New convergence results on the generalized Richardson extrapolation process GREP$^{(1)}$ for logarithmic sequences
HTML articles powered by AMS MathViewer
- by Avram Sidi PDF
- Math. Comp. 71 (2002), 1569-1596 Request permission
Abstract:
Let $a(t)\sim A+\varphi (t)\sum ^\infty _{i=0}\beta _it^i$ as $t\to 0+$, where $a(t)$ and $\varphi (t)$ are known for $0<t\leq c$ for some $c>0$, but $A$ and the $\beta _i$ are not known. The generalized Richardson extrapolation process GREP$^{(1)}$ is used in obtaining good approximations to $A$, the limit or antilimit of $a(t)$ as $t\to 0+$. The approximations $A^{(j)}_n$ to $A$ obtained via GREP$^{(1)}$ are defined by the linear systems $a(t_l)=A^{(j)}_n+\varphi (t_l) \sum ^{n-1}_{i=0}\bar {\beta }_it_l^i$, $l=j,j+1,\ldots ,j+n$, where $\{t_l\}^\infty _{l=0}$ is a decreasing positive sequence with limit zero. The study of GREP$^{(1)}$ for slowly varying functions $a(t)$ was begun in two recent papers by the author. For such $a(t)$ we have $\varphi (t)\sim \alpha t^\delta$ as $t\to 0+$ with $\delta$ possibly complex and $\delta \neq 0$, $-1$, $-2$, …. In the present work we continue to study the convergence and stability of GREP$^{(1)}$ as it is applied to such $a(t)$ with different sets of collocation points $t_l$ that have been used in practical situations. In particular, we consider the cases in which (i) $t_l$ are arbitrary, (ii) $\lim _{l\to \infty }t_{l+1}/t_l=1$, (iii) $t_l\sim cl^{-q}$ as $l\to \infty$ for some $c, q>0$, (iv) $t_{l+1}/t_l\leq \omega \in (0,1)$ for all $l$, (v) $\lim _{l\to \infty }t_{l+1}/t_l= \omega \in (0,1)$, and (vi) $t_{l+1}/t_l=\omega \in (0,1)$ for all $l$.References
- Kendall E. Atkinson, An introduction to numerical analysis, John Wiley & Sons, New York-Chichester-Brisbane, 1978. MR 504339
- Claude Brezinski, Accélération de suites à convergence logarithmique, C. R. Acad. Sci. Paris Sér. A-B 273 (1971), A727–A730 (French). MR 305544
- Roland Bulirsch and Josef Stoer, Fehlerabschätzungen und Extrapolation mit rationalen Funktionen bei Verfahren vom Richardson-Typus, Numer. Math. 6 (1964), 413–427. MR 176589, DOI 10.1007/BF01386092
- William F. Ford and Avram Sidi, An algorithm for a generalization of the Richardson extrapolation process, SIAM J. Numer. Anal. 24 (1987), no. 5, 1212–1232. MR 909075, DOI 10.1137/0724080
- Morgan Ward and R. P. Dilworth, The lattice theory of ova, Ann. of Math. (2) 40 (1939), 600–608. MR 11, DOI 10.2307/1968944
- Pierre-Jean Laurent, Un théorème de convergence pour le procédé d’extrapolation de Richardson, C. R. Acad. Sci. Paris 256 (1963), 1435–1437 (French). MR 146948
- D. P. Laurie, Propagation of initial rounding error in Romberg-like quadrature, Nordisk Tidskr. Informationsbehandling (BIT) 15 (1975), no. 3, 277–282. MR 395172, DOI 10.1007/bf01933660
- David Levin, Development of non-linear transformations of improving convergence of sequences, Internat. J. Comput. Math. 3 (1973), 371–388. MR 359261, DOI 10.1080/00207167308803075
- David Levin and Avram Sidi, Two new classes of nonlinear transformations for accelerating the convergence of infinite integrals and series, Appl. Math. Comput. 9 (1981), no. 3, 175–215. MR 650681, DOI 10.1016/0096-3003(81)90028-X
- R.E. Powell and S.M. Shah, Summability Theory and Its Applications, Van Nostrand Rheinhold, London, 1972.
- A. Sidi, Some properties of a generalization of the Richardson extrapolation process, J. Inst. Math. Appl. 24 (1979), no. 3, 327–346. MR 550478
- Avram Sidi, Convergence properties of some nonlinear sequence transformations, Math. Comp. 33 (1979), no. 145, 315–326. MR 514827, DOI 10.1090/S0025-5718-1979-0514827-6
- Avram Sidi, Analysis of convergence of the $T$-transformation for power series, Math. Comp. 35 (1980), no. 151, 833–850. MR 572860, DOI 10.1090/S0025-5718-1980-0572860-0
- Avram Sidi, An algorithm for a special case of generalization of the Richardson extrapolation process, Numer. Math. 38 (1981/82), no. 3, 299–307. MR 654099, DOI 10.1007/BF01396434
- Avram Sidi, On a generalization of the Richardson extrapolation process, Numer. Math. 57 (1990), no. 4, 365–377. MR 1062359, DOI 10.1007/BF01386416
- Avram Sidi, Convergence analysis for a generalized Richardson extrapolation process with an application to the $d^{(1)}$-transformation on convergent and divergent logarithmic sequences, Math. Comp. 64 (1995), no. 212, 1627–1657. MR 1312099, DOI 10.1090/S0025-5718-1995-1312099-5
- Avram Sidi, Further convergence and stability results for the generalized Richardson extrapolation process $\rm GREP^{(1)}$ with an application to the $D^{(1)}$-transformation for infinite integrals, J. Comput. Appl. Math. 112 (1999), no. 1-2, 269–290. Numerical evaluation of integrals. MR 1728465, DOI 10.1016/S0377-0427(99)90226-1
- Avram Sidi, The Richardson extrapolation process with a harmonic sequence of collocation points, SIAM J. Numer. Anal. 37 (2000), no. 5, 1729–1746. MR 1759914, DOI 10.1137/S0036142998340137
- David A. Smith and William F. Ford, Acceleration of linear and logarithmic convergence, SIAM J. Numer. Anal. 16 (1979), no. 2, 223–240. MR 526486, DOI 10.1137/0716017
- David A. Smith and William F. Ford, Numerical comparisons of nonlinear convergence accelerators, Math. Comp. 38 (1982), no. 158, 481–499. MR 645665, DOI 10.1090/S0025-5718-1982-0645665-1
- T.N. Thiele, Interpolationsrechnung, Teubner, Leipzig, 1909.
- Andrew H. Van Tuyl, Acceleration of convergence of a family of logarithmically convergent sequences, Math. Comp. 63 (1994), no. 207, 229–246. MR 1234428, DOI 10.1090/S0025-5718-1994-1234428-2
- Cahit Arf, Untersuchungen über reinverzweigte Erweiterungen diskret bewerteter perfekter Körper, J. Reine Angew. Math. 181 (1939), 1–44 (German). MR 18, DOI 10.1515/crll.1940.181.1
Additional Information
- Avram Sidi
- Affiliation: Computer Science Department, Technion—Israel Institute of Technology, Haifa 32000, Israel
- Email: asidi@cs.technion.ac.il
- Received by editor(s): October 3, 2000
- Published electronically: November 28, 2001
- © Copyright 2001 American Mathematical Society
- Journal: Math. Comp. 71 (2002), 1569-1596
- MSC (2000): Primary 65B05, 65B10, 40A05, 41A60
- DOI: https://doi.org/10.1090/S0025-5718-01-01384-9
- MathSciNet review: 1933045