emerge as prospective challengers to the current dominance of BDF methods for stiff problems.

The book is marred by an excessive number of minor errors, both typographical and substantive. The author’s attempt to simplify the presentation is at times potentially misleading, though not seriously so, since the implications and lacunae may not be readily apparent to the uninitiated reader. The cost of being more precise, without excessive rigor or generality, would be more than repaid by added clarity in some instances.