Preliminary review / Publisher’s description: Mit diesem Buch wollen wir verschiedene Teilgebiete der Mathematik aus algorithmischer Perspektive vorstellen und dabei auch Implementierungs- und Laufzeitaspekte diskutieren. Gleichzeitig möchten wir, bei einer verkürzten Grundausbildung in Mathematik in naturwissenschaftlichen und informatischen Studiengängen, möglichst viele Teilaspekte der Mathematik vorstellen und vielleicht zu einer vertiefenden Beschäftigung mit dem einen oder anderen Aspekt anregen. Unser Ziel ist es dabei nicht, den Leser zu einem versierten Anwender der besprochenen Algorithmen auszubilden, sondern wir wollen, immer ausgehend von konkreten Problemen, Analyse- und Lösungsstrategien in den Mittelpunkt stellen. Hierbei spielen insbesondere Beweise und Beweistechniken eine zentrale Rolle.

The author presents a book based on an algorithmic view of mathematics. The reason is algorithmic considerations of mathematical problems, having been more important in the past, are nowadays of minor importance in presenting and teaching mathematical techniques and solution schemes. Two reasons might be responsible for the development: first, the increasing importance of engineering in the development of software, and secondly, the shortened base courses at universities due to the so called Bologna process. Therefore, the author is interested in teaching and introducing a broad range of mathematical fields to stimulate interest in one or the other field, including algorithmic mathematics. After a brief introduction the book starts with a presentation of elementary counting problems and discrete probability theory. Chapter three contains graphs and the application of graphs, followed by chapter four on trees and matchings. Numerical problems and linear algebra are presented in chapter five, including eigenvalues, eigenvectors, and the Cholesky decomposition. Chapters six and seven deal with non-linear optimization, which is used as an application of algorithms in the field of analysis. More detailed, in chapter six the author discusses the necessary and sufficient conditions for extreme values, using graphical tools to show visually the content and meaning of the sentence of implicit defined functions. The graphical approach is also used in the following chapter which discusses numerical procedures as a solution strategy for nonlinear optimization problems. The chapter eight gives an introduction into linear optimization based on the Kuhn-Tucker conditions. In addition, the author presents the Simplex algorithm from a geometrical point of view. Furthermore, the author shows how geometric ideas can be efficiently translated into the tableau scheme of the Simplex algorithm. The final chapter nine contains solutions for the exercises given in the previous chapters. The textbook can be considered as a useful extension of mathematical knowledge beyond elementary standards and is suitable for applied scientists and mathematicians as well.
Reviewer: Herbert S. Buscher (Halle, Saale)