December 2, 2013 - 12:08 — Anonymous

Publisher:

Birkhäuser

Year:

2013

ISBN:

978-1-4614-7033-5, 978-1-4614-7048-9, 978-1-4614-7131-8 (hbk)

Price (tentative):

116.59 €, 148.99 €, 148.99 €

Short description:

The major part of these three thick volumes consists of reprints from a large part of the work of Walter Gautschi. He is a renowned numerical analyst whose main contributions are in the domain of numerical integration and orthogonal polynomials. The reprints are organized in 13 different topics. Each topic is commented and summarized by different specialists.

URL for publisher, author, or book:

www.springer.com/birkhauser/mathematics/book/[978-1-4614-7033-5 | 978-1-4614-7048-9 | 978-1-4614-7131-8]

MSC main category:

65 Numerical analysis

MSC category:

65-06

Other MSC categories:

65Dxx, 65Lxx, 65Axx, 65Yxx, 01Axx

Review:

Walter Gautschi got a Ph.D in 1953 under A.M. Ostrowski in Basel. So, being active in the early years of numerical analysis, he has helped shaping the field of wat is now a broad subject. With his three authored books, several others edited, and around 200 journal papers and book chapters he is now globally recognized as a respected numerical analyst. He is best known for his major work about quadrature formulas and orthogonal polynomials.

The larger part of these three thick volumes (2375 pages in total) consist of reprints of his own selection of his papers. The first volume starts with a very short biography by the editors and an original contribution by Gautschi himself in which he summarizes his work and cathegorizes it in 13 different domains, and this survey is followed by a complete bibliography.

This subdivision in 13 subjects is maintained in the rest of the volumes: topics 1-3 go into volume 1, 4-8 in volume 2 and 8-13 in volume 3. In each topic it goes in two steps: first there is a part in which well known specialists comment and summarize the contributions of Gautschi each of the topics in the volume and in a second part the papers classified in this topic follow. The layout in which they were originally published is maintained. Here is the list of the topics and in brackets are the authors who wrote the commentaries.

- numerical conditioning (N.J. Higham)
- special functions (J. Segura)
- interpolation and approximation (M.M. Spalević)
- orthogonal polynomials on the real line (G.V. Milovanović)
- orthogonal polynomilas on the semicircle (L. Reichel)
- Chebyshev quadrature (J. Korevaar)
- Kronrod and other quadratures (G. Monegato)
- Gauss quadrature (W. Van Assche)
- linear difference equations (L. Lorentzen)
- ordinary differential equation (J. Butcher)
- computer algorthms and softwarepackages (G.V. Milovanović)
- history and biography (G. Wanner)
- miscelenea (M.J. Gander)

Walter Gautschi is a very kind person and the commentators are not only world leading specialists, but they are also friends of Walter. So some of the comments also contain some personal reminiscences. Since his official retirement from Purdue University in 2000 to which he has been associated since 1963, Gautschi has remained active ever since. He published his book on *Orthogonal Polynomials* with Oxford University Press in 2004 and his book on *Numerical Analysis* published by Birkhäuser Boston got a second edition in 2012.

Of course the topics listed above are not disjunct and there is obvious overlap. The conditioning for example is about Vandermonde matrices and polynomials, important for interpolation and orthogonal polynomials for quadrature nodes. The recursive computation of these polynomials and special functions links to the stability in using recurrence relations, and special functions often have an integral representation that can be evaluated using quadratures. His work is mainly constructive and practical which resulted in two chapters in Abramowitz and Stegun's *Handbook of Mathematical Functions* and in software packages such as ORTHOPOL (orthogonal polynomias) and several other routines available on his website.

Somewhat separate from the rest is the section on history and biography. There we find his work about Euler (2007 was the Euler year and Gautschi gave a main lecture about Euler at the ICIAM Congress in Zürich). More historical work is about his thesis adviser Ostrowski, and about the Bieberbach conjecture. Also a survey paper is devoted to the work of E. Christoffel, while others are written as a tribute to colleagues he has known and that have passed away: Y. Luke, P. Rabinowitz, L. Gatteschi, G. Golub, and one for his friend G. Milovanović on the occasion of his 60th birthday in 2008.

The commentaties are usually rather short and do not extend or give additional results. They just summarize what the papers are about with little personal comments. The more important sections like Orthogonal Polynomials and Quadrature have more papers and hence also longer introductions.

Volume 3 has an extra third part. Walter Gautschi had a twin brother Werner, also a mathematician, who died in 1959. This part reprints 5 of Werner's papers, and two orbituaries, and a link to the Springer website where one may find a recording of 35 minutes in which Werner performs Schubert's Trout Quitet.

With this series *Contemporary Mathematicians*, Birkhäuser has started to publish the collected works (or at least a selection) of contemporary mathematicians whose publications are sometimes scattered and/or may be difficult to find. Bringing them together in one, or as in this case, several volumes is an excellent service to the scientific community. In the e-book version, the chapters are downloadable separately which is of course a highly desirable feature. Libraries will find these hard copies on their shelves to be of lasting significance.

Reviewer:

A. Bultheel

Affiliation:

KU Leuven

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