Asymptotic correction of Numerov's eigenvalue estimates with general boundary conditions

The error in the estimate of the kth eigenvalue of ?y??+qy=?y, y(0)=y(?)=0, obtained by Numerov's method with uniform step length h, is O(k 6 h 4 ). The author and J. Paine showed that a correction technique of Paine, de Hoog and Anderssen reduced this to O(k 4 h 5 /sin(kh)), with negligible extra effort. Later the author extended the method to deal with boundary conditions of the form y?(a)=0. This paper shows how a similar increase in accuracy can be obtained, with a little more effort, for problems with one or more boundary conditions of the form y?(a)=?y(a) where ? ? 0