中文摘要：

设A为一实对称正定的严格对角占优矩阵.设A=D-B为A的Jacobi分裂.为了求解线性方程组Ax=b,在新提出的预处理子的基础上,我们采用预处理共轭梯度方法（PCG）来求解该问题.新提出的预处理子Pv=D＋νvv~T,其中v=｜B｜e,e=（1,...,1）~T,ν=v~TBv/｜｜v｜｜_2~4,且ν使｜｜cvv~T-B｜｜_F达到极小.我们得到了预处理矩阵P_v~（-1）A特征值的上下界,它的界比JIN提出的预处理子的界简单紧凑.数值结果表明我们的预处理子的有效性.

英文摘要：

Let A be real symmetric positive definite and strictly diagonally dominant. Let A = D - B be the Jacobi splitting of A. We propose a new preconditioner in the form of P,, = D ＋ vvvT for solving the linear system Ax = b by the preconditioned conjugate gradient （PCG） method, where v is chosen to be v = ｜B｜e with e -- （1,..., 1）T and v = vTBv/｜｜v｜｜42 minimizing ｜｜cvvT - B｜｜E We obtain lower and upper bounds of the eigenvalues of preconditioned matrix Pv-1A, which are sharper and simpler than Jin＇s preconditioner. Numerical results demonstrate the effectiveness of our preconditioners.