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Recently, an approach known as relaxation has been developed for preservingthe correct evolution of a functional in the numerical solution ofinitial-value problems, using Runge-Kutta methods. We generalize this approachto multistep methods, including all general linear methods of order two orhigher, and many other classes of schemes. We prove the existence of a validrelaxation parameter and high-order accuracy of the resulting method, in thecontext of general equations, including but not limited to conservative ordissipative systems. The theory is illustrated with several numerical examples.

General Relaxation Methods for Initial-Value Problems with Application to Multistep Schemes