Remark on algorithm 162: Near-minimax polynomial approximations and partitioning of intervals

ABSTRACT

A method of near-minimax polynomial approximation is described. As a by-product, this method provides a formula for an estimate of the maximum error associated with a given degree of approximation. Using this formula, a partitioning algorithm is obtained for dividing a basic interval into sub-intervals for which approximations of equal degree give equal maximum error.

REFERENCES

• 1: CLENSHAW, C. W. Chebyshev series for mathematical functions. In Mathematical Tables, Vol. 5, National Physical Laboratory, Her Majesty's Stationery Office, London, 1962.
• 2: W. Fraser , J. F. Hart, On the computation of rational approximations to continuous functions, Communications of the ACM, v.5 n.7, p.401-403, July 1962 [doi>10.1145/368273.368578]
• 3: LANCZOS, C. Trigonometric interpolation of empirical and analytic functions. J. Math. Phys. 17 (1938), 123-199.
• 4: --. Applied Analysis. Prentice-Hall, New York, 1956.
• 5: C. L. Lawson, Numerical analysis I: Segmented minmax approximation, Proceedings of the 1962 ACM national conference on Digest of technical papers, p.62, September 01-01, 1962 [doi>10.1145/800198.806109]
• 6: Hans J. Maehly, Methods for Fitting Rational Approximations, Parts II and III, Journal of the ACM (JACM), v.10 n.3, p.257-277, July 1963 [doi>10.1145/321172.321173]
• 7: MURNAGHAN, F. D., AND WRENCH, J. W. Rep. No. 1175, David Taylor Model Basin, May, 1960.
• 8: NOVODVORSKII, E. P., AND PINSKER, I. S. The process of equating maxima. Uspehi Mat. Nauk 6 (1951), 174-181. English translation by A. Shenitzer.

INDEX TERMS

Primary Classification:
  G. Mathematics of Computing
  G.1 NUMERICAL ANALYSIS
      G.1.2 Approximation
          Subjects: Spline and piecewise polynomial approximation

Additional Classification:
  G. Mathematics of Computing
  G.1 NUMERICAL ANALYSIS
      G.1.2 Approximation
          Subjects: Minimax approximation and algorithms

General Terms:
Algorithms, Design, Theory