Corless and Fillion’s A Graduate Introduction to Numerical Methods from the Viewpoint of Backward Error Analysis

I acquired this book when it first came out in 2013 and have been dipping into it from time to time ever since. At 868 pages long, the book contains a lot of material and I have only sampled a small part of it. In this post I will not attempt to give a detailed review but rather will explain the distinctive features of the book and why I like it.

As the title suggests, the book is pitched at graduate level, but it will also be useful for advanced undergraduate courses. The book covers all the main topics of an introductory numerical analysis course: floating point arithmetic, interpolation, nonlinear equations, numerical linear algebra, quadrature, numerical solution of differential equations, and more.

In order to stand out in the crowded market of numerical analysis textbooks, a book needs to offer something different. This one certainly does.

• The concepts of backward error and conditioning are used throughout—not just in the numerical linear algebra chapters.
• Complex analysis, and particularly the residue theorem, is exploited throughout the book, with contour integration used as a fundamental tool in deriving interpolation formulas. I was pleased to see section 11.3.2 on the complex step approximation to the derivative of a real-valued function, which provides an interesting alternative to finite differences. Appendix B, titled “Complex Numbers”, provides in just 8 pages excellent advice on the practical usage of complex numbers and functions of a complex variable that would be hard to find in complex analysis texts. For example, it has a clear discussion of branch cuts, making use of Kahan’s counterclockwise continuity principle (eschewing Riemann surfaces, which have “almost no traction in the computer world”), and makes use of the unwinding number introduced by Corless, Hare, and Jeffrey in 1996.
• The barycentric formulation of Lagrange interpolation is used extensively, possibly for the first time in a numerical analysis text. This approach was popularized by Berrut and Trefethen in their 2004 SIAM Review paper, and my proof of the numerical stability of the formulas has helped it to gain popularity. Polynomial interpolation and rational interpolation are both covered.
• Both numerical and symbolic computation are employed—whichever is the most appropriate tool for the topic or problem at hand. Corless is well known for his contributions to symbolic computation and to Maple, but he is equally at home in the world of numerics. Chebfun is also used in a number of places. In addition, section 11.7 gives a 2-page treatment of automatic differentiation.

This is a book that one can dip into at any page and quickly find something that is interesting and beyond standard textbook content. Not many numerical analysis textbooks include the Lambert W function, a topic on which Corless is uniquely qualified to write. (I note that Corless and Jeffrey wrote an excellent article on the Lambert W function for the Princeton Computation to Applied Mathematics.) And not so many use pseudospectra.

I like Notes and References sections and this book has lots of them, with plenty of detail, including references that I was unaware of.

As regards the differential equation content, it includes initial and boundary value problems for ODEs, as well as delay differential equations (DDEs) and PDEs. The DDE chapter uses the MATLAB dde23 and ddesd functions for illustration and, like the other differential equation chapters, discusses conditioning.

The book would probably have benefited from editing to reduce its length. The index is thorough, but many long entries need breaking up into subentries. Navigation of the book would be easier if algorithms, theorems, definitions, remarks, etc., had been numbered in one sequence instead of as separate sequences.

Part of the book’s charm is its sometimes unexpected content. How many numerical analysis textbooks recommend reading a book on the programming language Forth (a small, reverse Polish notation-based language popular on microcomputers when I was a student)? And how many would point out the 1994 “rediscovery” of the trapezoidal rule in an article in the journal Diabetes Care (Google “Tai’s model” for some interesting responses to that article).

I bought the book from SpringerLink via the MyCopy feature, whereby any book available electronically via my institution’s subscription can be bought in (monochrome) hard copy for 24.99 euros, dollars, or pounds (the same price in each currency!).

I give the last word to John Butcher, who concludes the Foreword with “I love this book.”