A systematic way of extending a general fixed-stepsize multistep formula to a minimum storage variable-stepsize formula has been discovered that encompasses fixed-coefficient (interpolatory), variable-coefficient (variable step), and fixed leading coefficient as special cases. In particular, it is shown that the " interpolatory" stepsize changing technique of Nordsieck leads to a truly variable...

There is considerable evidence suggesting that for Hamiltonian systems of ordinary differential equations it is better to use numerical integrators that preserve the symplectic property of the ow of the system, at least for long-time integrations. We present what we believe is a practical way of doing symplectic integration with variable stepsize. Another idea, orthogonal to variable stepsize, ...

The stability of variable stepsize LMS (VSLMS) algorithms with uncorrelated stationary Gaussian data is studied. It is found that when the stepsize is determined by the past data, the boundedness of the stepsize by the usual stability condition of xed stepsize LMS is su cient for the stability of VSLMS. When the stepsize is also related to the current data, the above constraint is no longer su ...

Second derivative general linear methods (SGLMs) have been already implemented in a variable stepsize environment using Nordsieck technique. In this paper, we introduce SGLMs directly on nonuniform grid. By deriving the order conditions of proposed $p$ and stage $q=p$, some explicit examples these up to four are given. numerical experiments, show efficiency solving nonstiff problems confirm the...