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Quality and Value in Mathematical Science Literature

SIAM is proud to be adding Functions of Matrices: Theory and Computation by Nicholas J. Higham and Numerical Methods in Scientific Computing, Volume I by Germund Dahlquist and Åke Björck to its list of outstanding numerical analysis books. Inside SIAM spoke to the authors about why they decided to pursue careers in math, their influential teachers, the impetus behind their books, and how they have benefited from their association with SIAM.

Up to the age of about 16 I felt that English was my best subject, but I realized that science was what I wanted to specialize in. Mathematics was a natural choice for my degree and once I started research as a graduate student it became clear I wanted to pursue mathematics as a career.

I’ve been fortunate to have had inspiring teachers at all levels.

Among those who had an early influence on what I do today, Ian Gladwell showed me how to “be a researcher,” Pete Stewart’s writings piqued my interest in rounding error analysis, Beresford Parlett’s book showed me that technical writing can be elegant and articulate, and the first edition of Golub and Van Loan’s Matrix Computations was a revelation that transformed my view of what was to become my area of research.

I’ve been working on matrix functions since I was a graduate student. In 2003 I was prompted to start a book on the subject by the launch of the SIAM Fundamentals of Algorithms book series, which publishes short, practically oriented guides to numerical methods.

However, it rapidly became clear that the book I needed to write was a fully-fledged research monograph. I also soon realized that there were some gaps in my understanding of the subject, and in the literature, that needed to be filled before I could complete the book.

As a result, this has been the longest in gestation of my four books and certainly the hardest to finish, not least because interest in matrix functions has grown significantly in the last few years, with the burgeoning literature making it a challenge to keep up.

For matrices for which factorizations can be computed, we now have good numerical methods for general functions as well as ones tailored for specific functions.

Less well developed are methods for estimating the sensitivity of matrix functions to perturbations in the matrix, where sensitivity is measured by condition numbers defined in terms of the Fréchet derivative.

Many open questions also remain concerning how to exploit structure in f(A) problems. And there is much more to be done on the increasingly important problem of computing f(A)b for a vector b (without first computing f(A)), where A is large and sparse.

For me, SIAM is a terrific vehicle for interacting with like-minded researchers and keeping abreast of developments, through conferences, SIAM News, and the books and journals. SIAM is my favorite place to publish, not only because it has some of the leading journals and books in my field, but also because it’s always a pleasure working with the SIAM staff.

Nicholas J. Higham, FRS, is Richardson Professor of Applied Mathematics at The University of Manchester, UK. He is the author of more than 100 publications and of the books Accuracy and Stability of Numerical Algorithms (SIAM, 2nd ed., 2002), Handbook of Writing for the Mathematical Sciences, (SIAM, 2nd ed., 1998), and MATLAB Guide, (with Desmond J. Higham, SIAM, 2nd ed., 2005).

Functions of Matrices: Theory and Computation
Nicholas J. Higham

“This superb book is timely and is written with great attention paid to detail…”

— Alan Laub, Professor, University of California, Los Angeles

The only book devoted exclusively to matrix functions, this research monograph gives a thorough treatment of the theory of matrix functions and numerical methods for computing them. The author’s elegant presentation focuses on the equivalent definitions of f(A) via the Jordan canonical form, polynomial interpolation, and the Cauchy integral formula, and features an emphasis on results of practical interest and an extensive collection of problems and solutions. More than just a monograph on matrix functions, its wide-ranging content—including an overview of applications, historical references, and miscellaneous results, tricks, and techniques with an f(A) connection—makes it useful as a general reference in numerical linear algebra.

Available April 2008 · Approx. xx + 425 pages · Hardcover · ISBN 978-0-898716-46-7
List Price $59.00 · SIAM Member Price $41.30 · Order Code OT104

 

 

I did my undergraduate studies in technical physics at the Royal Institute of Technology (KTH) in Stockholm. In 1957 I did some programming (in machine code) for the first Swedish electronic computer, BESK. I found it very exciting and it influenced my decision to continue my studies in applied mathematics.

In 1963 a new department for Numerical Analysis and Information Processing (NADA) was created at KTH with Germund Dahlquist as chair. I jumped at the opportunity to join as a junior lecturer and graduate student.

I’ve had several influential teachers. During my undergraduate years it was Carl-Gustaf Esseen, whose field was mathematical statistics. His wide scope of knowledge was impressive and he had high standards. He gave well organized courses in classical numerical analysis. Heinz Kreiss was another inspiring teacher. Kreiss had come to Sweden in 1955 to do numerical work in oceanography on BESK. I wrote my licentiat thesis for him on boundary value problems for hyperbolic equations.

My advisor, Germund Dahlquist, came from the Swedish National Board for Computing Machinery, where he had been Head of the Department of Mathematical Analysis and Programming Development from 1956 to 1959. In 1958 he published his seminal dissertation, Stability and Error Bounds in the Numerical Integration of Ordinary Differential Equations. Germund was not only a great mathematician but a very kind and considerate person.

Lars Hörmander gave a series of lectures at Stockholm University. Although not much older than myself, he was world famous—he had been awarded the Fields medal in 1962 at the age of 31.

I was much influenced by Jim Wilkinson and read all of his work. Gene Golub visited KTH in 1966 and came to have a big influence on the direction of my research. He was very encouraging and arranged a postdoc year in 1968–1969, for me at Stanford and Berkeley. At Stanford I was fortunate to meet George Forsythe before his untimely death in 1972.

In 1969 Germund and I published a Swedish textbook on numerical methods, which Forsythe encouraged us to translate into English. A slightly extended English version was published by Prentice-Hall in 1974. In the mid 1980s Prentice-Hall asked us to do a new edition, but eventually we laid it aside.

We took the book up again with SIAM and aimed for a multi-volume book because the subject had expanded. We had progressed quite far with the first two volumes when sadly Germund died in 2005 shortly after his 80th birthday.

The challenge in matrix computations comes both from changes in computer architecture and (as it should) from new applications, for example, image processing and data mining.

An active area still in its early development is tensor methods for multidimensional analysis. A lot remains to be done in structured and sparse matrix computations.

SIAM conferences and journals have been essential for me to keep abreast with what is going on. I have been on the editorial boards of SIAP and SISC. My all-time favorite conference is the SIAM 30th Anniversary Meeting held at Stanford in 1982.

Åke Björck is a Professor in the Department of Mathematics at Linköping University in Sweden. His book Numerical Methods for Least Squares Problems was published by SIAM in 1996. He served as managing editor of the journal BIT Numerical Mathematics from 1993 to 2003.

Numerical Methods in Scientific Computing, Volume I
Germund Dahlquist and Åke Björck

This new book from the authors of the highly successful classic Numerical Methods (Prentice-Hall, 1974) addresses the increasingly important role of numerical methods in science and engineering. While treating traditional and well-developed topics, it also emphasizes concepts and ideas of importance to the design of accurate and efficient algorithms with applications to scientific computing. Although this volume is self-contained, more comprehensive treatments of matrix computations will be given in a forthcoming volume.

Available May 2008 · xxviii + 717 pages · Hardcover · ISBN 978-0-898716-44-3
List Price $109.00 · SIAM Member Price $76.30 · Order Code OT103

 

 

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