Exponential fitting BDF algorithms and their properties

We present two families of explicit and implicit BDF formulas, capable of the exact integration (with only round-off errors) of differential equations whose solutions belong to the space generated by the linear combinations of exponential of matrices, products of the exponentials by polynomials and products of those matrices by ordinary polynomials. Those methods are suitable for stiff and highly oscillatory problems, then we will study their properties of local truncation error and stability. Plots of their 0-stability regions in terms of Ah are provided. Plots of their regions of absolute stability that include all the negative real axis in the implicit case are provided. Exponential fitting algorithms depend on a parameter Ah, how can we find this parameter is a big question, here we show different ways to find a good parameter. Finally, numerical examples show the efficiency of the proposed codes, specially when we are integrating stiff problems where the jacobian of the function has complex eigenvalues or problems where the jacobian has positive eigenvalues but the solutions of the problems have not positive exponentials.