In 1962, Nordsieck presented a series of numerical methods for solving ordinary differential equations which rely on Taylor's series, but which are equivalent to the correctors in the Adams methods. In a method of Nordsieck, simplifications, such as in the often desirable process of changing step size, are made, since it is unnecessary to store more than one previous step. In 1964, Gragg and Stetter published a series of numerical methods for solving ordinary differential equations which rely on very accurate correctors, using a “nonstep” point within the interval of integration. The method in the present paper is equivalent, for uniform step size, to one of these very accurate correctors but is in the form of Nordsieck. The strengths and weaknesses of the method are discussed.

In a previous paper the authors suggested that the accurate correctors proposed by Gragg and Stetter for solving ordinary differential equations should be accompanied by similar predictors. In each method in that paper the corrector and one of the predictors use one “nonstop” point within the interval of integration. In the present paper a corrector is dealt with in which two “nonstep” points are used, and in which, to some degree, the authors have “balanced” the contributions of the errors in the predictors to the total local truncation error, a technique due to Butcher.

In April, 1964, Gragg and Stetter published a set of numerical methods for solving y′ = f ( x , y ) which rely on some very accurate correctors, using a “nonstep” point within the interval of integration. However, the present authors found that this accuracy is largely lost unless one also uses accurate predictors, also using a “nonstep” point within an interval of integration, of the same order of accuracy as that of the correctors. Numerical examples indicate that on the average those differential equations tested can be solved by the method of the sixth order proposed in this paper in about two-thirds the time required by an Adams method of the same order.