Summary: The paper deals with the method of quasilinearization as applied to the solution of two-point boundary-value (TPBV) problems associated with a system \(y''=f(t,y)\) of second-order differential equations. It is supposed that the equations, linearized about an approximate solution \(y^{(k)}(t)\), are integrated numerically and that the linear TPBV problem is solved by a scheme such as the method of complementary functions. The principal concern is the effect that the approximate interpolation, used to represent \(y^{(k)}(t)\), has upon the order of convergence and the order of accuracy of the final converged solution. For example, it is shown that, if Runge-Kutta (RK) integration is employed, then there is an interpolation formula which gives second-order convergence. It is also proven that, if the order of the integration is q (suppose q is even), then the order of the interpolation need to be only (q-2)/2 to give (q-1)th order global accuracy in the converged solution y(t). A particular method, based upon the earlier analysis, is applied to two numerical problems. It makes use of RK integration with automatic stepsize control, thereby avoiding the problem of determining mesh points. Moreover, there is no limit on the order of the RK formula which may be employed. The problem of regenerating earlier solutions at each step is also simplified.