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中国工业与应用数学学会会刊
主管：中华人民共和国教育部
主办：西安交通大学
ISSN 1005-3085  CN 61-1269/O1
工程数学学报

工程数学学报 ›› 2015, Vol. 32 ›› Issue (2): 251-260.doi: 10.3969/j.issn.1005-3085.2015.02.009

• • 上一篇    下一篇

非奇异$H$-矩阵的一组新判定方法

王磊磊1,2,   薛   媛1,   刘建州1   

  1. 1- 湘潭大学数学与计算科学学院，湘潭 411105
    2- 内蒙古民族大学数学学院，通辽 028043
  • 收稿日期:2013-10-11 接受日期:2014-04-17 出版日期:2015-04-15 发布日期:2015-06-15
  • 基金资助:
    国家自然科学基金 (11361038; 11471279)；国家自然科学基金重大研究计划重点支持项目 (91430213)；湖南省自然科学基金 (2015JJ2134)；湖南省教育厅重点项目 (12A137)；湖南省研究生科研创新项目 (CX2014B254)；长江学者和创新团队发展计划 (IRT1179).

A Set of New Criterion for Judging Nonsingular $H$-matrices

WANG Lei-lei1,2,   XUE Yuan1,   LIU Jian-zhou1   

  1. 1- School of Mathematics and Computer Science, Xiangtan University, Xiangtan 411105
    2- School of Mathematics, Inner Mongolia University for Nationalities, Tongliao 028043
  • Received:2013-10-11 Accepted:2014-04-17 Online:2015-04-15 Published:2015-06-15
  • Supported by:
    The National Natural Science Foundation of China (11361038; 11471279); the National Natural Science Foundation for Major Reseach Plan Key Support Project of China (91430213); the Natural Science Foundation of Hunan Province (2015JJ2134); the Key Project of Education Depar-tment of Hunan Province (12A137); the Innovation Foundation for Postgraduate of Hunan Province (CX2014B254); the Program for Changjiang Scholars and Innovative Research Team in University of China (IRT1179).

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摘要： 非奇异$H$-矩阵在控制理论、科学计算和工程应用中具有重要的作用，但在实际中要判定一给定矩阵为非奇异$H$-矩阵是有难度的．本文通过研究给定矩阵元素的性质，对矩阵元素的航标集进行分割，巧妙地构造正对角矩阵和运用不等式的放缩方法，给出了非奇异$H$-矩阵的一组新的实用性新判定方法．进一步，将相关结果推广到不可约和具有非零元素链的情形．最后，我们改进和推广了相关的结果，并举例说明了所得方法的优越性．

关键词: 非奇异$H$-矩阵, 对角占优矩阵, 不可约, 非零元素链

Abstract:

Nonsingular $H$-matrix plays a significant role in the control theory, the scientific computation and the applications in engineering. However, it is difficult to specify a non-singular $H$-matrix in practice. In this paper, we partition the row index set by studying the elements of a matrix, and construct a positive diagonal matrix. Then, we apply some techniques in inequalities to obtain a new criterion for nonsingular $H$-matrices. We also obtain several similar results in the cases of irreducible matrices and matrices with non-zero elements chains. These consequences improve and generalize the related results, and the advantage of the proposed consequences are illustrated by several numerical examples.

Key words: nonsingular $H$-matrix, diagonally dominant matrix, irreducibility, non-zero elements chain

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