### Abstract:

The background for this paper is the use of quadrature formulas
for the solution of ordinary differential equations. If we know the
values of the dependent variable for which we are solving, and its
derivative, at several equally spaced points, i.e., at values of the independent
variable separated by equal intervals, we may use a quadrature
formula to integrate the values of the derivative, so that we
may obtain an approximate value of the dependent variable at the next
point. The differential equation is then used to evaluate the derivative
at the new point. This procedure is then repeated to evaluate the
dependent variable and its derivative at point after point. The accuracy
of this method is limited by the accuracy of the quadrature formula
used.
In order to improve the accuracy of the solution one may use an open-type quadrature formula to "predict" the value of the dependent variable at the next point, then calculate the derivative, and now use
a more accurate closed-type formula to "correct" the value of the
dependent variable. This procedure is the basis of "Milne's method".
It has been shown that an error introduced at a step propagates
itself approximately according to a linear combination of the solutions
of a linear difference equation associated with the corrector . The
solutions of this difference equation consist of an approximation to the
solution of the differential equation and in some cases one or more
extraneous solutions. If one or more of the latter increases as the
process is repeated from step to step, the method is called instable.
Remedies for instability include periodic use of special quadrature
formulas called "stabilizers". This has been treated in the case of
fifth-order formulas by Milne and Reynolds. In this paper the idea
is extended to formulas of seventh order.