Constructive Approximation

, Volume 2, Issue 1, pp 263–289

Vector-valued, rational interpolants III

  • P. R. Graves-Morris
  • C. D. Jenkins
Article

Abstract

In 1963, Wynn proposed a method for rational interpolation of vector-valued quantities given on a set of distinct interpolation points. He used continued fractions, and generalized inverses for the reciprocals of vector-valued quantities. In this paper, we present an axiomatic approach to vector-valued rational interpolation. Uniquely defined interpolants are constructed for vector-valued data so that the components of the resulting vector-valued rational interpolant share a common denominator polynomial. An explicit determinantal formula is given for the denominator polynomial for the cases of (i) vector-valued rational interpolation on distinct real or complex points and (ii) vector-valued Padé approximation. We derive the connection with theε-algorithm of Wynn and Claessens, and we establish a five-term recurrence relation for the denominator polynomials.

AMS classification

41A21 

Key words and phrases

Rational interpolation Vector-valued data Continued fractions 

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References

  1. C. Brezinski (1974):Some results in the theory of the vector ε-algorithm. Linear Algebra Appl.,8:77–86.Google Scholar
  2. C. Brezinski (1975):Numerical stability of a quadratic method for solving systems of non-linear equations. Computing,14:205–211.Google Scholar
  3. C. Brezinski (1976):Computation of Padé approximants and continued fractions. J. Comput. Appl. Math.,2:113–123.Google Scholar
  4. C. Brezinski (1980): Padé-Type Approximations and General Orthogonal Polynomials. Basel: Birkhäuser.Google Scholar
  5. C. Brezinski, A. C. Rieu (1974):The solution of systems of equations using the ε-algorithm, and an application to boundary-value problems. Math. Comp.,28:731–741.Google Scholar
  6. G. Claessens (1978):A useful identity for the rational hermite interpolation table. Numer. Math.,29:227–231.Google Scholar
  7. E. Gekeler (1972):On the solution of systems of equations by the epsilon algorithm of Wynn. Math. Comp.,26:427–436.Google Scholar
  8. J. Gilewicz (1978): Approximants de Padé, Lecture Notes in Mathematics, vol 667. Heidelberg: Springer-Verlag, pp. 231–245.Google Scholar
  9. P. R. Graves-Morris (1981):Efficient, reliable rational interpolation. In: Padé Approximation and its Applications, Amsterdam 1980 (M. G. de Bruin, H. van Rossum eds.). New York; Springer-Verlag.Google Scholar
  10. P. R. Graves-Morris (1983):Vector-valued rational interpolants I. Numer. Math.,41:331–348.Google Scholar
  11. P. R. Graves-Morris, C. Jenkins (1984):Generalised inverse vector valued rational interpolation. In: Padé Approximation and its Applications, BadHonnef, 1983 (H. Werner, H. J. Hünger eds.). Lectures Notes in Mathematics, vol. 1071. Heidelberg: Springer-Verlag, pp. 144–156.Google Scholar
  12. P. R. Graves-Morris, C. D. Jenkins (1986): In preparation.Google Scholar
  13. J. B. McCleod (1971):A Note of the ε-algorithm. Computing,7:17–24.Google Scholar
  14. J. Nuttall (1970):Convergence of Padé approximants of meromorphic functions. J. Math. Anal. Appl.,31:147–153.Google Scholar
  15. D. E. Roberts, P. R. Graves-Morris (1986):The Application of Generalised Inverse Rational Interpolants in the Modal Analysis of Vibrating Structures I. In: Proceedings of the IMA Shrivenham Conference, 1985 (M. G. Cox, J. C. Mason eds.). Oxford: Oxford University Press.Google Scholar
  16. D. A. Smith, W. F. Ford, A. Sidi (1985):Extrapolation methods for vector sequences. Duke University reprint.Google Scholar
  17. H. C. Thacher, J. W. Tukey (1960): Rational Interpolation Made Easy by a Recursive Algorithm. Unpublished.Google Scholar
  18. P. Wynn (1962):Acceleration techniques for iterated vector and matrix problems. Math. Comp.,16:301–322.Google Scholar
  19. P. Wynn (1963):Continued fractions whose coefficients obey a non-commutative law of multiplication. Arch. Rational Mech. Anal.,12:273–312.Google Scholar

Copyright information

© Springer-Verlag New York Inc. 1986

Authors and Affiliations

  • P. R. Graves-Morris
    • 1
  • C. D. Jenkins
    • 1
  1. 1.Mathematical InstituteUniversity of KentCanterburyEngland

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