Proof of a conjecture of Yuan and Zontini on preconditioned methods for M-matrices

Systems of linear equations are ubiquitous in science and engineering, and iterative methods are indispensable for the numerical treatment of such systems. When we apprehend what properties of the coefficient matrix account for the rate of convergence, we may multiply the original system by some nonsingular matrix, called a preconditioner, so that the new coefficient matrix possesses better properties. Recently, some scholars presented several preconditioners and based on numerical tests proposed some conjectures for preconditioned iterative methods. In this paper, we prove one conjecture on the preconditioned Gauss–Seidel iterative method for solving linear systems whose coefficient matrix is an M-matrix.