Conversion of a power to a series of Chebyshev polynomials

ABSTRACT

Even slowly convergent power series can be rearranged as series in Chebyshev polynomials if appropriate sequence transformations are used in evaluating the coefficients. The method is illustrated by computing the coefficients for the expansion of the logarithm and dilogarithm.

REFERENCES

• 1: CLENSHAW, C. W. Chebyshev series for mathematical functions. National Physical Laboratory Math. Tables, Vol. 5, London, HMSO, 1962.
• 2: LANCZOS, C. Trigonometric interpolation of empirical and analytical functions. J. Math. Phys. 17 (1938), 123-199.
• 3: SHANKS, D. Nonlinear transformations of divergent and slowly convergent series. J. Math. Phys. 34 (1955), 1-42.
• 4: WYNN, P. On a device for computing the e(S) transformation. MTAC 10 (April 56), 91-96.
• 5: Peter Naur , J. W. Backus , F. L. Bauer , J. Green , C. Katz , J. McCarthy , A. J. Perlis , H. Rutishauser , K. Samelson , B. Vauquois , J. H. Wegstein , A. van Wijngaarden , M. Woodger, Report on the algorithmic language ALGOL 60, Communications of the ACM, v.3 n.5, p.299-314, May 1960 [doi>10.1145/367236.367262]
• 6: LEWIN, L. Dilogarithms and associated functions. Macdonald, London, 1958.

INDEX TERMS

Primary Classification:
  G. Mathematics of Computing
  G.1 NUMERICAL ANALYSIS
      G.1.5 Roots of Nonlinear Equations
          Subjects: Polynomials, methods for

Additional Classification:
  F. Theory of Computation
  F.2 ANALYSIS OF ALGORITHMS AND PROBLEM COMPLEXITY
      F.2.1 Numerical Algorithms and Problems
          Subjects: Computations on polynomials

  G. Mathematics of Computing
  G.1 NUMERICAL ANALYSIS
      G.1.5 Roots of Nonlinear Equations
          Subjects: Convergence

General Terms:
Design, Theory