线性方程组二级迭代法的收敛速度

Convergence Rates of Two-stage Iterative Methods for Linear Systems

二级迭代法由内、外迭代和内迭代次数三部分组成。给出了线性方程组二级迭代法R1-收敛因子的一个上界,这个上界由内、外迭代的R1-收敛因子和内迭代次数所决定,其主部为外迭代的R1-收敛因子。在矩阵单调性条件下,对于任何内迭代方法和任意内迭代次数,证明了外迭代的R1-收敛因子也是二级迭代法R1-收敛因子的下界。所得结果反映了内、外迭代的收敛速度以及内迭代次数对于二级迭代法收敛速度的综合影响。

Two-stage iterative method is comprised by the inner and outer iteration and the numbers of inner iteration. We find out an upper bound of R1-factor of two-stage iterative methods for solution of linear systems. And the upper bound is expressed applying the R1-factors of the inner and outer iteration and the numbers of the inner iteration; the premier parts of the upper bound is the R1-factor of outer iteration. Furthermore, for any inner iteration method and any numbers of the inner iteration, it is also showed that the lower bound of the R1-factor of the two-stage iterative method can also be given by the R1-factor of the outer iteration in the case of monotone matrices. The results indicate the effects of convergence rates of the inner and outer iteration and the numbers of inner iteration on the convergence rates of the two-stage iterative method.