Abstract
We define rational Hermite interpolants to vector-valued functions and show that, in the context of Clifford algebras, the numerator and denominator polynomials belong to a complex extension of the Lipschitz group. We also discuss the problem of constructing an algebraic representation for the generalized inverse of a vector, which is at the heart of the usual development of vector rational approximation.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
P. Wynn, “Acceleration techniques for iterative vector and matrix problems,”Math. Comp. 16, 301–322 (1962).
D. A. Smith, W. F. Ford and A. Sidi, “Extrapolation methods for vector sequences,”SIAM Rev. 29, 199–233 (1987).
D. E. Roberts and P. R. Graves-Morris, “The application of generalised inverse rational intepolants in the modal analysis of vibrating structures I,” inAlgorithms for Approximation, J. C. Mason and M. G. Cox, eds. (Oxford University Press, Oxford, 1987).
P. R. Graves-Morris, “Solution of integral equations using generalised inverse, function-valued Padé approximants I,”J. Comput. Appl. Math. 32, 117–134 (1990).
P. R. Graves-Morris and R. Thukral, “Solution of integral equations using function-valued Padé approximants II,”Numerical Algorithms 3, 223–234 (1992).
J. B. McLeod, “A note on the ɛ-algorithm,”Computing 7, 17–24 (1971).
P. Wynn, “Vector continued factions,”Linear Algebra Appl. 1, 357–395 (1968).
P. R. Graves-Morris, “Vector-valued rational interpolants I,”Numer. Math. 42, 331–348 (1983).
P. R. Graves-Morris and C. D. Jenkins, “Generalised inverse vector-valued rational interpolation,”Padé Approximation and Its Applications, Bad Honnef, 1983, H. Werner and H. J. Bünger, eds. (Springer, Berlin, 1984), pp. 144–156.
P. R. Graves-Morris and C. D. Jenkins, “Vector-valued rational interpolants III,”Constr. Approx. 2, 263–289 (1986).
D. E. Roberts, “Clifford algebras and vector-valued rational forms I,”Proc. R. Soc. London A. 431, 285–300 (1990).
D. E. Roberts, “Clifford algebras and vector-valued rational forms II,”Numerical Algorithms 3, 371–382 (1992).
P. Lounesto, “Clifford algebras and Hestenes spinors,”Found. Phys. 23, 1203–1237 (1993).
G. A. Baker, Jr., and P. R. Graves-Morris,Padé Approximants (Encyclopaedia of Mathematics and Its Applications13 and14) (Addison-Wesley, Reading, Massachusetts, 1981).
P. R. Graves-Morris and D. E. Roberts, “From matrix to vector Padé approximants,”J. Comput. Appl. Math., in press.
L. V. Ahlfors and P. Lounesto, “Some remarks on Clifford algebras,”Complex Variables 12, 201–209 (1989).
R. D. Schafer,An Introduction to Non-Associative Algebras (Academic, New York, 1966).
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Roberts, D.E. Vector-valued rational forms. Found Phys 23, 1521–1533 (1993). https://doi.org/10.1007/BF01243944
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF01243944