Insights from the relationship between the multistage Wiener filter and the method of conjugate gradients

This paper demonstrates that, under certain conditions, the method of conjugate gradients (CG) and the multistage Wiener filter (MWF) produce equivalent solutions. Equivalence follows from the fact that both algorithms minimize the same cost function in the same subspace, namely a Krylov subspace. Motivation for Krylov subspaces and their properties are developed herein. New insights into both algorithms follow from the equivalence including previously unpublished results on the convergence of the multistage Wiener filter as a function of rank. Furthermore a new perspective on CG is developed where CG can now be viewed as a reduced rank algorithm for the solution of the Wiener-Hopf equations.