Elements of Numerical Analysis with Mathematica®

This is a text for a one-semester beginning course in numerical analysis. Despite some weaknesses, it manages to give a useful and well-balanced coverage of numerical analysis in about 150 pages. The book is aimed at upper-division undergraduates and beginning graduate students. Prerequisites are multivariable calculus and linear algebra; some exposure to real analysis may be helpful.

The book uses Mathematica to do the drudge work; the usage is very straightforward and does not go into any depth, so this should not be thought of as a course in Mathematica. The author says (p. vii) “we use as little of the programming language as possible”.

The book covers the usual problem areas, namely linear systems, optimization, interpolation, quadrature, partial and ordinary differential equations (in that order), and the Monte Carlo method (for integrals and for simulating stochastic processes). The chapter on numerical differentiation is about partial differential equations and the finite differences method. The numerical integration chapter covers a wide variety of quadrature methods: the common trapezoid rule, midpoint rule, and Simpson’s rule, along with the less-common Hermitian, Gaussian, Chebyshev, and Laguerre quadratures.

Being such a short course, the book cannot go into a great deal of detail on any subject, although it does hit a few unusual areas. These include Krylov spaces, a variety of splines, Hermite interpolation, Neumann stability, and pseudo-random number generation. The linear systems coverage is slanted toward Big Data applications and emphasizes approximate techniques that work for large and sparse matrices rather than traditional exact methods.

There are a number of applications scattered through the book. These are brief, but they do give the student some idea of why these methods are useful.

The exercises are reasonable for a text at this level and also emphasize problems that require a lot of computation rather than those that could be solved by hand. The book quotes many theorems but proves only a few; many of the exercises are to prove some of the remaining theorems.

The book is weak on error estimates; it barely mentions round-off error, and doesn’t give any estimates for any of the quadrature methods. There’s a little bit about speed of convergence for iterative methods.

The editing and production of the book leave something to be desired. There are a number of typographical errors, and a few incomplete sentences and some repetitious areas. There are a number of inaccuracies in the description of Mathematica, but these are probably not harmful for the portions of the language that are used here.

Allen Stenger is a math hobbyist and retired software developer. He is an editor of the Missouri Journal of Mathematical Sciences. His personal web page is allenstenger.com. His mathematical interests are number theory and classical analysis.