Numerical analysis. A mathematical introduction. Transl. from the French by John Taylor (Q2753184)

The title of this book is well chosen: it is indeed a book on the mathematical background of numerical analysis and not on programming, software or scientific computing. Not even the BLAS or LAPACK are mentioned. Matlab is briefly mentioned but there is no discussion of its merits compared to compiled code. On the other hand, the physical detail in which many problems are introduced is one of the strong points of the book. Another strong point is the mathematical rigour which is definitely above the average for a numerical analysis textbook. Also, the author is obviously interested in the history of her subject and she inserts several colourful historical remarks.NEWLINENEWLINENEWLINEThe choice of the subjects is classical but good; the book contains all the numerical material that is typically taught in a mathematics curriculum. A research mathematician will need to study more specialized works, several of which are in the list of references.NEWLINENEWLINENEWLINEThe first part of the book (The entrance fee) is introductory. It gives a taste of the need for numerical analysis, showing the problems that arise in numerical computations. Furthermore, it gives the algebraic (non-numerical) preliminaries of numerical linear algebra.NEWLINENEWLINENEWLINEPart II (Polynomial and trigonometic approximation) contains chapters on interpolation and finite differences, least-squares approximation for polynomials, and on splines. Chapter 7 on Fourier's world contains a good discussion of convergence issues, illustrated nicely by the Gibbs phenomenon. Chapter 8 (quadrature) is fairly extensive and detailed.NEWLINENEWLINENEWLINENumerical linear algebra appears rather late in the book, in Part III. It is not discussed extensively but the approach to stability, error bounds, pivoting etcetera in Chapter 9 (Gauss's world) is excellent. Chapter 10 is a theoretical interlude on norms, the spectral radius and continuity. The classical iterative methods, including conjugate gradients are discussed in Chapter 11 which even contains an introduction to multigrid. Remarkably, the orthogonal decompositions come last, in Chapter 12 (Pythagoras' world).NEWLINENEWLINENEWLINEPart IV (Nonlinear problems) starts with Chapter 13 on Spectra which is in fact on the power and QR methods for eigenvalues and eigenvectors of general matrices.NEWLINENEWLINENEWLINEChapter 14 (nonlinear equations and systems) is rather incomplete. It is more about equations than about systems and there is little discussion of important issues such as Broyden updates.NEWLINENEWLINENEWLINEThe numerical solution of differential equations gets more attention in three Chapters, the first of which (Chapter 15, Solving differential equations) is theoretical. Chapter 16 is devoted to single-step schemes, in particular explicit and implicit Euler and Runge-Kutta. Chapter 17 discusses linear multistep methods. Stiffness is mentioned only briefly.NEWLINENEWLINENEWLINEChapter 18 on partial differential equations is modest (40 pages) but provides a nice introduction to the subject, more precisely to the advection equation, the one-dimensional wave equation and the heat equation.NEWLINENEWLINENEWLINEThere is no discussion of optimization methods, linear programming, boundary value methods, continuation methods. In numerical linear algebra Krylov subspaces, GMRES and the Arnoldi method are conspiciously absent. The list of references (79) is short for a textbook but the choice is good.NEWLINENEWLINENEWLINEThe global conclusion is that this is a good textbook for the numerical analysis courses in an undergraduate mathematics curriculum. It should be supplemented by practical instruction on computer languages and software. For engineers and computer scientists the approach is probably too abstract.NEWLINENEWLINENEWLINEThe presentation of the book is very good and sometimes original; in spite of teaching numerical courses for several years I found a few good ideas to improve my courses. I recommend the book for those who value mathematical rigour in numerical analysis.NEWLINENEWLINENEWLINE[For the French original edition (1991) see Zbl 0752.65001].