Åke Björk and Germund Dahlquist are well known for their first textbook on numerical analysis, which was published in 1974. The focus on key principles of numerical analysis and the particularly challenging exercises made it a classic textbook and reference. The book was so successful that it was reprinted by Dover in 2003.
The authors' new book (a posthumous one for Dahlquist, who passed away in 2005) is the first volume in a two volume set that takes a similar approach to the subject while incorporating important developments that have occurred over the past 30 years. The first volume covers the properties of floating point arithmetic, interpolation, approximation of functions, summation of series, quadrature, and root finding. Three online appendices contain additional material on linear algebra, a MATLAB package for high precision arithmetic, and a survey of the literature on numerical analysis. A forthcoming second volume will cover numerical linear algebra in much greater detail.
Many recent textbooks on this subject focus on the presentation of numerical methods in cookbook fashion, with an emphasis on implementation of the methods rather than mathematical analysis of the accuracy and convergence of the methods. Like its predecessor, the new book takes a theoretical approach that includes rigorous error bounds and proofs of convergence. For example, you won't find the formulas for the Runge-Kutta-Fehlberg method in this book, but you will find a proof of Chebyshev's equioscillation theorem. The greatest strength of this book is in the well written proofs and the accompanying theoretical exercises.
Students studying numerical analysis also need to see computational examples and write their own programs to gain practical experience. In the authors' first book, Fortran was used in some examples and exercises. The authors of many recent textbooks have tied their books to MATLAB. Unfortunately, programming environments change quickly, and books based on a particular language can quickly become dated. In their new book, Dahlquist and Björk have wisely kept the presentation independent of any particular programming language. However, there are still many exercises that ask the students to write computer programs without specifying the language to be used.
Like its predecessor, this book is likely to be widely used as a textbook in graduate level numerical analysis courses. It will also be useful as a reference for graduate students and researchers in numerical analysis.
Germund Dahlqist and Åke Björk. Numerical Methods. Prentice-Hall, 1974. Reprinted by Dover, 2003.
Brian Borchers is a professor of Mathematics at the New Mexico Institute of Mining and Technology. His interests are in optimization and applications of optimization in parameter estimation and inverse problems.