严格对角占优M-矩阵的逆矩阵的无穷大范数的上界估计
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摘要
M-矩阵作为特殊矩阵类在高阶稀疏线性方程组的迭代法求解中有重要作用,尤其是M-矩阵的逆矩阵的无穷大范数的上界估计在数值代数中具有重要意义。如许多代数方程组问题的收敛性条件、条件数等需要计算‖A~(-1)‖_∞,但当M-矩阵A的阶数较大时,其逆矩阵很难求,因此‖A~(-1)‖_∞估计是十分重要的问题。首先引入一组新的记号,给出严格对角占优M-矩阵A的逆矩阵A~(-1)的元素满足的两个不等式;此外得到了‖A~(-1)‖_∞的上界新估计式,这些估计式避免了求逆矩阵A~(-1)而直接利用矩阵A的元素表示,最后给出矩阵A的最小特征值q(A)下界的新估计式。理论分析和数值算例表明新估计式改进了相关结果。
M-Matrix as a special class of matrices which plays an important role in solving high order sparse linear equations,especially estimation for upper bounds of the infinity norms for matrix inverse of M-Matrices,has important significance in numerical algebra. Usually ‖A~(-1)‖_∞is used to calculate such as the convergence conditions,condition number and so on of many algebraic equations. But when order of the M-matrix A is much larger,the inverse matrix is very difficult to find,so estimation of ‖A~(-1)‖_∞is a very important problem. Firstly,some new marks are introduced,and two inequalities are given,which elements of inverse matrix of strictly diagonally dominant M-matrix A satisfy condition; The new upper bound estimates are obtained in addition,and these estimates,represented by elements of the matrix A directly,avoid calculating the inverse matrix A~(-1); the bound is applied to obtain a lower bound for the smallest eigenvalue of A. Theoretical analysis and numerical examples show that the new bounds improve related results.
M-Matrix as a special class of matrices which plays an important role in solving high order sparse linear equations,especially estimation for upper bounds of the infinity norms for matrix inverse of M-Matrices,has important significance in numerical algebra. Usually ‖A~(-1)‖_∞is used to calculate such as the convergence conditions,condition number and so on of many algebraic equations. But when order of the M-matrix A is much larger,the inverse matrix is very difficult to find,so estimation of ‖A~(-1)‖_∞is a very important problem. Firstly,some new marks are introduced,and two inequalities are given,which elements of inverse matrix of strictly diagonally dominant M-matrix A satisfy condition; The new upper bound estimates are obtained in addition,and these estimates,represented by elements of the matrix A directly,avoid calculating the inverse matrix A~(-1); the bound is applied to obtain a lower bound for the smallest eigenvalue of A. Theoretical analysis and numerical examples show that the new bounds improve related results.
引文
[1]WANG P.An upper bound for A-1∞of strictly diagonally dominant M-matrices[J].Linear Algebra Appl,2009,431:667-673.
[2]李艳艳,李耀堂.严格对角占优M-矩阵的逆矩阵的无穷大范数上界的估计[J].云南民族大学学报:自然科学版,2012,21(1):52-56.
[3]陈公宁.矩阵理论与应用[M].北京:科学出版社,2007.
[4]SHIVAKUMAR P N,WILLIAMS J J,YE Q,et al.On two-sided bounds related to weakly diagonally dominant M-matrces with application to digita dynamics[J].Matrix Anal.Appl,1996,17(2):298-312.
[5]王峰.非奇异M-矩阵的逆矩阵和M-矩阵的Hadamard积的最小特征值下界估计[J].应用数学,2013,26(2):341-345.
[6]潘淑珍,陈神灿.弱链对角占优矩阵A-1∞的上界估计[J].福州大学学报:自然科学版,2012,25(3):36-39.
[7]赵仁庆,熊昌明,李耀堂.块H-矩阵的判定及其逆底无穷大范数的上界[J].云南大学学报:自然科学版,2011,33(2):125-130.
[8]刘新,杨晓英.严格对角占优M-矩阵A的A-1∞的新上界[J].北华大学学报:自然科学版,2014,15(2):184-187.
[9]王永.严格对角占优M-矩阵的逆矩阵的无穷大范数上界的估计的新上界[J].工程数学报,2015(5):719-725.
[10]赵仁庆,刘鹏.弱链对角占优M-矩阵的逆矩阵的无穷大范数的上界估计[J].楚雄师范学院学报,2014,29(3):5-10.
[11]赵云平,李朝迁.广义M-矩阵的逆矩阵范数的估计[J].贵州大学学报:自然科学版,2014,31(4):8-10.
[12]WEN Li.The intinity norm bound for the inverse of nonsingular diagonal dominant matrices[J].Appl.Math.Lett.,2008,21:258-263.
[13]LI Y T,CHEN F B,WANG D F.New lower bounds on eigenvalue of the Hadamard product of an M-matrix and its inverses[J].Linear Algebra Appl,2009,430:1423-1431.
[14]HUANG T Z,ZHU Y.Estimation of A-1∞for weakly chained diagonally dominant M-matrice[J].Linear Algebra Appl.,2009,432:670-667.
[15]赵仁庆,钟振华,刘鹏.GS-SDD矩阵的逆矩阵的无穷范数和最小奇异值的估计[J].楚雄师范学院学报,2014,29(6):1-5.
[2]李艳艳,李耀堂.严格对角占优M-矩阵的逆矩阵的无穷大范数上界的估计[J].云南民族大学学报:自然科学版,2012,21(1):52-56.
[3]陈公宁.矩阵理论与应用[M].北京:科学出版社,2007.
[4]SHIVAKUMAR P N,WILLIAMS J J,YE Q,et al.On two-sided bounds related to weakly diagonally dominant M-matrces with application to digita dynamics[J].Matrix Anal.Appl,1996,17(2):298-312.
[5]王峰.非奇异M-矩阵的逆矩阵和M-矩阵的Hadamard积的最小特征值下界估计[J].应用数学,2013,26(2):341-345.
[6]潘淑珍,陈神灿.弱链对角占优矩阵A-1∞的上界估计[J].福州大学学报:自然科学版,2012,25(3):36-39.
[7]赵仁庆,熊昌明,李耀堂.块H-矩阵的判定及其逆底无穷大范数的上界[J].云南大学学报:自然科学版,2011,33(2):125-130.
[8]刘新,杨晓英.严格对角占优M-矩阵A的A-1∞的新上界[J].北华大学学报:自然科学版,2014,15(2):184-187.
[9]王永.严格对角占优M-矩阵的逆矩阵的无穷大范数上界的估计的新上界[J].工程数学报,2015(5):719-725.
[10]赵仁庆,刘鹏.弱链对角占优M-矩阵的逆矩阵的无穷大范数的上界估计[J].楚雄师范学院学报,2014,29(3):5-10.
[11]赵云平,李朝迁.广义M-矩阵的逆矩阵范数的估计[J].贵州大学学报:自然科学版,2014,31(4):8-10.
[12]WEN Li.The intinity norm bound for the inverse of nonsingular diagonal dominant matrices[J].Appl.Math.Lett.,2008,21:258-263.
[13]LI Y T,CHEN F B,WANG D F.New lower bounds on eigenvalue of the Hadamard product of an M-matrix and its inverses[J].Linear Algebra Appl,2009,430:1423-1431.
[14]HUANG T Z,ZHU Y.Estimation of A-1∞for weakly chained diagonally dominant M-matrice[J].Linear Algebra Appl.,2009,432:670-667.
[15]赵仁庆,钟振华,刘鹏.GS-SDD矩阵的逆矩阵的无穷范数和最小奇异值的估计[J].楚雄师范学院学报,2014,29(6):1-5.