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Mathematical Analysis and the Mathematics of Computation

Werner Römisch and Thomas Zeugmann
Publication Date: 
Number of Pages: 
[Reviewed by
Allen Stenger
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This is a text for a first real analysis course. Its selling point is that it presents a wide variety of topics with a uniform notation and terminology, and the same level of abstraction. It also presents an integrated view of approximation and mathematical analysis.

Although the preface states that it covers classical analysis, the presentation is much more general and abstract that you normally see in classical analysis books. For example, the intermediate value theorem is stated and proved for real-valued functions on a metric space. This shifts the focus from completeness to connectedness. The traditional version is given as an example. For another example, it states but does not prove the Weierstrass approximation theorem (polynomials approximating continuous functions uniformly). It instead states and proves the Stone-Weierstrass theorem in full generality (although it mysteriously claims that the collection of continuous functions is a field rather than an algebra). There’s no construction of the real numbers; this is abstract too, and the book lays out axioms (including the least-upper-bound axiom) for the reals and then hypothesizes that a field with these properties exists.

The computational part covers fairly standard material in a fairly standard way: root-finding, approximation of functions, quadrature, and numerical methods for differential equations. There are some less standard topics, such as interpolation by splines and Romberg integration (Richardson extrapolation). It focuses on algorithms and degrees of approximation (error estimates) and there is little actual numerical work, and no discussion of computation strategies and round-off errors. The presentation really is on the “mathematics of computation” as the title indicates, and is not as broad as numerical analysis. The arrangement of topics does work well; the computational topics are placed where they logically are needed in the exposition.

The book is skimpy on worked examples. There are items called exercises scattered through the text, that I think are intended as make-your-own examples. There are also problems at the end of each chapter; these focus on proofs and are reasonably numerous and difficult.

Overall I thought the book was overwhelming. I admire its goals and like its execution but I think the result, at 700 pages, is just too long, and few students will make it all the way through to benefit from these goals. If you don’t use the whole book, or if you use it as a supplement, you lose a lot of the benefit of the unified approach. I also thought the level of abstraction was too high for the intended market. As the next course after calculus this would be too much of a jump for many students.

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 His mathematical interests are number theory and classical analysis.

See the table of contents in the publisher's webpage.