An Iterative Method for Numerical Integration of Rational Functions

We describe a new method for numerical integration of rational functions on the real line. Given a rational integrand, we provide a new rational function preserving its integral on the line. The coefficients of the new function are explicit polynomials in the original ones. These transformations depend on the degree of the input and the desired order of the method. Both parameters are arbitrary. The formulas can be precomputed. Iteration yields an approximation of the desired integral, with m-th order convergence. Examples illustrating the automatic generation of these formulas and a comparison with standard numerical schemes are also presented.