中文摘要：

最近，黄雾建议了 block-counter-diagonal 和 block-counter-triangular preconditioning 矩阵到前提为解决线性方程从分布式的控制问题的 Galerkin 有限元素的 discretizations 产生在的结构化的系统的 GMRES 方法(计算 91 (2011 ) 379-395 ) 。他分析了光谱性质和特征值的导出的明确的表情和 preconditioned 矩阵的特徵向量。由使用 pre-conditioned 矩阵的特徵向量矩阵的特殊结构和性质，我们为特徵向量矩阵的 2 标准条件数字导出上面的界限并且与 block-counter-diagonal 和 block-counter-triangular pre- 调节器给 asymptotic preconditioned GMRES 方法的集中因素。试验性的结果证明集中分析与数字结果匹配很好。[从作者抽象]

英文摘要：

Recently, Bal proposed a block-counter-diagonal and a block-counter-triangular precon- ditioning matrices to precondition the GMRES method for solving the structured system of linear equations arising from the Galerkin finite-element discretizations of the distributed control problems in （Computing 91 （2011） 379-395）. He analyzed the spectral properties and derived explicit expressions of the eigenvalues and eigenvectors of the preconditioned matrices. By applying the special structures and properties of the eigenvector matrices of the preconditioned matrices, we derive upper bounds for the 2-norm condition numbers of the eigenvector matrices and give asymptotic convergence factors of the preconditioned GMRES methods with the block-counter-diagonal and the block-counter-triangular pre- conditioners. Experimental results show that the convergence analyses match well with the numerical results.