Integration of iterated integrals by multistep methods

Linear multistep methods with looking ahead points are discussed for the class of initial value problems:d ^r y/d x ^r =f(x). Upper bounds for the error order implied by the condition of stability are given for methods involving (i)f(x) only and (ii) bothf(x) andf(x). The methods discussed are more general than those treated by Dahlquist. It is seen that in case (i) the bounds depend uponr while in case (ii) these are independent ofr.Part of this work was carried out while the author held a postdoctoral fellowship at Dalhousie University.