quadgF

Integrating oscillatory functions in Matlab. II, In Part I [Int. J. Comput. Math. 88, No. 11, 2348–2358 (2011; Zbl 1229.65241)] we developed a Matlab program for the approximation of ∫ a b f(x)e iωx dx when ω is large. Here we study the more difficult task of approximating ∫ a b f(x)e ig(x) dx when g(x) is large on [a,b]. We propose a fundamentally different approach to the task – backward error analysis. Other approaches require users to supply the location and nature of critical points of g(x) and may require g ’ (x). With this new approach, the program quadgF merely asks a user to define the problem, i.e., to supply f(x), g(x), [a,b], and specify the desired accuracy. Though intended only for modest relative accuracy, quadgF is very easy to use and solves effectively a large class of problems. Of some independent interest is a vectorized Matlab function for evaluating Fresnel sine and cosine integrals.