Skip to main content
Log in

DCR2: An improved algorithm forl rational approximation on intervals

  • Published:
Numerische Mathematik Aims and scope Submit manuscript

Summary

This paper describes a new algorithm denoted DCR2, giving superior performance as a general purpose algorithm forl rational approximation on intervals. DCR2 combines the advantages of the Remez Second Algorithm (R2) and the Differential Correction Method (DC), achieving the best approximation when both DC and R2 either produce noticeably suboptimal results on the interval or fail to converge.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Similar content being viewed by others

References

  1. Acton, F.: Numerical methods that work. New York: Harper and Row 1970

    Google Scholar 

  2. Agmon, S.: Lecture notes on classical analysis [in Hebrew]. Jerusalem: Hebrew University Student Press 1965

    Google Scholar 

  3. Akhiezer, N.I.: The theory of approximation. New York: Ungar 1956

    Google Scholar 

  4. Anselone, P.M., Taylor, G.D.: The optimal exchange algorithm and comparisons with the generalized Remez algorithm. Applicable Anal.3, 7–27 (1973)

    Google Scholar 

  5. Barrodale, I., Powell, M.J., Roberts, F.D.: The differential correction algorithm for rational approximation. J. Numer. Anal.9, 493–504 (1972)

    Google Scholar 

  6. Barrodale, I.: Best rational approximation and quasi-convexity. J. Numer. Anal.10, 8–12 (1973)

    Google Scholar 

  7. Barrodale, I., Roberts, F.D.: Best approximation by rational functions. Math. Report No. 76, University of Victoria, Victoria, Canada, 1973

    Google Scholar 

  8. Belogus, D.: Minimax rational approximation on intervals. M.Sc. Thesis, Weizmann Institute of Science, Rehovot, Israel, 1975

    Google Scholar 

  9. Belogus, D.: Liron, N.: A condition number for near-degeneratel rational approximations: Theory and numerical study. Weizmann Institute of Science, Rehovot, Israel, 1977 (unpublished)

    Google Scholar 

  10. Boehm, B.: Functions whose best rationalT-approximations are polynomials. Numer. Math.6, 235–242 (1964)

    Google Scholar 

  11. Cheney, E.W.: Introduction to approximation theory. New York: McGraw-Hill 1966

    Google Scholar 

  12. Collatz, L.: Functional analysis and numerical methods. London-New York: Academic Press 1966

    Google Scholar 

  13. Dua, S.N., Loeb, H.L.: Further remarks on the DC algorithm. J. Numer. Anal.10, 123–126 (1972)

    Google Scholar 

  14. Dunham, C.B.: Existence of admissible rational Chebychev approximations on subsets. J. Approximation Theory12, 307–308 (1974)

    Google Scholar 

  15. Fike, C.T.: Computer evaluation of mathematical functions. Englewood Cliffs, N.J.: Prentice-Hall 1968

    Google Scholar 

  16. Gill, P.E., Murray, W.: A numerically stable form of the Simplex algorithm. Linear Algebra and Appl.7, 99–138 (1973)

    Google Scholar 

  17. Golub, G.H., Smith, L.B.: Chebychev approximation of continuous functions by a Chebychev system. Comm. ACM14, 737–746 (1971)

    Google Scholar 

  18. Isaacson, E., Keller, H.B.: Analysis of numerical methods. New York: Wiley 1966

    Google Scholar 

  19. Lee, C.M., Roberts, F.D.: A comparison of algorithms for rational approximation. Math. Comput.27, 111–121 (1973)

    Google Scholar 

  20. Meinardus, G.: Approximation of functions: Theory and numerical methods. Berlin-Heidelberg-New York: Spinger 1967

    Google Scholar 

  21. Rabinowitz, P.: Applications of linear programming to numerical analysis. SIAM Rev.10, 121–130 (1968)

    Google Scholar 

  22. Ralston, A.: Rational approximation by Remez' algorithms. Numer. Math.7, 322–330 (1965)

    Google Scholar 

  23. Ralston, A.: Rational Chebychev approximation. Mathematical methods for digital computers, Vol. II, pp. 264–284. New York: Wiley 1967

    Google Scholar 

  24. Rice, J.R.: The approximation of functions, Vol. II. Reading, Mass.: Addison-Wesley 1969

    Google Scholar 

  25. Rivlin, T.: An introduction to the approximation of functions. Waltham, Mass. Blaisdell 1969

    Google Scholar 

  26. Snyder, M.A.: Chebychev methods in numerical approximation. Englewood Cliffs, N.J.: Prentice Hall 1966

    Google Scholar 

  27. Szentirmai, G.: Computer-aided filter design. New York: IEEE Press 1973

    Google Scholar 

  28. Veidinger, L.: Numerical determination of the best approximation in the Chebychev sense. Numer. Math.2, 99–105 (1960)

    Google Scholar 

  29. Werner, H.: Die konstruktive Ermittlung derT-Approximation im Bereich der Rational-Funktionen. Arch. Rational Mech. Anal.11, 368–384 (1962)

    Article  Google Scholar 

  30. Werner, H.: Die Bedeutung der Normalität bei rationalerT-Approximation. Computing2, 34–52 (1967)

    Google Scholar 

  31. Werner, H., Stoer, J., Bommas, W.: Rational Chebychev approximation. Numer. Math.10, 289–306 (1967)

    Google Scholar 

  32. Werner, H.: Starting procedures for iterative calculation of rationalT-approximations. Information Processing, Vol. 68 (conference). Amsterdam: North-Holland 1969

    Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Rights and permissions

Reprints and permissions

About this article

Cite this article

Belogus, D., Liron, N. DCR2: An improved algorithm forl rational approximation on intervals. Numer. Math. 31, 17–29 (1978). https://doi.org/10.1007/BF01396011

Download citation

  • Received:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF01396011

Subject Classifications

Navigation