Weights of the exponential fitting multistep algorithms for first-order ODEs

We describe a numerical method for the calculation of the weights of the linear multistep algorithms for solving first-order differential equations. The main novelties are that (i) we admit nonequidistant mesh points in the partition and (ii) the weights are determined on the basis of the exponential functions exp(λix), i=1,2,3,… rather than on the power function set, as it is done for the classical weights. In this way the method allows computing not only the weights of the well-established algorithms but also those of new ones. Another novelty consists in the construction of a general scheme for the error analysis of this kind of algorithms. Some relevant numerical illustrations are given.