Gene H. Golub - Böcker

This computationally oriented book describes and explains the mathematical relationships among matrices, moments, orthogonal polynomials, quadrature rules, and the Lanczos and conjugate gradient algorithms. The book bridges different mathematical areas to obtain algorithms to estimate bilinear forms involving two vectors and a function of the matrix. The first part of the book provides the necessary mathematical background and explains the theory. The second part describes the applications and gives numerical examples of the algorithms and techniques developed in the first part. Applications addressed in the book include computing elements of functions of matrices; obtaining estimates of the error norm in iterative methods for solving linear systems and computing parameters in least squares and total least squares; and solving ill-posed problems using Tikhonov regularization. This book will interest researchers in numerical linear algebra and matrix computations, as well as scientists and engineers working on problems involving computation of bilinear forms.

The fourth edition of Gene H. Golub and Charles F. Van Loan's classic is an essential reference for computational scientists and engineers in addition to researchers in the numerical linear algebra community. Anyone whose work requires the solution to a matrix problem and an appreciation of its mathematical properties will find this book to be an indispensible tool. This revision is a cover-to-cover expansion and renovation of the third edition. It now includes an introduction to tensor computations and brand new sections on: fast transforms; parallel LU; discrete Poisson solvers; pseudospectra; structured linear equation problems; structured eigenvalue problems; large-scale SVD methods; and, polynomial eigenvalue problems. Matrix Computations is packed with challenging problems, insightful derivations, and pointers to the literature-everything needed to become a matrix-savvy developer of numerical methods and software.

The workshop on Applications and Computation of Orthogonal Polynomials took place March 22-28, 1998 at the Oberwolfach Mathematical Research Institute. It was the first workshop on this topic ever held at Oberwolfach. There were 46 participants from 13 countries, more than half coming from Germany and the United States, and a substantial number from Italy. A total of 23 plenary lectures were presented and 4 short informal talks. Open problems were discussed during an evening session. This volume contains refereed versions of 18 papers presented at, or submitted to, the conference. The theory of orthogonal polynomials, as a branch of classical analysis, is well established. But orthogonal polynomials play also an important role in many areas of scientific computing, such as least squares fitting, numerical integration, and solving linear algebraic systems. Though the basic tenets have their roots in 19th­ century mathematics, the use of modern computers has required the development and study of new algorithms that are accurate and robust. The computational methods and applications represented in this volume, of necessity, are incomplete, yet sufficiently varied to convey an impression of current activities in this area.

Numerical linear algebra, digital signal processing, and parallel algorithms are three disciplines with a great deal of activity in the last few years. The interaction between them has been growing to a level that merits an Advanced Study Institute dedicated to the three areas together. This volume gives an account of the main results in this interdisciplinary field. The following topics emerged as major themes of the meeting: - Singular value and eigenvalue decompositions, including applications, - Toeplitz matrices, including special algorithms and architectures, - Recursive least squares in linear algebra, digital signal processing and control, - Updating and downdating techniques in linear algebra and signal processing, - Stability and sensitivity analysis of special recursive least squares problems, - Special architectures for linear algebra and signal processing. This book contains tutorials on these topics given by leading scientists in each of the three areas. A consider- able number of new research results are presented in contributed papers. The tutorials and papers will be of value to anyone interested in the three disciplines.

This volume contains a collection of papers dealing with applications of orthogonal polynomials and methods for their computation. The applications address problems in applied mathematics as well as problems in engineering and the sciences. Prominent among the former are leastsquare approximations, Gauss and related quadrature, iterative methods in linear algebra, the detection of singularities, and integral equations.

In recent years, there has been a great interest in large-scale and real-time matrix compu- tationsj these computations arise in a variety of fields, such as computer graphics, imaging, speech and image processing, telecommunication, biomedical signal processing, optimiza- tion and so on. This volume gives an account of recent research advances in numerical techniques used in large-scale and real-time computations and their implementation on high performance computers. For anyone interested in any of the aforementioned disci- plines, this collection of papers is of value and provides state-of-the-art expositions as weil as new and important trends and directions for the future, motivated and illustrated by a wealth of scientific and engineering applications. The volume is an outgrowth of the NATO Advanced Study Institute "Linear Algebra for Large-Scale and Real-Time Applications," held at Leuven, Belgium, August 1992. We were quite fortunate to be able to gather such an exceilent group of researchers to participate in this Institute. We are indebted to all the participants who enriched the meeting through their many contributions.Special thanks are due to the invited speakers at the Institute, P. Bjlifrstad, Universitetet i Bergen -Norway, S. Boyd, Stanford University -U. S. A. , G. Cy- benko, Dartmouth College -U. S. A. , J. Demmel, University of California Berkeley -U. S. A, E. Deprettere, Technische Universiteit Delft -The Netherlands, P. Dewilde, Technische Uni- versiteit Delft -The Netherlands, R. Freund, AT & T -U. S. A. , M.