# Numerical continued fraction interpolation

### Abstract

UDC 517.524

We show that highly accurate approximations can often be obtained by constructing Thiele interpolating continued fractions by a Greedy selection of the interpolation points together with an early termination condition. The obtained results are comparable with the outcome of state-of-the-art rational interpolation techniques based on the barycentric form.

### References

B. Beckermann, J. Bisch, R. Luce, *On the rational approximation of Markov functions, with applications to the computation of Markov functions of Toeplitz matrices*, Numer. Algorithms, **91**, 109–144 (2022).

A. A. Cuyt, L. Jacobsen, B. M. Verdonk, *Instability and modification of thiele interpolating continued fractions*, Appl. Numer. Math., **4**, 253–262 (1988).

S.-I. Filip, Y. Nakatsukasa, L. N. Trefethen, B. Beckermann, *Rational minimax approximation via adaptive barycentric representations*, SIAM J. Sci. Comput., **40**, A2427–A2455 (2018).

P. R. Graves-Morris, *Practical, reliable, rational interpolation*, IMA J. Appl. Math., **25**, 267–286 (1980).

C. Hofreither, *An algorithm for best rational approximation based on barycentric rational interpolation*, Numer. Algorithms, **88**, 365–388 (2021).

G. Ion Victor, A. C. Athanasios, *Rational approximation of the absolute value function from measurements: a numerical study of recent methods*; arXiv (2020).

W. B. Jones, W. J. Thron, *Numerical stability in evaluating continued fractions*, Math. Comput., **28**, 795–810 (1974).

L. M. Milne-Thomson, *The calculus of finite differences*, Macmillan and Co. Ltd., London (1933).

Y. Nakatsukasa, O. Sète, L. Trefethen, *The AAA algorithm for rational approximation*, SIAM J. Sci. Comput., **40**, 1494–1522 (2018).

Y. Nakatsukasa, L. N. Trefethen, *An algorithm for real and complex rational minimax approximation*, SIAM J. Sci. Comput., **42**, A3157–A3179 (2020).

D. J. Newman, *Rational approximation to $|x|$*, Michigan Math. J., **11**, 11–14 (1964).

R. Pachón, *Algorithms for polynomial and rational approximation*, PhD Thesis, Univ. Oxford (2010).

R. Core Team, *R: a language and environment for statistical computing*, R Foundation for Statistical Computing, Vienna (2020); https://www.R-project.org/.

S. Robert Forsyth, *A treatise on the theory of determinants: and their applications in analysis and geometry*, Cambridge Univ. Press, Cambridge (1880).

R. Varga, A. Ruttan, A. Karpenter, *Numerical results on best numerical approximation of $|x|$ on $[-1, +1]$*, Math. USSR-Sb., **74**, 271–290 (1993).

J. Wallis, *Arithmetica infinitorum* (1655).

J. L. Walsh, *On approximation to an analytic function by rational functions of best approximation*, Math. Z., **38**, 163–176 (1934).

H. Werner, *A reliable method for rational interpolation*, Padé Approximation and Its Applications, L. Wuytack (ed.), Springer, Berlin, Heidelberg, (1979), p. 257–277.

T. F. Xie, S. P. Zhou, *The asymptotic property of approximation to $|x|$ by Newman's rational operators*, Acta Math. Hungar., **103**, 313–319 (2004).

*Ukrains’kyi Matematychnyi Zhurnal*, Vol. 76, no. 4, Apr. 2024, pp. 568 -0, doi:10.3842/umzh.v74i4.7349.

Copyright (c) 2024 Oliver Salazar Celis

This work is licensed under a Creative Commons Attribution 4.0 International License.