Extrapolation methods and derivatives

of limits of sequences

Author:
Avram Sidi

Journal:
Math. Comp. **69** (2000), 305-323

MSC (1991):
Primary 40A25, 41A60, 65B05, 65B10, 65D30

DOI:
https://doi.org/10.1090/S0025-5718-99-01169-2

Published electronically:
August 17, 1999

MathSciNet review:
1665967

Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Let be an infinite sequence whose limit or antilimit can be approximated very efficiently by applying a suitable extrapolation method E to . Assume that the and hence also are differentiable functions of some parameter , being the limit or antilimit of , and that we need to approximate . A direct way of achieving this would be by applying again a suitable extrapolation method E to the sequence , and this approach has often been used efficiently in various problems of practical importance. Unfortunately, as has been observed at least in some important cases, when and have essentially different asymptotic behaviors as , the approximations to produced by this approach, despite the fact that they are good, do not converge as quickly as those obtained for , and this is puzzling. In this paper we first give a rigorous mathematical explanation of this phenomenon for the cases in which E is the Richardson extrapolation process and E is a generalization of it, thus showing that the phenomenon has very little to do with numerics. Following that, we propose a procedure that amounts to first applying the extrapolation method E to and then differentiating the resulting approximations to , and we provide a thorough convergence and stability analysis in conjunction with the Richardson extrapolation process. It follows from this analysis that the new procedure for has practically the same convergence properties as E for . We show that a very efficient way of implementing the new procedure is by actually differentiating the recursion relations satisfied by the extrapolation method used, and we derive the necessary algorithm for the Richardson extrapolation process. We demonstrate the effectiveness of the new approach with numerical examples that also support the theory. We discuss the application of this approach to numerical integration in the presence of endpoint singularities. We also discuss briefly its application in conjunction with other extrapolation methods.

**[BRS]**F.L. Bauer, H. Rutishauser, and E. Stiefel, New aspects in numerical quadrature, in*Experimental Arithmetic, High Speed Computing, and Mathematics*, Proc. Sympos. Appl. Math., vol. 15, AMS, Providence, Rhode Island, 1963, pp.199-218. MR**30:4384****[B1]**C. Brezinski, Accélération de suites à convergence logarithmique,*C.R. Acad. Sci. Paris,***273**(1971), pp. A727-A730. MR**46:4674****[B2]**C. Brezinski, A general extrapolation algorithm,*Numer. Math.*,**35**(1980), pp. 305-323. MR**81j:65015****[BS]**R. Bulirsch and J. Stoer, Fehlerabschätzungen und Extrapolation mit rationalen Funktionen bei Verfahren vom Richardson-Typus,*Numer. Math.*,**6**(1964), pp. 305-323. MR**31:861****[DR]**P.J. Davis and P. Rabinowitz,*Methods of Numerical Integration*, 2nd edition, Academic Press, New York, 1984. MR**86d:65004****[FS]**W.F. Ford and A. Sidi, An algorithm for a generalization of the Richardson extrapolation process,*SIAM J. Numer. Anal.*,**24**(1987), pp. 305-323. MR**89a:65006****[H]**T. Håvie, Generalized Neville type extrapolation schemes,*BIT*,**19**(1979), pp. 305-323. MR**80f:65005****[L]**J.N. Lyness, An error functional expansion for -dimensional quadrature with an integrand function singular at a point,*Math. Comp.*,**30**(1976), pp. 305-323. MR**53:11976****[LM]**J.N. Lyness and G. Monegato, Quadrature error functional expansions for the simplex when the integrand function has singularities at vertices,*Math. Comp.*,**34**(1980), pp. 305-323. MR**80m:65017****[LN]**J.N. Lyness and B.W. Ninham, Numerical quadrature and asymptotic expansions,*Math. Comp.*,**21**(1967), pp. 305-323. MR**37:1081****[N1]**I. Navot, An extension of the Euler-Maclaurin summation formula to functions with a branch singularity,*J. Math. and Phys.*,**40**(1961), pp. 305-323. MR**25:4290****[N2]**I. Navot, A further extension of the Euler-Maclaurin summation formula,*J. Math. and Phys.*,**41**(1962), pp. 305-323.**[Sc]**C. Schneider, Vereinfachte Rekursionen zur Richardson-Extrapolation in Spezialfällen,*Numer. Math.*,**24**(1975), pp. 305-323. MR**51:14510****[Sh]**D. Shanks, Non-linear transformations of divergent and slowly convergent sequences,*J. Math. and Phys.*,**34**(1955), pp. 305-323. MR**16:961c****[Si1]**A. Sidi, Euler-Maclaurin expansions for integrals over triangles and squares of functions having algebraic/logarithmic singularities along an edge,*J. Approx. Th.*,**39**(1983), pp. 305-323. MR**84j:65026****[Si2]**A. Sidi, Generalizations of Richardson extrapolation with applications to numerical integration, in*Numerical Integration III*, ISNM Vol. 85, H. Brass and G. Hämmerlin, eds., Birkhäuser, Basel, Switzerland, 1988, pp. 237-250. MR**90k:65076****[Si3]**A. Sidi, A complete convergence and stability theory for a generalized Richardson extrapolation process,*SIAM J. Numer. Anal.*,**34**(1997), pp. 305-323. MR**98g:65012****[W]**P. Wynn, On a device for computing the transformation,*Math. Comp.*,**10**(1956), pp. 305-323. MR**18:801e**

Retrieve articles in *Mathematics of Computation*
with MSC (1991):
40A25,
41A60,
65B05,
65B10,
65D30

Retrieve articles in all journals with MSC (1991): 40A25, 41A60, 65B05, 65B10, 65D30

Additional Information

**Avram Sidi**

Affiliation:
Computer Science Department, Technion—Israel Institute of Technology, Haifa 32000, Israel

Email:
asidi@cs.technion.ac.il

DOI:
https://doi.org/10.1090/S0025-5718-99-01169-2

Received by editor(s):
March 26, 1998

Published electronically:
August 17, 1999

Article copyright:
© Copyright 1999
American Mathematical Society