Abstract
We present a novel algorithm for computing best uniform rational approximations to real scalar functions in the setting of zero defect. The method, dubbed BRASIL (best rational approximation by successive interval length adjustment), is based on the observation that the best rational approximation r to a function f must interpolate f at a certain number of interpolation nodes (xj). Furthermore, the sequence of local maximum errors per interval (xj− 1,xj) must equioscillate. The proposed algorithm iteratively rescales the lengths of the intervals with the goal of equilibrating the local errors. The required rational interpolants are computed in a stable way using the barycentric rational formula. The BRASIL algorithm may be viewed as a fixed-point iteration for the interpolation nodes and converges linearly. We demonstrate that a suitably designed rescaled and restarted Anderson acceleration (RAA) method significantly improves its convergence rate. The new algorithm exhibits excellent numerical stability and computes best rational approximations of high degree to many functions in a few seconds, using only standard IEEE double-precision arithmetic. A free and open-source implementation in Python is provided. We validate the algorithm by comparing to results from the literature. We also demonstrate that it converges quickly in some situations where the current state-of-the-art method, the minimax function from the Chebfun package which implements a barycentric variant of the Remez algorithm, fails.
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References
Achieser, N.I.: Theory of Approximation. Dover books on advanced mathematics. Dover Publications, ISBN 9780486671291 (1992)
Anderson, D.G.: Iterative procedures for nonlinear integral equations. J. ACM (JACM) 12(4), 547–560 (1965). https://doi.org/10.1145/321296.321305
Berrut, J.-P., Trefethen, L.N.: Barycentric Lagrange interpolation. SIAM Rev. 46 (3), 501–517 (2004). https://doi.org/10.1137/s0036144502417715
Berrut, J.-P., Baltensperger, R., Mittelmann, H.D.: Recent developments in barycentric rational interpolation. In: Trends and Applications in Constructive Approximation. https://doi.org/10.1007/3-7643-7356-3_3, pp 27–51. Birkhäuser, Basel (2005)
Braess, D.: Nonlinear Approximation Theory. Springer, Berlin (1986). ISBN 978-3-642-64883-0. https://doi.org/10.1007/978-3-642-61609-9
Brezinski, C., Redivo-Zaglia, M.: Padé-type rational and barycentric interpolation. Numer. Math. 125(1), 89–113 (2013). https://doi.org/10.1007/s00211-013-0535-7
Brezinski, C., Redivo-Zaglia, M.: New representations of padé padé-type, and partial padé approximants. J. Comput. Appl. Math. 284, 69–77 (2015). https://doi.org/10.1016/j.cam.2014.07.007
Brezinski, C., Redivo-Zaglia, M., Saad, Y.: Shanks sequence transformations and Anderson acceleration. SIAM Rev. 60 (3), 646–669 (2018). https://doi.org/10.1137/17m1120725
Carpenter, A.J., Ruttan, A., Varga, R.S.: Extended numerical computations on the 1/9 conjecture in rational approximation theory. In: Rational Approximation and Interpolation. https://doi.org/10.1007/bfb0072427, pp 383–411. Springer, Berlin (1984)
Cheney, E.W., Loeb, H.L.: Two new algorithms for rational approximation. Numer. Math. 3(1), 72–75 (1961). https://doi.org/10.1007/bf01386002
Demmel, J., Gu, M., Eisenstat, S., Slapničar, I., Veselič, K., Drmač, Z.: Computing the singular value decomposition with high relative accuracy. Linear Algebra Appl. 299(1-3), 21–80 (1999). https://doi.org/10.1016/s0024-3795(99)00134-2
Driscoll, T.A., Hale, N., Trefethen, L.N.: Chebfun Guide. Pafnuty Publications. http://www.chebfun.org/docs/guide/ (2014)
Fang, H., Saad, Y.: Two classes of multisecant methods for nonlinear acceleration. Numer. Linear Algebra Appl. 16(3), 197–221 (2009). https://doi.org/10.1002/nla.617
Filip, S.-I., Nakatsukasa, Y., Trefethen, L.N., Beckermann, B.: Rational minimax approximation via adaptive barycentric representations. SIAM J. Sci. Comput. 40(4), A2427–A2455 (2018). https://doi.org/10.1137/17m1132409
Golub, G.H., Van Loan, C.F.: Matrix Computations. Johns Hopkins University Press, 4th edn. ISBN 9781421408590 (2012)
Harizanov, S., Lazarov, R., Margenov, S., Marinov, P.: The best uniform rational approximation: Applications to solving equations involving fractional powers of elliptic operators. arXiv:1910.13865 (2019)
Harizanov, S., Lazarov, R., Margenov, S., Marinov, P., Pasciak, J.: Analysis of numerical methods for spectral fractional elliptic equations based on the best uniform rational approximation. J. Comput. Phys. https://doi.org/10.1016/j.jcp.2020.109285. Available online (2020)
Higham, N.J.: QR Factorization with complete pivoting and accurate computation of the SVD. Linear Algebra Appl. 309(1-3), 153–174 (2000). https://doi.org/10.1016/s0024-3795(99)00230-x
Higham, N.J.: Accuracy and Stability of Numerical Algorithms, 2nd edn. Society for Industrial and Applied Mathematics, Philadelphia (2002). ISBN 0-89871-521-0
Higham, N.J.: The numerical stability of barycentric Lagrange interpolation. IMA J. Numer. Anal. 24(4), 547–556 (2004). https://doi.org/10.1093/imanum/24.4.547
Higham, N.J., Strabic̀, N.: Anderson acceleration of the alternating projections method for computing the nearest correlation matrix. Numer. Algorithms 72(4), 1021–1042 (2015). https://doi.org/10.1007/s11075-015-0078-3
Hofreither, C.: A unified view of some numerical methods for fractional diffusion. Comput Math Appl 80(2), 332–350 (2020). https://doi.org/10.1016/j.camwa.2019.07.025
Ionięă, A.C.: Lagrange rational interpolation and its applications to approximation of large-scale dynamical systems. PhD thesis, Rice University, Houston, TY (2013)
Knockaert, L.: A simple and accurate algorithm for barycentric rational interpolation. IEEE Signal Process. Lett. 15, 154–157 (2008). https://doi.org/10.1109/lsp.2007.913583
Press, W.H., Teukolsky, S.A., Vetterling, W.T., Flannery, B.P.: Numerical Recipes in C, 2nd edn. Cambridge University Press, Cambridge (1992)
Schneider, C., Werner, W.: Some new aspects of rational interpolation. Math. Comput. 47(175), 285–285 (1986). https://doi.org/10.1090/s0025-5718-1986-0842136-8
Stahl, H.R.: Best uniform rational approximation of xα on [0, 1]. Acta Mathematica 190(2), 241–306 (2003). https://doi.org/10.1007/bf02392691
Toth, A., Kelley, C.T.: Convergence analysis for Anderson acceleration. SIAM J. Numer. Anal. 53(2), 805–819 (2015). https://doi.org/10.1137/130919398
Trefethen, L.N.: Approximation Theory and Approximation Practice. Other Titles in Applied Mathematics. SIAM, 2013. ISBN 9781611972405
Varga, R.S., Carpenter, A.J.: Some numerical results on best uniform rational approximation of xα on [0, 1]. Numer. Algorithms 2(2), 171–185 (1992). https://doi.org/10.1007/bf02145384
Varga, R.S., Ruttan, A., Carpenter, A.D.: Numerical results on best uniform rational approximation of |x| on [− 1, 1]. Mathematics of the USSR-Sbornik 74 (2), 271–290 (1993). https://doi.org/10.1070/sm1993v074n02abeh003347
Walker, H.F., Ni, P.: Anderson acceleration for fixed-point iterations. SIAM J. Numer. Anal. 49(4), 1715–1735 (2011). https://doi.org/10.1137/10078356x
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Hofreither, C. An algorithm for best rational approximation based on barycentric rational interpolation. Numer Algor 88, 365–388 (2021). https://doi.org/10.1007/s11075-020-01042-0
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DOI: https://doi.org/10.1007/s11075-020-01042-0