Preconditioned diagonal dominant matrices

Abstract

It is well-known that most iterative methods converge for linear systems whose coefficient matrix is diagonal dominant. However, most matrices of nonsingular linear systems are not diagonal dominant. In this case, iterative methods have breakdown problem. Here, we try to overcome the trouble by preconditioned techniques. It is shown here that there do exist preconditioned matrices such that every nonsingular matrix can be transformed to diagonal dominant matrix, that is, the product of preconditioned matrices and the original matrix is diagonal dominant. Therefore, iterative methods converge for the preconditioned system. Some sufficient conditions for such preconditioned matrix are given. The p-cyclic systems are also studied. For the p-cyclic systems, the preconditioner can be very simple, just a lower bidiagonal matrix.