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Walter Gautschi : selected works with commentaries (3 vols.)
Brezinski C., Sameh A., Springer, New York, NY, 2014. 669 pp.  Type: Book (978-1-461470-33-5)
Date Reviewed: May 6 2014

Walter Gautschi is one of the leading numerical analysts of the second half of the 20th and the beginning of the 21st century. He has contributed to almost all areas of numerical analysis, and many of his results have proven to be highly significant and timeless. In view of this fact, Brezinski and Sameh have undertaken to summarize Gautschi’s work in a generally accessible form, using the format of a three-volume series [1,2,3].

The structure of this set of books has been carefully chosen in a way that is appropriate for the contents. Specifically, Gautschi himself picked about 130 particularly important papers from his list of publications, which includes more than 200 entries, and he split up this work into 13 topically arranged subcategories of numerical analysis. These 13 principal parts of the book series were then distributed across the three volumes. Volume 1 consists of reprints of the papers selected by Gautschi that belong to the three areas of numerical conditioning (13 papers), special functions (21 papers), and interpolation and approximation (10 papers). Volume 2 is devoted to orthogonal polynomials and numerical integration; more precisely, it contains Gautschi’s selected papers from the five areas of orthogonal polynomials on the real line (14 papers), orthogonal polynomials on the semicircle (three papers), Chebyshev quadrature (four papers), Kronrod and other quadrature (14 papers), and Gauss-type quadrature (19 papers). Finally, the third volume covers the remaining five areas, namely, linear recurrence relations (six papers), ordinary differential equations (three papers), computer algorithms and software packages (two papers), history and biography (13 papers), and miscellanea (seven papers).

In addition to these main parts of the contents, which are well worth reading, some additional gems are provided as well. In particular, the first volume actually begins with a brief biography of Gautschi, which is followed by a short chapter where Gautschi himself recollects his scientific work--the first part of which was published in 1951, and which is still ongoing. In this document, he also explains why he chose the particular papers contained in these selected works, describes the relations between his papers, and discusses how his results influenced other work. Gautschi’s assessment of his own work is followed by a table giving the complete bibliographical data for all of his publications.

Furthermore, each of the 13 principal parts is accompanied by commentary from a corresponding renowned specialist, who gives an independent account of Gautschi’s work in the respective area. Most of these summaries are more detailed than Gautschi’s own brief assessment given in the first volume; in fact, they give very interesting details about the usefulness of Gautschi’s work for the respective fields in general and for certain special problems in particular. Moreover, they also nicely illustrate the interconnections between the 13 areas that become evident from Gautschi’s work.

Finally, the third volume concludes with reprints of five papers by Walter Gautschi’s twin brother Werner, a highly gifted mathematician himself who unfortunately passed away, in 1959, at the age of 31, together with two obituaries for Werner Gautschi.

It may appear somewhat premature to publish the selected works of a researcher who is still as active as Gautschi, but it has the advantage that he himself is still available to provide useful comments and additional insight. I also believe the commentaries about Gautschi’s publications given by the other researchers are an extremely useful feature of this set of books. These comments are written from today’s perspective and assess the implications that Gautschi’s work has had, and still has, in many different branches of mathematics, ranging from his contributions to de Branges’ proof of the Bieberbach conjecture in complex analysis, an area usually considered to belong to pure mathematics, to the development of mathematical software that remains reliable under the adverse influence of the finite precision of today’s standard hardware. Many of the consequences of Gautschi’s results are very far reaching and were certainly not expected at the time that the corresponding papers were written, and there are plenty of instances where research on currently important problems can greatly benefit from results that Gautschi obtained many years ago.

Thus, this multi-volume work is not only for readers who are interested in the recent history of numerical analysis, but also for those who want to gain a general overview of this vast field. Also, anyone who is engaged in one of Gautschi’s core areas of interest--for example, numerical integration, the numerical evaluation of special functions, or the design of numerically stable mathematical software algorithms in general--is likely to substantially profit from getting acquainted with his techniques, and this set of books provides a most convenient path to this goal.

Reviewer:  Kai Diethelm Review #: CR142254 (1407-0507)
1) Brezinski, C.; Sameh, A. (Eds.) Walter Gautschi: selected works with commentaries (vol. 1). Springer, New York, NY, 2014.
2) Brezinski, C.; Sameh, A. (Eds.) Walter Gautschi: selected works with commentaries (vol. 2). Springer, New York, NY, 2014.
3) Brezinski, C.; Sameh, A. (Eds.) Walter Gautschi: selected works with commentaries (vol. 3). Springer, New York, NY, 2014.
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