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...
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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.
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