Applications of a modified FFT to product type integration

Abstract An automatic integration scheme is proposed for evaluating the so-called product type (indefinite) integral Q(K, ƒ) = ∫ y x >K(t)ƒ(t) dt, −1 ⩽ x, y where ƒ(t) is assumed to be a smooth function and K(t) are some singular or badly-behaved functions. Typical examples for K(t) are 1n|t − c|, |t − cα, α > − 1, Cauchy principal value 1/(t − c) and eiωt, |ω| ≫ 1. The function ƒ(t) is approximated by a truncated Chebyshev series pN(t) of degree N, whose coefficients are efficiently computed using the FFT. The approximation QN (K, ƒ) to the integral Q(K, ƒ) is given by Q(K, pN. The sequence {pN(t)} is recursively generated until the required tolerance for the integral is satisfied. To enhance the efficiency of the automatic quadrature, the degree N is increased more slowly than doubling, which is usually the case. The evaluations of QN(K, ƒ)=Q(K, pN) for a set of {(x, y, c)} can be efficiently made by using recurrence relations for the singular kernels K(t) above. Numerical examples for the algebraic singular kernel K(t)=|t − c|α, α > − 1, are included.