Practical Extrapolation Methods

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 state-of-the art reference for mathematicians, scientists and engineers is concerned with the coherent treatment, including derivation, analysis, and applications, of the most useful scalar extrapolation methods. The methods discussed are geared toward common problems in scientific and engineering disciplines. It differs from existing books by concentrateing on the most powerful nonlinear methods, presenting in-depth treatments of them, and showing which methods are most effective for different classes of practical nontrivial problems.

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 Watsons 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.