An optimality property of approximated solution computed by Hessenberg method

Abstract

We  revisit the implementation of Krylov subspace method based on the Hessenberg process for solving general linear operator equations. More precisely, it is established that the computed approximate solution by corresponding approach  can be regarded as the minimizer of a certain norm of system's residual at each step.  Test problems are numerically examined to compare the performance of Hessenberg method with Krylov subspace method based on the Arnoldi process in conjunction with the Tikhonov regularization technique for solving tensor equations with cosine transform product arising from image restoration.