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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.
Preface Introduction Part I. The Richardson Extrapolation Process and Its Generalizations:
The richardson extrapolation process
Additional topics in Richardson extrapolation
First generalization of the Richardson extrapolation process
GREP: further generalization of the Richardson extrapolation process
The d-transformation: a GREP for infinite-range integrals
The d-transformation: a GREP for infinite series and sequences
Recursive algorithms for GREP
Analytic study of GREP (1): slowly varying A(y) ∈ F(1)
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
Reduction of the d-transformation for oscillatory infinite-range integrals: the D-, D-, W-, and mW-transformations
Acceleration of convergence of power series by the d-transformation: rational d-approximants
Acceleration of convergence of Fourier and generalized Fourier series by the d-transformation: the complex series approach with APS
Special topics in Richardson extrapolation Part II. Sequence Transformations:
The Euler transformation, Aitken Δ2-process, and Lubkin W-transformation
The Shanks transformation
The Padé table
Generalizations of Padé approximants
The Levin L- and S-transformations
The Wynn ρ- and Brezinski θ-algorithms
The g-transformation and its generalizations
The transformations of Overholt and Wimp
Confluent transformations
Formal theory of sequence transformations Part III. Further Applications:
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