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Manuscript Title: A Chebyshev series approximation to continuous functions.
Authors: M.A. Christie, K.J.M. Moriarty
Program title: CHEBY
Catalogue identifier: ABQJ_v1_0
Distribution format: gz
Journal reference: Comput. Phys. Commun. 23(1981)145
Programming language: Fortran.
Computer: CDC 6600, CDC 7600.
Operating system: CDC SCOPE.
RAM: 8K words
Word size: 60
Keywords: General purpose, Function, Chebyshev polynomials, Chebyshev series, Recurrence relations, Composite trapezoidal Rule integration, Uniform approximation, Minimax approximation, Error estimate.
Classification: 4.7.

Nature of problem:
When modelling a particular physical situation, a complicated function may be needed at many points. The function may well be very time consuming to calculate - e.g. the absorbed two-body scattering differential cross section for high-energy physics.

Solution method:
In this case it is more economical to use the original program to calculate the Chebyshev series approximation to the function. Subsequent calculations will use the Chebyshev series approximation and will be fast. The program described in the paper is designed to:
(i) evaluate N+1 approximate Chebyshev coefficients for a function defined on [-1,1], by using the values of that function at M+1 Chebyshev points, where M and N (<=M) are specified by the user;
(ii)sum a finite Chebyshev series by Clenshaw's method (also known as Goertzel's or Watt's method).
The program also contains subroutines to evaluate a particular function F and to calculate errors by comparing the exact value of F with the sum of the approximating finite Chebyshev series. An error estimate, based on the sum of the absolute value of the eight successive Chebyshev series coefficients beyond those in the finite Chebyshev series summation, is also presented.

Restrictions:
The only restrictions on the program are the number M of points used in the calculation of the Chebyshev series coefficients, and the number of terms N in the Chebyshev series approximation. For the accurate calculation of the coefficients, we must have N<=M. The program is dimensioned for N<=M<=400. This can be altered by the user. However, for an accurate calculation of the error bound we should have N+8<=M which gives the inequality N+8<=M<=400.