Computational Mathematics and Mathematical Physics , 44(5) (2004), 794-814

A theory of efficient computation of principal terms of the local and global errors for multistep methods with fixed coefficients is developed, and practical algorithms for their computation are presented. A linear mapping is found to approximate high derivatives of the exact solution of a system of differential equations, which are needed for finding estimates of the local and global errors of multistep methods, by a linear combination of easy-to-compute vectors with fixed coefficients. An extended Vandermonde matrix is introduced, and a formula for calculating its determinant is derived. Necessary and
sufficient conditions for the nonsingularity of this matrix are obtained. The theoretical results discussed in this paper are illustrated by numerical examples.