Hostname: page-component-8448b6f56d-jr42d Total loading time: 0 Render date: 2024-04-19T21:59:55.541Z Has data issue: false hasContentIssue false
  • English
  • Français

Matrix and vector sequence transformations revisited

Published online by Cambridge University Press:  20 January 2009

C. Brezinski  and
A. Salam
C. Brezinski
Affiliation:
Laboratoire D'Analyse Numerique et D'Optimisation, UFR IEEA-M3, Université des Sciences et Technologies de Lille, 59655 Villeneuve D'Ascq Cedex, France E-mail address: brezinsk@omega.univ-lillel.fr
A. Salam
Affiliation:
Laboratoire D'Analyse Numerique et D'Optimisation, UFR IEEA-M3, Université des Sciences et Technologies de Lille, 59655 Villeneuve D'Ascq Cedex, France E-mail address: brezinsk@omega.univ-lillel.fr
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Sequence transformations are extrapolation methods. They are used for the purpose of convergence acceleration. In the scalar case, such algorithms can be obtained by two different approaches which are equivalent. The first one is an elimination approach based on the solution of a system of linear equations and it makes use of determinants. The second approach is based on the notion of annihilation difference operators. In this paper, these two approaches are generalized to the matrix and the vector cases.

Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 38 , Issue 3 , October 1995 , pp. 495 - 510
Copyright
Copyright © Edinburgh Mathematical Society 1995

References

REFERENCES

1.Brezinski, C., A general extrapolation algorithm, Numer. Math. 35 (1980), 175–187.CrossRefGoogle Scholar
2.Brezinski, C., Some new convergence acceleration methods, Math. Comput. 39 (1982), 133–145.Google Scholar
3.Brezinski, C., Other manifestations of the Schur complement, Linear Algebra Appl. 111 (1988), 231–247.CrossRefGoogle Scholar
4.Brezinski, C. and Matos, A. C., A derivation of extrapolation algorithms based on error estimates, J. Comput. Appl. Math., to appear.Google Scholar
5.Brezinski, C. and Zaglia, M. Redivo, Extrapolation Methods. Theory and Practice (North-Holland, Amsterdam, 1991).Google Scholar
6.Brezinski, C. and Zaglia, M. Redivo, Construction of extrapolation processes, Appl. Numer. Math. 8 (1991), 11–23.CrossRefGoogle Scholar
7.Brezinski, C. and Zaglia, M. Redivo, A general extrapolation procedure revisited, Adv. Comput. Math, 2 (1994), 461–477.CrossRefGoogle Scholar
8.Brezinski, C. and Walz, G., Sequences of transformations and triangular recursion schemes with applications in numerical analysis, J. Comput. Appl. Math. 34 (1991), 361–383.CrossRefGoogle Scholar
9.Delahaye, J. P., Sequence Transformations (Springer-Verlag, Berlin, 1988).CrossRefGoogle Scholar
10.Dieudonne, J., Les déterminants sur un corps non commutatif, Bull. Soc. Math. Franc. 7 (1943), 27–45.CrossRefGoogle Scholar
11.Dyson, F. J., Quaternion determinants, Helv. Phys. Act. 45 (1972), 289–302.Google Scholar
12.Havie, T., Generalized Neville type extrapolation schemes, BI. 19 (1979), 204–213.Google Scholar
13.Henrici, P., Elements of Numerical Analysis (Wiley, New York, 1964).Google Scholar
14.Heyting, A., Die Theorie der linearen Gleichungen in einer Zahlenspezies mit nicht-kommutativer Multiplikation, Math. Ann. 98 (1927), 465–490.CrossRefGoogle Scholar
15.Mehta, M. L., Matrix Theory, Selected Topics and Useful Results (Les Editions de Physique, Les Ullis, 1989).Google Scholar
16.Sadok, H., About Henrici's transformation for accelerating vector sequences, J. Comput. Appl. Math. 29 (1990), 101–110.Google Scholar
17.Salam, A., Extrapolation: Extension el Nouveaux Resultats (Thèse, Université des Sciences et Technologies de Lille, 1993).Google Scholar
18.Salam, A., Non-commutative extrapolation algorithms, Numerical Algorithm. 7 (1994), 225–251.Google Scholar
19.Salam, A., On the vector-valued Padé approximants and the vector ε-algorithm, in Nonlinear Numerical Methods and Rational Approximation, II (Cuyt, A. ed., Kluwer, Dordrecht, 1994), 291–301.CrossRefGoogle Scholar
20.Weniger, E. J., Nonlinear sequence transformations for the acceleration of convergence and the summation of divergent series, Comput. Phys. Report. 10 (1989), 189–371.Google Scholar
21.Wimp, J., Sequence Transformations and Their Applications (Academic Press, New York, 1981).Google Scholar