Nothing Special   »   [go: up one dir, main page]
Skip to main content
SpringerLink
Log in

Fast barycentric rational interpolations for complex functions with some singularities

  • Published:
Calcolo Aims and scope Submit manuscript

Abstract

Based on Cauchy’s integral formula and conformal maps, this paper presents a new method for constructing barycentric rational interpolation formulae for complex functions, which may contain singularities such as poles, branch cuts, or essential singularities. The resulting interpolations are pole-free, exponentially convergent, and numerically stable, requiring only \({\mathcal {O}}(N)\) operations. Inspired by the logarithm equilibrium potential, we introduce a Möbius transform to concentrate nodes to the vicinity of singularity to get a spectacular improvement on approximation quality. A thorough convergence analysis is provided, alongside numerous numerical examples that illustrate the theoretical results and demonstrate the accuracy and efficiency of the methodology. Meanwhile, the paper also discusses some applications of the method including the numerical solutions of boundary value problems and the zero locations of holomorphic functions.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12
Fig. 13
Fig. 14
Fig. 15
Fig. 16
Fig. 17
Fig. 18
Fig. 19
Fig. 20
Fig. 21
Fig. 22

Similar content being viewed by others

References

  1. Oppenheim, A.V., Willsky, A.S., Nawab, S.H.: Signals and Systems, 2nd edn. Pearson Prentice-Hall, USA (1997)

    Google Scholar 

  2. Oppenheim, A.V., Schafer, R.W.: Discrete-Time Signal Processing, 3rd edn. Pearson Prentice-Hall, USA (2009)

    MATH  Google Scholar 

  3. Sierra, J.M.G.: Digital Signal Processing with Matlab Examples, Volume (1–3). Springer Science+Business Media, Singapore (2017)

    Book  Google Scholar 

  4. Gaier, D.: Vorlesungen über Approximation im Komplexen. Birkhäuser Basel, (1980)

  5. Webb, M., Trefethen, L. N.: Computing Complex Singularities of Differential Equations with Chebfun. Electronic Transactions on Numerical Analysis (2011)

  6. Guide, M.E., Miedlar, A., Saad, Y.: A rational approximation method for solving acoustic nonlinear eigenvalue problems. Eng. Anal. Bound. Element 111, 44–54 (2020). https://doi.org/10.1016/j.enganabound.2019.10.006

    Article  MathSciNet  MATH  Google Scholar 

  7. Gopal, A., Trefethen, L.N.: Solving Laplace problems with corner singularities via rational functions. IAM J. Numer. Anal. 57(5), 2074–2094 (2019). https://doi.org/10.1137/19M125947X

    Article  MathSciNet  MATH  Google Scholar 

  8. Taylor, W.J.: Method of Lagrangian curvilinear interpolation. J. Res. Natl. Bureau Standards 35, 151–155 (1945). https://doi.org/10.6028/JRES.035.006

    Article  MathSciNet  MATH  Google Scholar 

  9. Dupuy, M.: Le calcul numerique des fonctions par linterpolation barycentrique. Comptes Rendus Hebdomadaires Des Seances De L Acad 226, 158–159 (1948)

    MathSciNet  MATH  Google Scholar 

  10. Jacobi, C. G. J.: Disquisitiones analyticae de fractionibus simplicibus. Dissertation, Berlin (1825)

  11. Gautschi, W.: Numerical Analysis, 2nd edition. Birkhäuser (2012)

  12. Berrut, J.-P., Trefethen, L.N.: Barycentric lagrange interpolation. SIAM Rev. 46(3), 501–517 (2004). https://doi.org/10.1137/s0036144502417715

    Article  MathSciNet  MATH  Google Scholar 

  13. Trefethen, L.N.: Approximation Theory and Approximation Practice. Extended edition, SIAM Philadelphia (2020)

    MATH  Google Scholar 

  14. Wang, H., Xiang, S.: On the convergence rates of Legendre approximation. Math. Comput. 278(81), 861–877 (2012). https://doi.org/10.2307/23267976

    Article  MathSciNet  MATH  Google Scholar 

  15. Salzer, H.E.: Lagrangian interpolation at the chebyshev points \(x_n=\cos {v\pi /n}\), \(v = 0(1)n\): some unnoted advantages. Comput. J. (1972). https://doi.org/10.1093/comjnl/15.2.156

    Article  MATH  Google Scholar 

  16. Wang, H., Huybrechs, D., Vandewalle, S.: Explicit barycentric weights for polynomial interpolation in the roots or extrema of classical orthogonal polynomials. Math. Comput. 290(83), 2893–2914 (2014). https://doi.org/10.1090/S0025-5718-2014-02821-4

    Article  MathSciNet  MATH  Google Scholar 

  17. Higham, N.J.: The numerical stability of barycentric Lagrange interpolation. IMA J. Numer. Anal. 24, 547–556 (2004). https://doi.org/10.1093/imanum/24.4.547

    Article  MathSciNet  MATH  Google Scholar 

  18. Schneider, C., Werner, W.: Some new aspects of rational interpolation. Math. Comput. 47(175), 285–299 (1986). https://doi.org/10.2307/2008095

    Article  MathSciNet  MATH  Google Scholar 

  19. Berrut, J.-P.: Rational functions for guaranteed and experimentally well-conditioned global interpolation. Comput. Math. Appl. 15(1), 1–16 (1988). https://doi.org/10.1016/0898-1221(88)90067-3

    Article  MathSciNet  MATH  Google Scholar 

  20. Floater, M.S., Hormann, K.: Barycentric rational interpolation with no poles and high rates of approximation. Numer. Math. 107, 315–331 (2007). https://doi.org/10.1007/s00211-007-0093-y

    Article  MathSciNet  MATH  Google Scholar 

  21. Hormann, K.: Barycentric interpolation. Approximation Theory XIV: San Antonio 2013. Springer International Publishing, 197–218. (2014) https://doi.org/10.1007/978-3-319-06404-8-11

  22. Berrut, J.-P., Klein, G.: Recent advances in linear barycentric rational interpolation. J. Comput. Appl. Math. 259, 95–107 (2014). https://doi.org/10.1016/j.cam.2013.03.044

    Article  MathSciNet  MATH  Google Scholar 

  23. Berrut, J.-P., Mittelmann, H.D.: Optimized point shifts and poles in the linear rational pseudospectral method for boundary value problems. J. Comput. Phys. 204, 292–301 (2005). https://doi.org/10.1016/j.jcp.2004.10.009

    Article  MathSciNet  MATH  Google Scholar 

  24. Baltenspeger, R., Berrut, J.-P., Noël, B.: Exponential convergence of a linear rational interpolant between transformed Chebyshev points. Math. Comput. 227(68), 1109–1120 (1999). https://doi.org/10.1090/S0025-5718-99-01070-4

    Article  MathSciNet  MATH  Google Scholar 

  25. Berrut, J.-P., Mittelmann, H.D.: Adaptive point shifts in rational approximation with optimized denominator. J. Comput. Appl. Math. 164–165, 81–92 (2004). https://doi.org/10.1016/S0377-0427(03)00485-0

    Article  MathSciNet  MATH  Google Scholar 

  26. Berrut, J.-P., Elefante, G.: A periodic map for linear barycentric rational trigonometric interpolation. Appl. Math. Comput. 371, 124924 (2020). https://doi.org/10.1016/j.amc.2019.124924

    Article  MathSciNet  MATH  Google Scholar 

  27. Tee, T.W., Trefethen, L.N.: A Rational Spectral Collocation Method With Adaptively Transformed Chebyshev Grid Points. SIAM J. Sci. Comput. 28(5), 1798–1811 (2006). https://doi.org/10.1137/050641296

    Article  MathSciNet  MATH  Google Scholar 

  28. Baltensperger, R.: Some Results on Linear Rational Trigonometric Interpolation. Comput. Math. Appl. 43, 737–746 (2002). https://doi.org/10.1016/S0898-1221(01)00317-0

    Article  MathSciNet  MATH  Google Scholar 

  29. Berrut, J.-P., Elefante, G.: Bounding the Lebesgue constant for a barycentric rational trigonometric interpolant at periodic well-spaced nodes. J. Comput. Appl. Math. 398, 113664 (2021). https://doi.org/10.1016/j.cam.2021.113664

    Article  MathSciNet  MATH  Google Scholar 

  30. Kong, D., Xiang, S.: Fast rational spectral interpolation for singular functions via scaled transformations. arXiv preprint arXiv:2101.07949, (2021). https://doi.org/10.48550/arXiv.2101.07949

  31. Kosloff, D., Tal-Ezer, H.: A modified Chebyshev pseudospectral method with an \({\cal{O} }(N^{-1})\) time step restriction. J. Comput. Phys. 104, 457–469 (1993). https://doi.org/10.1006/jcph.1993.1044

    Article  MathSciNet  MATH  Google Scholar 

  32. Bayliss, A., Turkel, E.: Mappings and accuracy for Chebyshev pseudo-spectral approximations. J. Comput. Phys. 101, 349–359 (1992). https://doi.org/10.1016/0021-9991(92)90012-N

    Article  MathSciNet  MATH  Google Scholar 

  33. Austin, A.P., Kravanja, P., Trefethen, L.N.: Numerical Algorithms Based on Analytic Function Values at Roots of Unity. SIAM J. Numer. Anal. 52(4), 1795–1821 (2014). https://doi.org/10.1137/130931035

    Article  MathSciNet  MATH  Google Scholar 

  34. de Bruin, M.G., Dikshit, H.P., Sharma, A.: Birkhoff interpolation on unity and on M\(\ddot{{\rm o}}\)bius transform of the roots of unity. Numer. Algo. 23, 115–125 (2000). https://doi.org/10.1023/A:1019147900265

    Article  MATH  Google Scholar 

  35. Pachón, R., Gonnet, P., van Deun, J.: Fast and Stable Rational Interpolation in Roots of Unity and Chebyshev points. SIAM J. Numer. Anal. 50(3), 1713–1734 (2012). https://doi.org/10.1137/100797291

    Article  MathSciNet  MATH  Google Scholar 

  36. Gonnet, P., Pachón, R., Trefethen, L.N.: Robust rational interpolation and least-squares. Elect. Trans. Numer. Anal. 38, 146–167 (2011). https://doi.org/10.1142/S0217751X97001079

    Article  MathSciNet  MATH  Google Scholar 

  37. Henrici, P.: Applied and Computational Complex Analysis. Vol. 1. Wiley-Interscience (John Wiley & Sons), New York-London-Sydney (1974)

  38. Driscoll, T. A., Hale, N., Trefethen, L. N.: Chebfun Guide, http://www.chebfun.org/docs/guide/

  39. Trefethen, L.N., Weideman, J.A.C.: The exponentially convergent trapezoidal rule. SIAM Rev. 56(3), 385–458 (2014). https://doi.org/10.1137/130932132

    Article  MathSciNet  MATH  Google Scholar 

  40. Rizzardi, M.: Detection of the singularities of a complex function by numercical approximations of its Laurent coefficients. Numer. Algo. 77, 955–982 (2018). https://doi.org/10.1007/s11075-017-0349-2

    Article  MathSciNet  MATH  Google Scholar 

  41. Rizzardi, M.: http://dist.uniparthenope.it/mariarosaria.rizzardi, (2017)

  42. Lyness, J. N.: Numerical algorithms based on the theory of complex variables. In: Proceedings of the 1967 22nd National Conference, 125–133. (1967) https://doi.org/10.1145/800196.805983

  43. Bornemann, F.: Accuracy and stability of computing high-order derivatives of analytic functions by cauchy integrals. Found. Comput. Math. 11, 1–63 (2011). https://doi.org/10.1007/s10208-010-9075-z

    Article  MathSciNet  MATH  Google Scholar 

  44. Jentzsch, R.: Untersuchungen zur Theorie Analytischer Funktionen. Dissertation, Berlin (1914)

  45. Walsh, J.L.: The analogue for maximally convergent polynomials of Jentzsch’s theorem. Duke Math. J. 26, 605–616 (1959). https://doi.org/10.1215/S0012-7094-59-02658-4

    Article  MATH  Google Scholar 

  46. Wegert, E.: Visual Complex Functions: An Introduction with Phase Portraits. Springer, Basel (2012)

    Book  MATH  Google Scholar 

  47. Wegert, E.: Phase Plots of Complex Functions (https://www.mathworks.com/matlabcentral/fileexchange/44375-phase-plots-of-complex-functions), MATLAB Central File Exchange. Retrieved March 25, (2022)

  48. Lyness, J.N., Delves, L.M.: On numerical contour integration round a closed contour. Math. Comput. 21, 561–577 (1967). https://doi.org/10.2307/2005000

    Article  MathSciNet  MATH  Google Scholar 

  49. Kaup, L., Kaup, B.: Holomorphic Functions of Several Variables. de Gruyter, Berlin (1983)

    Book  MATH  Google Scholar 

  50. Levin, E., Saff, E.B.: Potential Theoretic Tools in Polynomial and Rational Approximation. In: Harmonic Analysis and Rational Approximation, Springer, Berlin Heidelberg 327, 71–94 (2006). https://doi.org/10.1007/11601609-5

  51. Saff, E. B., Totik, V.: Logarithmic Potentials with External Fields. Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], Springer, Berlin, 316: 505. (1997) https://doi.org/10.1007/978-3-662-03329-6

  52. Berrut, J.-P., Elefante, G.: A linear barycentric rational interpolant on starlike domains. Elect. Trans. Numer. Anal. 55, 726–74 (2022). https://doi.org/10.1553/etna_vol55s726

    Article  MathSciNet  MATH  Google Scholar 

  53. Kravanja, P., Barel, M.V.: Computing the Zeros of Analytic Functions. Springer-Verlag, Berlin Heidelberg (2000)

    Book  MATH  Google Scholar 

  54. Gautschi, W.: On the construction of Gaussian quadrature rules from modified moments. Mathematics of Computation 110(24), 245–260 (1970). https://doi.org/10.1090/S0025-5718-1970-0285117-6

    Article  MathSciNet  MATH  Google Scholar 

  55. Gautschi, W.: On generating orthogonal polynomials. SIAM J. Sci. Stat. Comput. 3(3), 289–317 (1982). https://doi.org/10.1137/0903018

    Article  MathSciNet  MATH  Google Scholar 

  56. Gautschi, W.: On the sensitivity of orthogonal polynomials to perturbations in the moments. Numer. Math. 48, 369–382 (1986). https://doi.org/10.1007/bf01389645

    Article  MathSciNet  MATH  Google Scholar 

  57. Beckermann, B., Bourreau, E.: How to choose modified moments? J. Comput. Appl. Math. 98, 81–98 (1998). https://doi.org/10.1016/S0377-0427(98)00116-2

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

The authors would like to thank Guidong Liu, Yanghao Wu and Desong Kong for their useful suggestions. The authors are grateful to the anonymous referees for their valuable comments and suggestions for improvement of this paper.

Author information

Authors and Affiliations

  1. School of Mathematics and Statistics, Central South University, Changsha, 410083, Hunan, P. R. China

    Shunfeng Yang & Shuhuang Xiang

  2. School of Mathematics and Physics, Southwest Forest University, Kunming, 650224, Yunnan, P. R. China

    Shunfeng Yang

Authors
  1. Shunfeng Yang
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Shuhuang Xiang
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Shuhuang Xiang.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

This work was supported by the National Natural Science Foundation of China (No. 12271528). The first author is supported by the Fundamental Research Funds for the Central Universities of Central South University (No.2020zzts030).

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Yang, S., Xiang, S. Fast barycentric rational interpolations for complex functions with some singularities. Calcolo 60, 55 (2023). https://doi.org/10.1007/s10092-023-00550-4

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s10092-023-00550-4

Keywords

Mathematics Subject Classification

Search

Navigation