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Algorithmic Mathematics

Stefan Hougardy and Jens Vygen
Publication Date: 
Number of Pages: 
[Reviewed by
Allen Stenger
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This is a brief introduction to some simple numerical and computer algorithms, including an analysis of their timing. It’s aimed at students with a good high-school math background, but has no specific prerequisites, and has enough material for a one-semester course. The present volume is a translation of a German-language work, Algorithmische Mathematik, that was published by Springer in 2015.

The title’s term “algorithmic mathematics” is not a familiar one. The meaning here seems to be a “strict and rigorous mathematical presentation” (p. v), that is, it means a method of study rather than a specific subject. This works fairly well, although because the book’s scope is so limited it mostly means rigorous definitions more than rigorous analysis or rigorous proof (although it has those too). It also include C++ programs for most of the algorithms, and a lot of the book is devoted to explaining these and to giving C++ tutorials, since the student is expected not to already know the language.

The mathematical subjects covered include radix representations and computations, floating-point arithmetic, the Euclidean algorithm for greatest common divisor, the sieve of Eratosthenes, Newton’s method (or Heron’s method) for calculating square roots (including a nice analysis of convergence), and an introduction to Gaussian elimination and LU decomposition (but there’s nothing about error analysis or even about how to pick the pivot). The computer science topics covered include sorting algorithms and a variety of topics related to networks, including a lot of material on graphs and trees.

The book works fairly well given its limited goals. The biggest weaknesses are that there are no exercises and there are only a few worked examples. There are a number of minor infelicities. The index of symbols (actually a glossary) on p. xiii only lists symbols the reader is supposed to know already, not symbols that are defined in the book. The primality test on p. 8 describes one algorithm in words, and a totally different algorithm in pseudocode, without mentioning that they are different. Many of the definitions are given only symbolically rather than in words, and are a hard-to-read mass of set notation and there-exists and for-every. There’s no use of the Master Theorem (popularized by Cormen et al.) for asymptotic analysis; every algorithm is figured independently, even though they all use the same method.

The book is much more shallow than more mainstream computer-science books such as Sedgewick & Flajolet’s An Introduction to the Analysis of Algorithms (Addison-Wesley, 2nd edition 2013; mostly about analysis) or Cormen et al.’s Introduction to Algorithms (mostly about algorithms, but with good analysis), but it’s aimed at a different audience. It’s still a good choice for a very early introduction to these topics.

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.