On a new criterion for the HH-matrix property

A new criterion for identifying generalized diagonally dominant matrices (GDDM, or HH-matrices) was given in [K. Ojiro, H. Niki, M. Usui, A note on a new criterion for the HH-matrix property, J. Comput. Appl. Math. 150 (2003) 293–302], where the criterion is effective when the given matrix is irreducible. However, when the matrix is reducible, it may lead to a false answer. In this work, to overcome this drawback, we present an improved version. The new method is always convergent and needs fewer iterations than the earlier one. An interesting sufficient and necessary condition for any weakly diagonally dominant matrix (WDDM) to be an HH-matrix is obtained. Finally, some numerical examples for the effectiveness of the proposed algorithm are presented.