关于具优势对称部分的不定线性代数方程组的分裂极小残量算法

１．引 言 考虑大型稀疏线性代数方程组 为利用系数矩阵的稀疏结构以尽可能减少存储空间和计算开销，Ｋｒｙｌｏｖ子空间迭代算法［１，１６，２３］及其预处理变型［６，８，１３，１８，１９］通常是求解（１）的有效而实用的方法．当系数矩阵对称正定时，共轭梯度法（ＣＧ（

SPLITTING-MINRES METHODS FOR LINEAR SYSTEMS WITH THE COEFFICIENT MATRIX WITH A DOMINANT INDEFINITE SYMMETRIC PART

For large sparse system of linear equations with the coefficient matrix with a dominant indefinite symmetric part, we present a class of splitting minimal resid- ual method, briefly called as SMINRES-method, by making use of the inner/outer iteration technique. The SMINRES-method is established by first transforming the linear system into an equivalent fixed-point problem based on the symmetric/skew- symmetric splitting of the coefficient matrix, and then utilizing the minimal resid- ual (MINRES) method as the inner iterate process to get a new approximation to the original system of linear equations at each of the outer iteration step. The MINRES can be replaced by a preconditioned MINRES (PMINRES) at the inner iterate of the SMINRES method, which resulting in the so-called preconditioned splitting minimal residual (PSMINRES) method. Under suitable conditions, we prove the convergence and derive the residual estimates of the new SMINRES and PSMINRES methods. Computations show that numerical behaviours of the SMIN- RES as well as its symmetric Gauss-Seidel (SGS) iteration preconditioned variant, SGS-SMINRES, are superior to those of some standard Krylov subspace meth- ods such as CGS, CMRES and their unsymmetric Gauss-Seidel (UGS) iteration preconditioned variants UGS-CGS and UGS-GMRES.