Note on starting the Numerov method more accurately by a hybrid formula of order four for an initial—value problem

In this note we report a hybrid formula of order four for starting the Numerov method applied to the initial—value problem for y ′ ′ = f ( x , y ) , over the recently obtained result of order three by two different papers (J. Pure Appl. Sci. 2(2) (2002) 1, Abacus 29(2) (2002) 92), based on two different approaches. We illustrate the accuracy of the methods by two numerical examples.