严格α2对角占优M矩阵A的■估计式的改进  

Improvements on the Estimation of ■for Strictly α2-Diagonally Dominant M-Matrix A

作　　者：周平[1] ZHOU Ping(College of Mathematics,Wenshan University,Wenshan 663099,China)

机构地区：[1]文山学院数学学院

出　　处：《四川理工学院学报（自然科学版）》2019年第6期76-81,共6页Journal of Sichuan University of Science & Engineering(Natural Science Edition)

基　　金：云南省教育厅科研基金项目(2019J0910)

摘　　要：严格α2对角占优M矩阵是矩阵理论中重要的特殊矩阵之一,它被广泛应用于计算数学、经济学、生物学、密码学和智能科学等领域,尤其是数值计算中迭代系统的收敛性,运筹学中的线性互补问题,数理经济学中的Leontief模型,一般平衡的稳定性分析,网络计算中离散系统是否稳定等问题。针对该矩阵A的||A^-1||∞的上界估计问题,首先介绍了它的相关定义、符号和性质引理,借助矩阵A的元素特征,通过矩阵分裂的方法将A表示成严格对角占优矩阵B和对角矩阵F之差的形式,其次结合||A^-1||∞的范围和矩阵范数的性质,给出了||A^-1||∞的一个新估计式,进一步获得了矩阵A的最小奇异值的新下界,用理论分析和数值示例说明了所得估计式比已有的几个结果提高了估计的精度,且计算简单易行。Strictly α2-diagonally dominant M-matrix is one of the important special matrices in matrix theory, which is widely used in Computational Mathematics, Economics, Biology, Cryptography and Intelligent Science, especially the convergence of iterative system in Numerical Calculation, linear complementarity problem in Operations Research, Leontief model in Mathematical Economics, stability analysis of general equilibrium, stability of discrete system in Network Computing, etc. Aiming at the estimation problem on the upper bound of ||A^-1||∞ for strictly α2-diagonally dominant matrix A. First of all, according to the element properties of matrix A, through the method of matrix splitting, the matrix A is expressed as the difference between the strictly diagonally dominant matrix B and the diagonally dominant matrix F. Then a new estimator of ||A^-1||∞ is given by combining the range of ||A^-1||∞ and some properties of Matrix norm, and the new lower bound of the minimum singular value of matrix A is obtained. The theoretical analysis and numerical examples to improve the accuracy of the estimation, and the calculation is simple and easy.

关 键 词：严格α2对角占优M矩阵 无穷范数 上界 最小奇异值 

分 类 号：O151[理学—数学]