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An important problem that arises in different disciplines of science and engineering is that of computing limits of sequences of vectors of very large dimension. Such sequences arise, for example, in the numerical solution of systems of linear and nonlinear equations by fixed-point iterative methods, and their limits are simply the required solutions to these systems. The convergence of these sequences, which is very slow in many cases, can be accelerated successfully by using suitable vector extrapolation methods. Vector Extrapolation Methods with Applications?is the first book fully dedicated to the subject of vector extrapolation methods. It is a self-contained, up-to-date, and state-of-the-art reference on the theory and practice of the most useful methods. It covers all aspects of the subject, including development of the methods, their convergence study, numerically stable algorithms for their implementation, and their various applications. It also provides complete proofs in most places. As an interesting application, the author shows how these methods give rise to rational approximation procedures for vector-valued functions in the complex plane, a subject of importance in model reduction problems among others. This book is intended for numerical analysts, applied mathematicians, and computational scientists and engineers in fields such as computational fluid dynamics, structures, and mechanical and electrical engineering, to name a few. Since it provides complete proofs in most places, it can also serve as a textbook in courses on acceleration of convergence of iterative vector processes, for example.
Mathematics of Computing -- General.
This book paints a fresco of the field of extrapolation and rational approximation over the last several centuries to the present through the works of their primary contributors. It can serve as an introduction to the topics covered, including extrapolation methods, Padé approximation, orthogonal polynomials, continued fractions, Lanczos-type methods etc.; it also provides in depth discussion of the many links between these subjects. A highlight of this book is the presentation of the human side of the fields discussed via personal testimonies from contemporary researchers, their anecdotes, and their exclusive remembrances of some of the “actors.” This book shows how research in this domain started and evolved. Biographies of other scholars encountered have also been included. An important branch of mathematics is described in its historical context, opening the way to new developments. After a mathematical introduction, the book contains a precise description of the mathematical landscape of these fields spanning from the 19th century to the first part of the 20th. After an analysis of the works produced after that period (in particular those of Richardson, Aitken, Shanks, Wynn, and others), the most recent developments and applications are reviewed.
Supercomputing is an important science and technology that enables the scientist or the engineer to simulate numerically very complex physical phenomena related to large-scale scientific, industrial and military applications. It has made considerable progress since the first NATO Workshop on High-Speed Computation in 1983 (Vol. 7 of the same series). This book is a collection of papers presented at the NATO Advanced Research Workshop held in Trondheim, Norway, in June 1989. It presents key research issues related to: - hardware systems, architecture and performance; - compilers and programming tools; - user environments and visualization; - algorithms and applications. Contributions include critical evaluations of the state-of-the-art and many original research results.
Applied Iterative Methods
A groundbreaking introduction to vectors, matrices, and least squares for engineering applications, offering a wealth of practical examples.
This volume is a self-contained, exhaustive exposition of the extrapolation methods theory, and of the various algorithms and procedures for accelerating the convergence of scalar and vector sequences. Many subroutines (written in FORTRAN 77) with instructions for their use are provided on a floppy disk in order to demonstrate to those working with sequences the advantages of the use of extrapolation methods. Many numerical examples showing the effectiveness of the procedures and a consequent chapter on applications are also provided – including some never before published results and applications. Although intended for researchers in the field, and for those using extrapolation methods for solving particular problems, this volume also provides a valuable resource for graduate courses on the subject.
Table of contents
Numerical Linear Algebra with Applications is designed for those who want to gain a practical knowledge of modern computational techniques for the numerical solution of linear algebra problems, using MATLAB as the vehicle for computation. The book contains all the material necessary for a first year graduate or advanced undergraduate course on numerical linear algebra with numerous applications to engineering and science. With a unified presentation of computation, basic algorithm analysis, and numerical methods to compute solutions, this book is ideal for solving real-world problems. The text consists of six introductory chapters that thoroughly provide the required background for those who have not taken a course in applied or theoretical linear algebra. It explains in great detail the algorithms necessary for the accurate computation of the solution to the most frequently occurring problems in numerical linear algebra. In addition to examples from engineering and science applications, proofs of required results are provided without leaving out critical details. The Preface suggests ways in which the book can be used with or without an intensive study of proofs. This book will be a useful reference for graduate or advanced undergraduate students in engineering, science, and mathematics. It will also appeal to professionals in engineering and science, such as practicing engineers who want to see how numerical linear algebra problems can be solved using a programming language such as MATLAB, MAPLE, or Mathematica. - Six introductory chapters that thoroughly provide the required background for those who have not taken a course in applied or theoretical linear algebra - Detailed explanations and examples - A through discussion of the algorithms necessary for the accurate computation of the solution to the most frequently occurring problems in numerical linear algebra - Examples from engineering and science applications