BIT Numerical Mathematics

, 49:721

Barycentric-Remez algorithms for best polynomial approximation in the chebfun system

Article

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.

Keywords

Remez algorithm Best polynomial approximation Barycentric interpolation Chebfun system 

Mathematics Subject Classification (2000)

41A50 41A10 65D05 

References

  1. 1.
    Almacany, M., Dunham, C., Williams, J.: Discrete Chebyshev approximation by interpolating rationals. IMA J. Numer. Anal. 4, 467–477 (1984) MATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Barrodale, I., Phillips, C.: Solution of an overdetermined system of linear equations in the Chebyshev norm. ACM Trans. Math. Softw. 1, 264–270 (1975) MATHCrossRefGoogle Scholar
  3. 3.
    Battles, Z.: Numerical linear algebra for continuous functions. PhD thesis, University of Oxford (2005) Google Scholar
  4. 4.
    Battles, Z., Trefethen, L.N.: An extension of MATLAB to continuous functions and operators. SIAM J. Sci. Comput. 25(5), 1743–1770 (2004) MATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Bernstein, S.: Sur la meilleure approximation de |x| par des polynomes de degrés donnés. Acta Math. 37, 1–57 (1914) CrossRefMathSciNetGoogle Scholar
  6. 6.
    Berrut, J.P., Trefethen, L.N.: Barycentric Lagrange interpolation. SIAM Rev. 46, 501–517 (2004) MATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Boothroyd, J.: Algorithm 318: Chebyschev curve-fit. Commun. ACM 10(12), 801–803 (1967) CrossRefGoogle Scholar
  8. 8.
    Borel, E.: Leçons sur les fonctions de variables réelles. Gauthier-Villars, Paris (1905) MATHGoogle Scholar
  9. 9.
    Boyd, J.A.: Computing zeros on a real interval through Chebyshev expansion and polynomial rootfinding. SIAM J. Numer. Anal. 40(5), 1666–1682 (2002) MATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Brutman, L.: Lebesgue functions for polynomial interpolation—a survey. Ann. Numer. Math. 4, 111–128 (1997) MATHMathSciNetGoogle Scholar
  11. 11.
    Cheney, E.W.: Introduction to Approximation Theory. McGraw-Hill, New York (1966) MATHGoogle Scholar
  12. 12.
    Cody, W.J.: The FUNPACK package of special function subroutines. ACM Trans. Math. Softw. 1(1), 13–25 (1975) MATHCrossRefGoogle Scholar
  13. 13.
    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) MATHCrossRefGoogle Scholar
  14. 14.
    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) MATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    Davis, P.J.: Interpolation and Approximation. Dover, New York (1975) MATHGoogle Scholar
  16. 16.
    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) MATHCrossRefGoogle Scholar
  17. 17.
    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) Google Scholar
  18. 18.
    Dunham, C.B.: Choice of basis for Chebyshev approximation. ACM Trans. Math. Softw. 8(1), 21–25 (1982) MATHCrossRefMathSciNetGoogle Scholar
  19. 19.
    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) CrossRefGoogle Scholar
  20. 20.
    Good, I.J.: The colleague matrix, a Chebyshev analogue of the companion matrix. Q. J. Math. 12, 61–68 (1961) MATHCrossRefMathSciNetGoogle Scholar
  21. 21.
    Gutknecht, M.H., Trefethen, L.N.: Real polynomial Chebyshev approximation by the Carathéodory-Fejér method. SIAM J. Numer. Anal. 19, 358–371 (1982) MATHCrossRefMathSciNetGoogle Scholar
  22. 22.
    Higham, N.J.: Accuracy and Stability of Numerical Algorithms, 2nd edn. SIAM, Philadelphia (2002) MATHGoogle Scholar
  23. 23.
    Higham, N.J.: The numerical stability of barycentric Lagrange interpolation. IMA J. Numer. Anal. 24, 547–556 (2004) MATHCrossRefMathSciNetGoogle Scholar
  24. 24.
    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) MATHCrossRefMathSciNetGoogle Scholar
  25. 25.
    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) Google Scholar
  26. 26.
    Lorentz, G.G.: Approximation of Functions. Holt, Rinehart and Winston (1966) Google Scholar
  27. 27.
    MATLAB: User’s Guide. The MathWorks Inc., Natick, Massachusetts Google Scholar
  28. 28.
    McClellan, J.H., Parks, T.W.: A personal history of the Parks-McClellan algorithm. IEEE Signal Process. Mag. 22, 82–86 (2005) CrossRefGoogle Scholar
  29. 29.
    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) CrossRefGoogle Scholar
  30. 30.
    Meinardus, G.: Approximation of Functions: Theory and Numerical Methods. Springer, Heidelberg (1967) MATHGoogle Scholar
  31. 31.
    Mhaskar, H.N., Pai, D.V.: Fundamentals of Approximation Theory. Narosa Publishing House, New Delhi (2000) MATHGoogle Scholar
  32. 32.
    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) MATHCrossRefMathSciNetGoogle Scholar
  33. 33.
    NAG: Library, Manual. The Numerical Algorithms Group, Ltd., Oxford, UK Google Scholar
  34. 34.
    Numerical Libraries, I.M.S.L.: Technical Documentation. Visual Numerics Inc., Houston Google Scholar
  35. 35.
    Pachón, R., Platte, R., Trefethen, L.N.: Piecewise smooth chebfuns. IMA J. Numer. Anal. (to appear) Google Scholar
  36. 36.
    Parks, T.W., McClellan, J.H.: Chebyshev approximation for nonrecursive digital filters with linear phase. IEEE Trans. Circuit Theory 19, 189–194 (1972) CrossRefGoogle Scholar
  37. 37.
    Powell, M.J.D.: Approximation Theory and Methods. Cambridge University Press, Cambridge (1981) MATHGoogle Scholar
  38. 38.
    Rabinowitz, P.: Applications of linear programming to numerical analysis. SIAM Rev. 10, 121–159 (1968) MATHCrossRefMathSciNetGoogle Scholar
  39. 39.
    Remes, E.: Sur le calcul effectif des polynomes d’approximation de Tchebychef. C. R. Acad. Sci. 199, 337–340 (1934) Google Scholar
  40. 40.
    Remes, E.: Sur un procédé convergent d’approximations successives pour déterminer les polynomes d’approximation. C. R. Acad. Sci. 198, 2063–2065 (1934) Google Scholar
  41. 41.
    Remes, E.: Sur la détermination des polynomes d’approximation de degré donnée. Commun. Soc. Math. Kharkov 10 (1934) Google Scholar
  42. 42.
    Rice, J.R.: The Approximation of Functions, vol. 1. Addison-Wesley, Reading (1964) MATHGoogle Scholar
  43. 43.
    Sauer, F.W.: Algorithm 604: A FORTRAN program for the calculation of an extremal polynomial. ACM Trans. Math. Softw. 9(3), 381–383 (1983) MATHCrossRefMathSciNetGoogle Scholar
  44. 44.
    Schmitt, H.: Algorithm 409, discrete Chebychev curve fit. Commun. ACM 14, 355–356 (1971) CrossRefGoogle Scholar
  45. 45.
    Simpson, J.C.: Fortran translation of algorithm 409, Discrete Chebychev curve fit. ACM Trans. Math. Softw. 2, 95–97 (1976) MATHCrossRefGoogle Scholar
  46. 46.
    Specht, W.: Die Lage der Nullstellen eines Polynoms, IV. Math. Nachr. 21, 201–222 (1960) MATHCrossRefMathSciNetGoogle Scholar
  47. 47.
    Steffens, K.G.: The History of Approximation Theory: From Euler to Bernstein. Birkhäuser, Boston (2006) Google Scholar
  48. 48.
    Stiefel, E.L.: Numerical methods of Tchebycheff approximation. In: Langer, R. (ed.) On Numerical Approximation, pp. 217–232. University of Wisconsin Press, Madison (1959) Google Scholar
  49. 49.
    Taylor, R., Totik, V.: Lebesgue constants for Leja points. IMA J. Numer. Anal. (to appear) Google Scholar
  50. 50.
    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) CrossRefGoogle Scholar
  51. 51.
    Trefethen, L.N.: Spectral Methods in MATLAB. SIAM, Philadelphia (2000) MATHGoogle Scholar
  52. 52.
    Varga, R.S., Carpenter, A.J.: On the Bernstein conjecture in approximation theory. Constr. Approx. 1, 333–348 (1985) MATHCrossRefMathSciNetGoogle Scholar
  53. 53.
    Veidinger, L.: On the numerical determination of the best approximation in the Chebyshev sense. Numer. Math. 2, 99–105 (1960) MATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer Science + Business Media B.V. 2009

Authors and Affiliations

  1. 1.Computing LaboratoryUniversity of OxfordOxfordUK

Personalised recommendations