Abstract
The Remez algorithm, 75 years old, is a famous method for computing minimax polynomial approximations. Most implementations of this algorithm date to an era when tractable degrees were in the dozens, whereas today, degrees of hundreds or thousands are not a problem. We present a 21st-century update of the Remez ideas in the context of the chebfun software system, which carries out numerical computing with functions rather than numbers. A crucial feature of the new method is its use of chebfun global rootfinding to locate extrema at each iterative step, based on a recursive algorithm combining ideas of Specht, Good, Boyd, and Battles. Another important feature is the use of the barycentric interpolation formula to represent the trial polynomials, which points the way to generalizations for rational approximations. We comment on available software for minimax approximation and its scientific context, arguing that its greatest importance these days is probably for fundamental studies rather than applications.
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References
Almacany, M., Dunham, C., Williams, J.: Discrete Chebyshev approximation by interpolating rationals. IMA J. Numer. Anal. 4, 467–477 (1984)
Barrodale, I., Phillips, C.: Solution of an overdetermined system of linear equations in the Chebyshev norm. ACM Trans. Math. Softw. 1, 264–270 (1975)
Battles, Z.: Numerical linear algebra for continuous functions. PhD thesis, University of Oxford (2005)
Battles, Z., Trefethen, L.N.: An extension of MATLAB to continuous functions and operators. SIAM J. Sci. Comput. 25(5), 1743–1770 (2004)
Bernstein, S.: Sur la meilleure approximation de |x| par des polynomes de degrés donnés. Acta Math. 37, 1–57 (1914)
Berrut, J.P., Trefethen, L.N.: Barycentric Lagrange interpolation. SIAM Rev. 46, 501–517 (2004)
Boothroyd, J.: Algorithm 318: Chebyschev curve-fit. Commun. ACM 10(12), 801–803 (1967)
Borel, E.: Leçons sur les fonctions de variables réelles. Gauthier-Villars, Paris (1905)
Boyd, J.A.: Computing zeros on a real interval through Chebyshev expansion and polynomial rootfinding. SIAM J. Numer. Anal. 40(5), 1666–1682 (2002)
Brutman, L.: Lebesgue functions for polynomial interpolation—a survey. Ann. Numer. Math. 4, 111–128 (1997)
Cheney, E.W.: Introduction to Approximation Theory. McGraw-Hill, New York (1966)
Cody, W.J.: The FUNPACK package of special function subroutines. ACM Trans. Math. Softw. 1(1), 13–25 (1975)
Cody, W.J.: Algorithm 715: SPECFUN—a portable FORTRAN package of special function routines and test drivers. ACM Trans. Math. Softw. 19(1), 22–30 (1993)
Curtis, P.C., Frank, W.L.: An algorithm for the determination of the polynomial of best minimax approximation to a function defined on a finite point set. J. ACM 6, 395–404 (1959)
Davis, P.J.: Interpolation and Approximation. Dover, New York (1975)
de Boor, C., Rice, J.R.: Extremal polynomials with application to Richardson iteration for indefinite linear systems. SIAM J. Sci. Stat. Comput. 3, 47–57 (1982)
de la Vallée Poussin, C.J.: Sur les polynomes d’approximation et la représentation approchée d’un angle. Acad. R. Belg., Bull. Cl. Sci. 12 (1910)
Dunham, C.B.: Choice of basis for Chebyshev approximation. ACM Trans. Math. Softw. 8(1), 21–25 (1982)
Golub, G.H., Smith, L.B.: Algorithm 414: Chebyshev approximation of continuous functions by a Chebyshev system of functions. Commun. ACM 14(11), 737–746 (1971)
Good, I.J.: The colleague matrix, a Chebyshev analogue of the companion matrix. Q. J. Math. 12, 61–68 (1961)
Gutknecht, M.H., Trefethen, L.N.: Real polynomial Chebyshev approximation by the Carathéodory-Fejér method. SIAM J. Numer. Anal. 19, 358–371 (1982)
Higham, N.J.: Accuracy and Stability of Numerical Algorithms, 2nd edn. SIAM, Philadelphia (2002)
Higham, N.J.: The numerical stability of barycentric Lagrange interpolation. IMA J. Numer. Anal. 24, 547–556 (2004)
Kaufman, Jr., E.H., Leeming, D.J., Taylor, G.D.: Uniform rational approximation by differential correction and Remes-differential correction. Int. J. Numer. Methods Eng. 17, 1273–1278 (1981)
Le Bailly, B., Thiran, J.P.: Computing complex polynomial Chebyshev approximants on the unit circle by the real Remez algorithm. SIAM J. Numer. Anal. 36, 1858–1877 (1999)
Lorentz, G.G.: Approximation of Functions. Holt, Rinehart and Winston (1966)
MATLAB: User’s Guide. The MathWorks Inc., Natick, Massachusetts
McClellan, J.H., Parks, T.W.: A personal history of the Parks-McClellan algorithm. IEEE Signal Process. Mag. 22, 82–86 (2005)
McClellan, J.H., Parks, T.W., Rabiner, L.R.: A computer program for designing optimum FIR linear phase digital filters. IEEE Trans. Audio Electroacoust. 21, 506–526 (1973)
Meinardus, G.: Approximation of Functions: Theory and Numerical Methods. Springer, Heidelberg (1967)
Mhaskar, H.N., Pai, D.V.: Fundamentals of Approximation Theory. Narosa Publishing House, New Delhi (2000)
Murnaghan, F.D., Wrench, J.W.: J.: The determination of the Chebyshev approximating polynomial for a differentiable function. Math. Tables Aids Comput. 13, 185–193 (1959)
NAG: Library, Manual. The Numerical Algorithms Group, Ltd., Oxford, UK
Numerical Libraries, I.M.S.L.: Technical Documentation. Visual Numerics Inc., Houston
Pachón, R., Platte, R., Trefethen, L.N.: Piecewise smooth chebfuns. IMA J. Numer. Anal. (to appear)
Parks, T.W., McClellan, J.H.: Chebyshev approximation for nonrecursive digital filters with linear phase. IEEE Trans. Circuit Theory 19, 189–194 (1972)
Powell, M.J.D.: Approximation Theory and Methods. Cambridge University Press, Cambridge (1981)
Rabinowitz, P.: Applications of linear programming to numerical analysis. SIAM Rev. 10, 121–159 (1968)
Remes, E.: Sur le calcul effectif des polynomes d’approximation de Tchebychef. C. R. Acad. Sci. 199, 337–340 (1934)
Remes, E.: Sur un procédé convergent d’approximations successives pour déterminer les polynomes d’approximation. C. R. Acad. Sci. 198, 2063–2065 (1934)
Remes, E.: Sur la détermination des polynomes d’approximation de degré donnée. Commun. Soc. Math. Kharkov 10 (1934)
Rice, J.R.: The Approximation of Functions, vol. 1. Addison-Wesley, Reading (1964)
Sauer, F.W.: Algorithm 604: A FORTRAN program for the calculation of an extremal polynomial. ACM Trans. Math. Softw. 9(3), 381–383 (1983)
Schmitt, H.: Algorithm 409, discrete Chebychev curve fit. Commun. ACM 14, 355–356 (1971)
Simpson, J.C.: Fortran translation of algorithm 409, Discrete Chebychev curve fit. ACM Trans. Math. Softw. 2, 95–97 (1976)
Specht, W.: Die Lage der Nullstellen eines Polynoms, IV. Math. Nachr. 21, 201–222 (1960)
Steffens, K.G.: The History of Approximation Theory: From Euler to Bernstein. Birkhäuser, Boston (2006)
Stiefel, E.L.: Numerical methods of Tchebycheff approximation. In: Langer, R. (ed.) On Numerical Approximation, pp. 217–232. University of Wisconsin Press, Madison (1959)
Taylor, R., Totik, V.: Lebesgue constants for Leja points. IMA J. Numer. Anal. (to appear)
Trefethen, L.N.: Square blocks and equioscillation in the Padé, Walsh, and CF tables. In: Graves-Morris, P., Saff, E., Varga, R. (eds.) Rational Approximation and Interpolation. Lect. Notes in Math., vol. 1105. Springer, Berlin (1984)
Trefethen, L.N.: Spectral Methods in MATLAB. SIAM, Philadelphia (2000)
Varga, R.S., Carpenter, A.J.: On the Bernstein conjecture in approximation theory. Constr. Approx. 1, 333–348 (1985)
Veidinger, L.: On the numerical determination of the best approximation in the Chebyshev sense. Numer. Math. 2, 99–105 (1960)
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Communicated by Hans Petter Langtangen.
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Pachón, R., Trefethen, L.N. Barycentric-Remez algorithms for best polynomial approximation in the chebfun system. Bit Numer Math 49, 721–741 (2009). https://doi.org/10.1007/s10543-009-0240-1
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DOI: https://doi.org/10.1007/s10543-009-0240-1