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  

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

作　　者：白中治[1] 仇寿霞[1] 

机构地区：[1]中国科学院数学与系统科学研究院计算数学与科学工程计算研究所科学与工程计算国家重点实验室,北京100080

出　　处：《计算数学》2002年第1期113-128,共16页Mathematica Numerica Sinica

基　　金：国家重点基础研究项目"大规模科学计算研究(G1999032803)"专项经费资助课题

摘　　要：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.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.

关 键 词：线性代数方程组 不定线性方程组 内外迭代法 分裂极小残量算法 收敛性 

分 类 号：O241.6[理学—计算数学]