Attainable order of Adams type linear multistep multiderivative method with nonnegative coefficients for solving initial value problems

Abstract

The attainable order of the Adams type linear multistep multiderivative method for solving initial value problem is considered. The method to be considered is of the form

$$y_{n + k} = y_{n + k - 1} + \sum\limits_{i = 0}^k { \sum\limits_{j = 1}^l {h^j \beta _{ij} f_{n + i}^{(j - 1)} ,} } $$

and the coefficients are subject to the constraint

$$\begin{gathered} ( - 1)^{j - 1} \beta _{ij} \geqslant 0, i = 0, \ldots ,k - 1, ( - 1)^{j - 1} \beta _{kj} > 0, \hfill \\ j = 1, \ldots ,l, \hfill \\ \end{gathered} $$

which prevents the cancellation of significant digits. It is proved that the attainable order with the method isl + 1 under this condition. It is also proved that forl = 1, i.e., for the usual Adams type linear multistep method, the Trapezoidal rule is the most accurate formula in the family of the Adams type formulae with nonnegative coefficients.