Generalised product integration

This thesis is concerned with the construction of quadrature rules for the numerical evaluation of integrals of the form ‚Äöv†¬¥ba w(x)f(x)K(x; ˜í¬™)dx, where the functions w, f and K are required to a possess certain distinct characteristics. Initially, we describe the construction and implementation of a quadrature rule when K(x; ˜í¬™) = (x - ˜í¬™) -1 , a < ˜í¬™ < b, and the above integral is taken to be a Cauchy principal value integral. Convergence of this quadrature rule is investigated when f is a Holder continuous function and when f is an analytic function. In the former case sufficient conditions for convergence of the rule are established, and in the latter case the remainder is expressed as a contour integral from which asymptotic estimates of the remainder may be derived. Particular attention is given to the case when w is a Jacobi weight function. The generalised product integration rule for arbitrary K is developed in Chapter 5. Sufficient conditions for convergence of the generalised rule are established for continuous functions f and K. The implementation and convergence of the rule is further investigated for three functions K of considerable interest, namely: K(x; ˜í¬™) = exp(ix˜í¬™), ˜í¬™ real, I˜í¬™I large; K(x; ˜í¬™) = lnIx - ˜í¬™I,-1 < ˜í¬™ < 1; K(x; ˜í¬™) = Ix - ˜í¬™Is, s > -I, -I < ˜í¬™ < 1. In each case we obtain conditions sufficient to ensure convergence of the rule when w is a Jacobi weight function.