# The need for knowledge and reliability in numeric computation: Case study of multivariate Padé approximation

- 31 Downloads
- 4 Citations

## Abstract

In this paper we review and link the numeric research projects carried out at the Department of Mathematics and Computer Science of the University of Antwerp since 1978. Results have and are being obtained in various areas. A lot of effort has been put in the theoretical investigation of the multivariate Padé approximation problem using different definitions (see Sections 3 and 7). The numerical implementation raises two delicate issues. First, there is the need to see the wood for the trees again: switching from one to many variables greatly increases the number of choices to be made on the way (see Sections 1 and 5). Second, there is the typical problem of breakdown when computing ratios of determinants: the added value of interval arithmetic combined with defect correction turns out to be significant (see Sections 2 and 4). In Section 6 these two problems are thoroughly illustrated and the interested reader is taken by the hand and guided through a typical computation session. On the way some open problems are indicated which motivate us to continue our research mainly in the area of gathering and offering more knowledge about the problem domain on one hand, and improving the arithmetic tools and numerical routines for a reliable computation of the approximants on the other hand.

## Mathematics subject classifications (1991)

41A21 65D20## Key words

Padé approximation multivariate approximation## Preview

Unable to display preview. Download preview PDF.

## References

- 1.A. Beardon, The convergence of Padé approximants,
*J. Math. Anal. Appl.***21**(1968), pp. 344–346.Google Scholar - 2.C. Brezinski, and M. Redivo-Zaglia, Breakdowns in the computation of orthogonal polynomials, In A. Cuyt [17] (to appear).Google Scholar
- 3.Kevin A. Broughan, SENAC: A high-level interface for the NAG library,
*ACM Transactions on Mathematical Software***17**No. 4 (1991), pp. 462–480.Google Scholar - 4.C. W. Cryer, The ESPRIT project FOCUS, In P. W. Gaffney and E. N. Houstis [21], pp. 371–380.Google Scholar
- 5.A. Cuyt, The epsilon-algorithm and multivariate Padé approximants,
*Numer. Math.***40**(1982), pp. 39–46.Google Scholar - 6.A. Cuyt, A comparison of some multivariate Padé approximants,
*SIAM J. Math. Anal.***14**(1983), pp. 195–202.Google Scholar - 7.A. Cuyt, The QD-algorithm and multivariate Padé approximants,
*Numer. Math***42**(1983), pp. 259–269.Google Scholar - 8.A. Cuyt,
*Padé Approximants for Operators: Theory and Applications*, Lecture Notes in Mathematics**1065**, Springer-Verlag, Berlin, 1984.Google Scholar - 9.A. Cuyt, A Montessus de Ballore theorem for multivariate Padé approximants,
*J. Approx. Theory***43**(1985), pp. 43–52.Google Scholar - 10.A. Cuyt, A review of multivariate Padé approximation theory,
*J. Comput. Appl. Math.***12/13**(1985), pp. 221–232.Google Scholar - 11.A. Cuyt, General order multivariate rational Hermite interpolants, Monograph, 1986, University of Antwerp (UIA).Google Scholar
- 12.
- 13.A. Cuyt, A recursive computation scheme for multivariate rational interpolants,
*SIAM J. Num. Anal.***24**(1987), pp. 228–238.Google Scholar - 14.
- 15.A. Cuyt, A multivariate convergence theorem of the “de Montessus de Ballore” type,
*J. Comp. Appl. Math.***32**(1990), pp. 47–57.Google Scholar - 16.A. Cuyt, Extension of “A multivariate convergence theorem of the de Montessus de Bailore type” to multipoles,
*J. Comp. Appl. Math.***41**(1992), pp. 323–330.Google Scholar - 17.A. Cuyt (ed.),
*Nonlinear Numerical Methods and Rational Approximation II*, Dordrecht, Kluwer Academic Publishers, 1994 (to appear).Google Scholar - 18.A. Cuyt, Where do the columns of the multivariate qd-algorithm go? A proof constructed with the aid of Mathematica,
*Numer. Math.*(1994) (submitted).Google Scholar - 19.A. Cuyt, and B. Verdonk, General order Newton-Padé approximants for multivariate functions,
*Numer. Math.***43**(1984), pp. 293–307.Google Scholar - 20.R. de Montessus de Ballore, Sur les fractions continues algébriques,
*Rend. Circ. Mat. Palermo***19**(1905), pp. 1–73.Google Scholar - 21.P. W. Gaffney, and E. N. Houstis (eds.),
*Programming Environments for High-Level Scientific Problem Solving*, Amsterdam, Elsevier Science Publishers, 1992.Google Scholar - 22.P. Henrici,
*Applied and Computational Complex Analysis, Vols. 1 & 2*, John Wiley, New York, 1974.Google Scholar - 23.R. Klatte, U. Kulish, M. Neaga, D. Ratz, and Ch. Ullrich,
*Pascal-XSC: Language Reference with Examples*, Springer-Verlag, Berlin, 1992.Google Scholar - 24.U. Kulish, and W. L. Miranker (eds.),
*A New Approach to Scientific Computation*, Academic Press, Orlando, 1983.Google Scholar - 25.D. Levin, General-order Padé-type approximants defined from double power series,
*J. Inst. Maths. Appl***18**(1976), pp. 1–8.Google Scholar - 26.J. Nuttall, The convergence of Padé approximants of meromorphic functions,
*J. Math. Anal Appl***31**(1970), pp. 147–153.Google Scholar - 27.O. Perron,
*Die Lehre von den Kettenbruchen II*, Teubner, Stuttgart, 1977.Google Scholar - 28.S. Rump, Solving algebraic probems with high accuracy, In U. Kulish, and W. L. Miranker [24], pp. 51–120.Google Scholar
- 29.G. Schumacher, and J. Wolff von Gudenberg, Highly accurate numerical algorithms, In Ch. Ullrich, and J. Wolff von Gudenberg [30], pp. 1–58.Google Scholar
- 30.Ch. Ullrich, and J. Wolff von Gudenberg (eds.),
*Accurate Numerical Algorithms, A Collection of Research Papers*, Research Reports ESPRIT, Vol. 1, Springer-Verlag, Berlin, 1989.Google Scholar