Abstract
We present a method for asymptotically monitoring poles to a rational interpolant written in barycentric form. Theoretical and numerical results are given to show the potential of the proposed interpolant.
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Schneider, C., Werner, W.: Some new aspects of rational interpolation. Math. Comput. 47, 285–299 (1986)
Berrut, J.-P.: Rational functions for guaranteed and experimentally well-conditioned global interpolation. Comput. Math. Appl. 15, 1–16 (1988)
Berrut, J.-P., Mittelmann, H.D.: Lebesgue constant minimizing linear rational interpolation of continuous functions over the interval. Comput. Math. Appl. 33, 77–86 (1997)
Berrut, J.-P., Baltensperger, R., Mittelmann, H.D.: Recent developments in barycentric rational interpolation. Trends and applications in constructive approximation. Internat. Ser. Numer. Math. (ISNM) 151, 27–51 (2005)
Berrut, J.-P., Trefethen, L.N.: Barycentric lagrange interpolation. SIAM Rev. 46, 501–517 (2004)
Higham, N.: The numerical stability of barycentric Lagrange interpolation. IMA J. Numer. Anal. 24, 547–556 (2004)
Salzer, H.E.: Lagrangian interpolation at the Chebyshev points x n,ν = cos(νπ/n); some unnoted advantages. Comput. J. 15, 156–159 (1972)
Henrici, P.: Essentials of Numerical Analysis with Pocket Calculator Demonstrations. Wiley, New York (1982)
Battles, Z., Trefethen, L.N.: An extension of MATLAB to continuous functions and operators. SIAM J. Sci. Comput. 25, 1743–1770 (2004)
Trefethen, L.N.: Spectral Methods in MATLAB. SIAM, Philadelphia (2001)
Fornberg, B.: A Practical Guide to Pseudospectral Methods. Cambridge University Press, Cambridge (1996)
Berrut, J.-P.: The barycentric weights of rational interpolation with prescribed poles. J. Comput. Appl. Math. 86, 45–52 (1997)
Berrut, J.-P., Mittelmann, H.D.: Rational interpolation through the optimal attachment of poles to the interpolating polynomial. Numer. Algorithms 23, 315–328 (2000)
Baltensperger, R., Berrut, J.-P., Dubey, Y.: The linear rational pseudospectral method with preassigned poles. Numer. Algorithms 33, 53–63 (2003)
Floater, M.S., Hormann, K.: Barycentric rational interpolation with no poles and high rates of approximation. Numer. Math. 107, 315-331 (2007)
Baltensperger, R., Berrut, J.-P., Noël, B.: Exponential convergence of a linear rational interpolant between transformed Chebyshev points. Math. Comput. 68, 1109–1120 (1999)
Kosloff, D., Tal-Ezer, H.: A modified Chebyshev pseudospectral method with an \(\mathcal{O}(N^{-1})\) time step restriction. J. Comput. Phys. 104, 457–469 (1993)
Baltensperger, R., Berrut, J.-P.: The linear rational collocation method. J. Comput. Appl. Math. 134, 243–258 (2001)
Berrut, J.-P., Baltensperger, R.: The linear rational pseudospectral method for boundary value problems. BIT 41, 868–879 (2001)
Bayliss, A., Turkel, E.: Mappings and accuracy for Chebyshev pseudo-spectral approximations. J. Comput. Phys. 101, 349–359 (1992)
Berrut, J.-P., Mittelmann, H.D.: Adaptive point shifts in rational approximation with optimized denominator. J. Comput. Appl. Math. 164–165, 81–92 (2004)
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)
Tee, T.W., Trefethen, L.N.: A rational spectral collocation method with adaptively transformed Chebyshev grid points. SIAM J. Sci. Comput. 28, 1798–1811 (2006)
Hale, N., Tee, T.W.: Conformal maps to multiply-slit domains and applications. SIAM J. Sci. Comput. 31, 3195–3215 (2009)
Berrut, J.-P., Mittelmann, H.D.: Point shift in rational interpolation with optimized denominator. In: Algorithms for Approximation IV: Proceedings of the 2001 International Symposium, pp. 420–427 (2002)
Tee, T.W.: An adaptive rational spectral method for differential equations with rapidly varying solutions. Ph.D. Thesis, University of Oxford (2006)
Abramowitz, M., Stegun, I.A.: Handbook of Mathematical Functions with Formulas, Graphs and Mathematical Tables. Dover, New York (1964)
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Baltensperger, R. Barycentric rational interpolation with asymptotically monitored poles. Numer Algor 57, 67–81 (2011). https://doi.org/10.1007/s11075-010-9415-8
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DOI: https://doi.org/10.1007/s11075-010-9415-8