解线性方程组的预条件AOR迭代法.pdfVIP

Improving AOR method for consistent linear systems ∗ Yan-lei CHANG † Guo-feng ZHANG‡ Jing-yu ZHAO§ School of Mathematics and statistics, Lanzhou University, Lanzhou 730000, P.R.China Abstract For solving the linear system AX = b, different preconditioned Gauss-Seidel methods have been proposed by many authors. In this paper, we consider preconditioned AOR iterative methods with preconditioners (I + R) , (I + S) and (I + S + R). In addition, the convergence and comparison theorems of two preconditioned AOR methods with preconditioners (I + R) and (I + S + R), respectively, are established. Numerical example is also given to illustrate our results. Keywords: Preconditioner; L-matrix; M-matrix; AOR method; SOR method; Mathematics Subject Classification(2000): 65F10,65F50 1 Introduction Consider the following linear system: Ax = b, (1.1) where A ∈ Rn ×n , b ∈ Rn are given and x ∈ Rn is unknown. For any splitting, A = M − N with det(M ) = 0, the basic iterative method for solving (1.1) is x(i+1) = M −1Nxi + M −1b, i = 0, 1, ... For simplicity, without loss of generality, we assume throughout this paper that A = I − L − U, (1.2) where I is the identity matrix, L and U are strictly lower and upper triangular matrices obtained from A, respectively. Then the iteration matrices of the classical AOR iterative method in [4] is defined: Lrw = (I − rL)−1 [(1 − w)I + (w − r)L + wU ], (1.3) where w and r are real parameters