The text is an excellent introduction to numerical methods for upper undergraduate and beginning graduate students with a good mathematical background. The first section of the text (Chapters 1-7) covers methods for ordinary differential equations. This includes a very well written section on solving nonlinear systems using iteration. The second section (Chapters 8-15) of the text covers methods for solving the Poisson equation in one and two dimensions. Included are finite difference, finite element and spectral discretization methods as well as chapters on how to solve the resulting linear systems that arise. The final section (Chapters 16-17) discuss how to solve the heat, wave and Burgers equations. All chapters include pointers to relevant recent monographs on the specific contents of each chapter.

Using differential equations as a common thread to teach numerical analysis is clever. Similar level texts for teaching a course in numerical analysis are Numerical Analysis by L.R. Scott; A Friendly Introduction to Numerical Analysis by B. Bradie; and Numerical Mathematics by A. Quarteroni, R. Sacco and F.Saleri. All of these other texts discuss finite precision arithmetic, whereas Iserles does not. For many mathematicians, this will probably be a newly encountered topic with important practical implications. In most situations, floating point arithmetic is good enough that once one has a convergent and consistent algorithm, then it is simply a matter of implementing the algorithm. However, this is not always true. I can imagine many computational number theorists who work rather hard to obtain a large number of digits of accuracy. It would be good for mathematicians to encounter some of the problems that may arise from using floating point arithmetic when they need to obtain a prediction from a numerical solution of a differential equation. Iserles includes a good chapter on geometric integration, a topic which is covered in only a handful of monographs and is usually only encountered in the journal literature. In comparison to Bradie or Quarteroni, Sacco and Saleri, explicit algorithms or programs are kept to a minimum. It is perhaps a result of the style of course that students may not get any practical exercises where they need to implement numerical methods. This may make the text hard to use on its own for numerical analysis courses at a majority of universities. It does however make a very good second text. It is not as encyclopaedic as Quarteroni, Sacco and Saleri and is much more concise than Bradie's text which is more suited to students with an engineering background. In choosing between Scott's text and Iserles' text, the primary differences are that Iserles' text does not mention floating point arithmetic, though it does have enough material for a year long course, whereas Scott's text does mention floating point arithmetic, has some historical background but does not cover partial differential equations and so would only be suitable for a one term/semester course. Of the four books mentioned, only Quarteroni, Sacco and Saleri seems to be available online through the reviewers university library. For institutions which have purchased the online version, the print version of the book is available at a very reasonable $25.00. This makes Quarteroni, Sacco and Saleri very easy to adopt as a course text.

To summarise, the current text book, is carefully and clearly presented and is well suited as a primary text for a first course on numerical methods for mathematics students or students with a good mathematical background. For such students it fills an important niche. For students with an applied or engineering background who are taking a course in numerical differential equations, it is excellent supplemental reading. The reviewer also highly recommends the text for those teaching courses on numerical methods or requiring a good reference to help choose appropriate time discretization methods to solve differential equations.