Practical Extrapolation Methods: Theory and Applications

ISBN-13: 9780521661591 / Angielski / Twarda / 2003 / 519 str.

Avram Sidi

Opis

An important problem that arises in many scientific and engineering applications is that of approximating limits of infinite sequences which in most instances converge very slowly. Thus, to approximate limits with reasonable accuracy, it is necessary to compute a large number of terms, and this is in general costly. These limits can be approximated economically and with high accuracy by applying suitable extrapolation (or convergence acceleration) methods to a small number of terms. This book is concerned with the coherent treatment, including derivation, analysis, and applications, of the most useful scalar extrapolation methods. The methods it discusses are geared toward problems that commonly arise in scientific and engineering disciplines. It differs from existing books on the subject in that it concentrates on the most powerful nonlinear methods, presents in-depth treatments of them, and shows which methods are most effective for different classes of practical nontrivial problems; it also shows how to fine-tune these methods to obtain the best numerical results. This state-of-the-art reference on the theory and practice of extrapolation methods will interest mathematicians interested in the theory of the relevant methods as well as giving applied scientists and engineers a practical guide to applying speed-up methods in the solution of difficult computational problems. Avram Sidi is Professor is Numerical Analysis in the Computer Science Department at the Technion-Israel Institute of Technology and holds the Technion Administration Chair in Computer Science. He has published extensively in various areas of numerical analysis and approximation theory and in journals such as Mathematics of Computation, SIAM Review, SIAM Journal on Numerical Analysis, Journal of Approximation Theory, Journal of Computational and Applied Mathematics, Numerische Mathematik, and Journal of Scientific Computing. Professor Sidi's work has involved the development of novel methods, their detailed mathematical analysis, design of efficient algorithms for their implementation, and their application to difficult practical problems. His methods and algorithms are successfully used in various scientific and engineering disciplines.

Recenzje

'The book is an excellent support for the theoretical and practical studies of the speed-up methods based on extrapolation. It is a useful book for mathematicians interested in this field of research, but it can be used successfully by computer scientists and engineers.' Zentralblatt MATH

Spis treści

Preface; Introduction; Part I. The Richardson Extrapolation Process and Its Generalizations: 1. The richardson extrapolation process; 2. Additional topics in Richardson extrapolation; 3. First generalization of the Richardson extrapolation process; 4. GREP: further generalization of the Richardson extrapolation process; 5. The d-transformation: a GREP for infinite-range integrals; 6. The d-transformation: a GREP for infinite series and sequences; 7. Recursive algorithms for GREP; 8. Analytic study of GREP (1): slowly varying A(y) ∈ F(1); 9. Analytic study of GREP(1): quickly varying A(y) ∈ F(1); 10: Efficient use of GREP(1): applications to the D(1)-, d(1)- and d(m)-transformations; 11. Reduction of the d-transformation for oscillatory infinite-range integrals: the D-, D-, W-, and mW-transformations; 12. Acceleration of convergence of power series by the d-transformation: rational d-approximants; 13. Acceleration of convergence of Fourier and generalized Fourier series by the d-transformation: the complex series approach with APS; 14. Special topics in Richardson extrapolation; Part II. Sequence Transformations: 15. The Euler transformation, Aitken Δ2-process, and Lubkin W-transformation; 16. The Shanks transformation; 17. The Padé table; 18. Generalizations of Padé approximants; 19. The Levin L- and S-transformations; 20. The Wynn ρ- and Brezinski θ-algorithms; 21. The g-transformation and its generalizations; 22. The transformations of Overholt and Wimp; 23. Confluent transformations; 24. Formal theory of sequence transformations; Part III. Further Applications: 25. Further applications of extrapolation methods and sequence transformations; Part IV. Appendices: A. review of basic asymptotics; B. The Laplace transform and Watson's lemma; C. The gamma function; D. Bernoulli numbers and polynomials and the Euler-Maclaurin formula; E. The Riemann zeta function; F. Some highlights of polynomial approximation theory; G. A compendium of sequence transformations; H. Efficient application of sequence transformations: Summary; I. FORTRAN 77 program for the d(m)-transformation.