Convergence properties of sequences related to the Ando–Li–Mathias construction and to the weighted Cheap mean

Abstract

Sequences defining a weighted matrix geometric mean are investigated and their convergence speed is analyzed. The superlinear convergence of a weighted mean based on the Ando–Li–Mathias (ALM) construction is proved. A weighted Cheap mean is defined and conditions on the weights for linear or superlinear convergence of order at least three are provided.