Abstract
The best approximation problem is a classical topic of the approximation theory and the Remez algorithm is one of the most famous methods for computing minimax polynomial approximations. We present a slight modification of the (second) Remez algorithm where a new approach to update the trial reference is considered. In particular at each step, given the local extrema of the error function of the trial polynomial, the proposed algorithm replaces all the points of the trial reference considering some “ad hoc” oscillating local extrema and the global extremum (with its adjacent) of the error function. Moreover at each step the new trial reference is chosen trying to preserve a sort of equidistribution of the nodes at the ends of the approximation interval. Experimentally we have that this method is particularly appropriate when the number of the local extrema of the error function is very large. Several numerical experiments are performed to assess the real performance of the proposed method in the approximation of continuous and Lipschitz continuous functions. In particular, we compare the performance of the proposed method for the computation of the best approximant with the algorithm proposed in [17] where an update of the Remez ideas for best polynomial approximation in the context of the chebfun software system is studied.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Atkinson, K.E.: An Introduction to Numerical Analysis, 2nd edn. Wiley, New York (1989)
Battles, Z., Trefethen, L.N.: An extension of MATLAB to continuous functions and operators. SIAM J. Sci. Comput. 25(5), 1743–1770 (2004)
Berrut, J.P., Trefethen, L.N.: Barycentric Lagrange interpolation. SIAM Rev. 46, 501–517 (2004)
Blichfeldt, H.F.: Note on the functions of the form \( f(x)\equiv \varphi (x)+a_{1}x^{n-1} + \cdots +a_{n}\). Trans. Am. Math. Soc. 2, 100–102 (1901)
Chebyshev, P.L.: Théorie des mécanismes connus sous le nom de parallélogrammes, Mémoires de l’Académie impériale des sciences de St. Pétersbourg, vol. 7, pp. 539–564 (1854)
Davis, P.: Interpolation and Approximation. Ginn (Blaisdell), Boston (1963)
De La Vallée Poussin, Ch.J.: Leçons sur l’approximation des fonctions d’une variable réelle. Gautier-Villars, Paris (1919)
Devore, R.A., Lorentz, G.G.: Constructive Approximation. Grundlehren der mathematischen Wissenschaften, vol. 303. Springer, Berlin (1993)
Dunham, C.B.: Choice of basis for Chebyshev approximation. ACM Trans. Math. Softw. 8, 21–25 (1982)
Egidi, N., Fatone, L., Misici, L.: A stable Remez algorithm for minimax approximation (in preparation)
Higham, N.J.: Accuracy and Stability of Numerical Algorithms, 2nd edn. SIAM, Philadelphia (2002)
Kirchberger, P.: Über Tschebysheff’sche Annäherungsmethoden, dissertation, Göttingen (1902)
Korneichuk, N.: Exact Constants in Approximation Theory, Encyclopedia of Mathematics and its Applications, vol. 38. Cambridge University Press, Cambridge (1991)
Lorentz, G.G.: Approximation of Functions. Holt, Rinehart and Winston, New York (1966)
Mastroianni, G., Milovanovic, G.V.: Interpolation Processes. Basic Theory and Applications. Springer Monographs in Mathematics. Springer, Berlin (2008). https://doi.org/10.1007/978-3-540-68349-0
Meinardus, G.: Approximation of Functions: Theory and Numerical Methods. Springer, New York (1967). https://doi.org/10.1007/978-3-642-85643-3. (transl. L. Schumaker)
Pachón, R., Trefethen, L.N.: Barycentric-Remez algorithms for best polynomial approximation in the chebfun system. BIT Numer. Math. 49(4), 721–741 (2009)
Petrushev, P.P., Popov, V.A.: Rational Approximation of Real Function, Encyclopedia of Mathematics and its Applications, vol. 28. Cambridge University Press, Cambridge (1987)
Powell, M.J.D.: Approximation Theory and Methods. Cambridge University Press, Cambridge (1981)
Remez, E.Y.: Sur la détermination des polynomes d’approximation de degré donné. Commun. Kharkov Math. Soc. 10, 41–63 (1934)
Remez, E.Y.: Sur le calcul effectif des polynomes d’approximation de Tchebychef. Comptes rendus de l’Académie des Sciences 199, 337–340 (1934)
Remez, E.Y.: Sur un procédé convergent d’approximations successives pour déterminer les polynomes d’approximation. Comptes rendus de l’Académie des Sciences 198, 2063–2065 (1934)
Veidinger, L.: On the numerical determination of the best approximation in the Chebyshev sense. Numer. Math. 2, 99–105 (1960)
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2020 Springer Nature Switzerland AG
About this paper
Cite this paper
Egidi, N., Fatone, L., Misici, L. (2020). A New Remez-Type Algorithm for Best Polynomial Approximation. In: Sergeyev, Y., Kvasov, D. (eds) Numerical Computations: Theory and Algorithms. NUMTA 2019. Lecture Notes in Computer Science(), vol 11973. Springer, Cham. https://doi.org/10.1007/978-3-030-39081-5_7
Download citation
DOI: https://doi.org/10.1007/978-3-030-39081-5_7
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-030-39080-8
Online ISBN: 978-3-030-39081-5
eBook Packages: Computer ScienceComputer Science (R0)