A Min-max Problem on Roots of Unity

Abstract. The worst-case residual norms of the GMRES method for linear algebraic systems [3] can, in case of a normal matrix, be characterized by a min-max approximation problem on the matrix eigenvalues. In [2] we derive a lower bound on this min-max value (worst-case residual norm) for each step of the GMRES iteration. We conjecture that the lower bound and the min-max value agree up to a factor of 4/π, i.e. that the lower bound multiplied by 4/π represents an upper bound. In this paper we prove for several different iteration steps that our conjecture is true for a special set of eigenvalues, namely the roots of unity. This case is of interest, since numerical experiments indicate that the ratio of the min-max value and our lower bound is maximal when the eigenvalues are the roots of unity.