An H2 error expression for the Lanczos procedure

The Lanczos procedure is widely used in model reduction of large-scale dynamical systems. In this note we introduce an exact expression for the H₂ norm of the error system in the Lanczos procedure. To our knowledge, this is the first such global result for the Lanczos procedure. Then we turn our attention to the rational Krylov method, which has the disadvantage that the selection of interpolation points is an ad-hoc process. The error expression for the Lanczos procedure suggests that the interpolation points be chosen at the negative of the poles of the to-be-reduced model. Numerical examples reveal that this choice of the interpolation points improves the approximation and reduces the error significantly.