Common wisdom demands that unstable algorithms be avoided when numerically solving problems given in a mathematical formulation. But what does “(un)stable” actually mean? The usual answer to this question is that stable algorithms do not amplify errors introduced due to, for example, rounding, finite precision of the arithmetic, imprecise knowledge of input data, or the approximation method underlying the algorithm by such an amount that the final result becomes meaningless. Evidently, this is not a very precise definition of stability. Hackbusch’s book attempts to provide a more rigorous answer to the above question. This endeavor turns out to be quite a challenge because there is no universally valid general stability concept; rather, one has to deal with different ways to approach the issue for the individual sub-areas of numerical mathematics.
Following some introductory words and illustrative examples in the first two chapters, Hackbusch dedicates each of chapters 3 through 8 of his book to one special area. In this process, he exclusively concentrates on numerical methods for problems arising in calculus and the neighboring fields.
Regarding algorithms for problems from linear algebra, stability has already been discussed in great detail in the seminal work of Higham [1], to which I consider Hackbusch’s book a natural companion. The fact that Hackbusch concentrates on questions from calculus and related fields allows him to stress similarities of these numerical methods that cannot be found in linear algebra; in particular, this applies to the consequences of the observation that all of the problems he discusses are essentially formulated in an infinite dimensional space and need to be discretized and thus transferred to an appropriate finite dimensional subspace.
Specifically, chapter 3 is devoted to numerical integration. This is a relatively simple and easily understandable application that nevertheless exhibits all the relevant features; hence, it is a very good point for starting the investigations and for introducing the required machinery, particularly the essential tools from functional analysis. The closely related subject of interpolation is then addressed in a similar way in chapter 4.
In chapter 5, the author turns his attention toward numerical methods for ordinary differential equations. Whereas the situation is uncomplicated in the case of one-step methods (essentially, all reasonable algorithms are stable), things are much more involved for multistep methods. Therefore, Hackbusch presents a detailed analysis of the stability properties of such methods, in particular providing a very thorough explanation of the by no means obvious connections between stability and the coefficients of the methods. It must be mentioned, however, that the standard literature in this area (for example, [2]) does not only talk about the basic version of stability, but also about many variants like A-stability, L-stability, and so on; regrettably, none of these variants is discussed in the present book.
Chapter 6 is then devoted to finite difference methods for parabolic and hyperbolic differential equations. The chapter begins with a lengthy presentation of the analytic properties of such equations and their solutions; it only becomes clear much later why this is important for the stability investigation of the numerical methods. Nevertheless, the relevant results are stated and explained very clearly.
Then, chapter 7 handles the stability questions for finite difference and finite element methods for elliptic differential equations, and chapter 8 briefly presents the corresponding analysis for various numerical approaches for solving integral equations.
In essentially all sub-areas of numerical calculus, the concept of stability is closely related to the notions of consistency and convergence, usually via the statements that convergence implies stability and that consistency and stability imply convergence. In the various chapters, Hackbusch also sheds light on these important relations and discusses the rare situations where this rule is not applicable.
There are a few instances where the notation is chosen in a slightly unfortunate way, for example, in chapter 6 where the symbol p stands for both a prolongation operator and the exponent of an ℓp or Lp norm. Apart from these minor deficiencies that do not limit the book’s usefulness, it is written in a nicely readable style. All chapters are self-contained, so a reader who is interested in just one special case only needs to consult the appropriate chapter. The contents are presented in a way that is accessible to graduate students who may use the book for self-study of the topic, and it can easily be used as a textbook for a corresponding lecture series. Moreover, advanced researchers in numerical mathematics are likely to benefit from reading it, in particular because the book provides interesting insight into how stability relates to areas other than their own particular specialization field. Although implementation issues are not discussed, it is also useful reading material for numerical software developers.
