Frequency evaluation in exponential fitting multistep algorithms for ODEs

We consider the linear multistep algorithms for first order ODEs and examine the problem of how the λ-frequencies should be tuned in order to obtain the maximal benefit from the exponential fitting versions of such algorithms. We find out that the key of the answer consists in analysing the behaviour of the error. On further investigating the simple case of two-step bdf algorithms we produce formulae for the optimal λʹs and show that, if the optimal λʹs are used, the order of the method is increased by one unit. The reported numerical illustrations suggest that further investigations along these lines deserve a real attention.