Convergence analysis of the preconditioned Gauss–Seidel method for HH-matrices

In 1997, Kohno et al. [Toshiyuki Kohno, Hisashi Kotakemori, Hiroshi Niki, Improving the modified Gauss–Seidel method for ZZ-matrices, Linear Algebra Appl. 267 (1997) 113–123] proved that the convergence rate of the preconditioned Gauss–Seidel method for irreducibly diagonally dominant ZZ-matrices with a preconditioner I+SαI+Sα is superior to that of the basic iterative method. In this paper, we present a new preconditioner I+KβI+Kβ which is different from the preconditioner given by Kohno et al. [Toshiyuki Kohno, Hisashi Kotakemori, Hiroshi Niki, Improving the modified Gauss–Seidel method for ZZ-matrices, Linear Algebra Appl. 267 (1997) 113–123] and prove the convergence theory about two preconditioned iterative methods when the coefficient matrix is an HH-matrix. Meanwhile, two novel sufficient conditions for guaranteeing the convergence of the preconditioned iterative methods are given.