An algorithm based on the FFT for a generalized Chebyshev interpolation

An algorithm for a generalized Chebyshev interpolation procedure, increasing the number of sample points more moderately than doubling, is pre- sented. The FFT for a real sequence is incorporated into the algorithm to enhance its efficiency. Numerical comparison with other existing algorithms is given.

[1]  By J. N. Lyness.  The calculation of Fourier coefficients by the Möbius inversion of the Poisson summation formula. I. Functions whose early derivatives are continuous , 1970 .

[2]  W. Morven Gentleman,et al.  Implementing Clenshaw-Curtis quadrature, II computing the cosine transformation , 1972, Commun. ACM.

[3]  Further symmetries of in-place FFTs , 1987 .

[4]  Francis Jack Smith,et al.  Error Estimation in the Clenshaw-Curtis Quadrature Formula , 1968, Comput. J..

[5]  Graeme Fairweather.  An investigation of Romberg quadrature , 1978, TOMS.

[6]  Hermann Engles,et al.  Numerical quadrature and cubature , 1980 .

[7]  David Elliott,et al.  Truncation Errors in Two Chebyshev Series Approximations , 1965 .

[8]  Paul N Swarztrauber.  Symmetric FFTs , 1986 .

[9]  C. W. Clenshaw,et al.  A method for numerical integration on an automatic computer , 1960 .

[10]  Robert Piessens,et al.  Numerical solution of integral equations of mathematical physics, using Chebyshev polynomials , 1976 .

[11]  J. Oliver.  A doubly-adaptive Clenshaw-Curtis quadrature method , 1972, Comput. J..

[12]  Ian Robinson,et al.  A comparison of numerical integration programs , 1979 .

[13]  William E. Smith,et al.  Product-integration with the Clenshaw-Curtis and related points , 1978 .

[14]  William E. Smith,et al.  Product integration with the Clenshaw-Curtis points: Implementation and error estimates , 1980 .

[15]  Robert Piessens,et al.  Modified Clenshaw-Curtis Integration and Applications to Numerical Computation of Integral Transforms , 1987 .

[16]  Tatsuo Torii,et al.  Generalized Chebyshev interpolation and its application to automatic quadrature , 1983 .

[17]  R. Bulirsch,et al.  Bemerkungen zur Romberg-Integration , 1964 .

[18]  R. Piessens,et al.  An extension of Clenshaw-Curtis quadrature , 1975 .

[19]  Peter D. Welch,et al.  The Fast Fourier Transform and Its Applications , 1969 .

[20]  E. D. Doncker-Kapenga.  Asymptotic Expansions and Their Applications in Numerical Integration , 1987 .

[21]  Ian H. Sloan,et al.  Properties of Interpolatory Product Integration Rules , 1982 .

[22]  W. Morven Gentleman.  Implementing Clenshaw-Curtis quadrature, I methodology and experience , 1972, CACM.

[23]  Philip Rabinowitz,et al.  Methods of Numerical Integration , 1985 .

[24]  Tatsuo Torii,et al.  Indefinite integration of oscillatory functions by the Chebyshev series expansion , 1987 .

[25]  W. M. Gentleman,et al.  Fast Fourier Transforms: for fun and profit , 1966, AFIPS '66 (Fall).