Fast Computation of Contour Integrals of Rational Functions

Abstract

This paper presents fast algorithms for computing numerical approximations for contour integrals of rational functions. Given the coefficients of two polynomials q and p∈C[z], a curve Γ in the complex plane, and an error bound ε, the integral ∫Γq(z)/p(z)dz is computed up to an error of ε. In the special case that the zeros of p lie in a small disc not intersected by Γ, the integral is computed by summing up the integrals of an initial segment of a suitable Laurent series of q/p. The general case is reduced to this special one by partial fraction decomposition as described by P. Kirrinnis (1998, Partial fraction decomposition in C(z) and simultaneous Newton iteration for factorization in C[z], J. Complexity14, 378–444. The algorithms are analyzed from the point of view of (serial) bit complexity. The running time of the algorithms is estimated in terms of the error bound prescribed for the result, the degree of the polynomials involved, and the condition of the problem, measured by a lower bound for the distance between the zeros of p and the points of Γ. This condition parameter need not be known in advance.