Integration Methods of Clenshaw-Curtis Type, Based on Four Kinds of Chebyshev Polynomials

Abstract

Previously Chebyshev polynomials of the first and second kinds have been used in interpolation methods by taking their zeros and possibly end points as abscissae, and these have then been adopted in quadrature methods of Clenshaw-Curtis and Gauss type. Here we extend these ideas to include Chebyshev polynomials of the third and fourth kinds showing that (at least) 14 discrete orthogonality formulae hold for the four kinds of Chebyshev polynomials at appropriate abscissae, the latter chosen from nine basic sets of abscissae, and each formula yields a general (double summation) Clenshaw-Curtis (C-C) integration formula for integrating k(x)f(x) from -1 to 1. For each of the 9 basic sets of abscissae, there is a particular weight function k(x) of Jacobi type for which the C-C formula is a Gauss formula and for which the double summation formula reduces to a single summation formula. Moreover error estimates have already been derived by Sloan and Smith, based on the equivalence at the abscissae of a Chebyshev polynomial of high degree to one of degree no more than the interpolation degree. We show that a similar equivalence can be established for all 14 choices of polynomials and abscissae. The validity of the formulae is illustrated in some numerical applications.