块H—矩阵的进一步推广

对Ｎａｂｂｅｎ提出的块Ｈ－矩阵做进一步推广，使得非对角块矩阵不必是Ｈｅｒｍｉｔｅ矩阵，但仍保留其基本特征不变。对块Ｈ－矩阵提出块Ｈ－分裂及块相容Ｈ－分裂。证明了矩阵的任意块相容Ｈ－分裂都是收敛分裂。     

圆盘定理的改进与弱连对角占优矩阵

本文对圆盘定理进行了改进，给出了特征值分布新的估计，由此引出了弱连对角占优矩阵，讨论了其基本性质，重点分析了该类矩阵的逆与分裂特征，证明了在该类矩阵条件下Ｈ－相容分裂是收敛分裂，并给出迭代矩阵谱半径的上界及ＳＯＲ算法中参数ω的选取范围。     

奇异M—矩阵的图相容弱正则分裂

设Ａ是奇异Ｍ－矩阵，Ａ＝Ｍ－Ｎ是Ａ的图相容弱正则分裂。本文研究迭代矩阵Ｍ＾－１Ｎ的谱性质，得到与迭代矩阵的指数有关的一个定理：ｉｎｄ０（Ａ）＝ｉｎｄ１（Ｍ＾－１Ｎ）．它推广了Ｈ．Ｓｃｈｎｅｉｄｅｒ和作者的结果。     

奇异M─矩阵的图相容弱正则分裂

设Ａ是奇异Ｍ－矩阵，Ａ＝Ｍ－Ｎ是Ａ的图相容弱正则分裂．本文研究迭代矩阵Ｍ－１Ｎ的谱性质，得到与迭代矩阵的指数有关的一个定理：ｉｎｄ０（Ａ）＝ｉｎｄ１（Ｍ－１Ｎ）．它推广了Ｈ．Ｓｃｈｎｅｉｄｅｒ和作者的结果．     

广义异步矩阵多分裂向前向后松驰算法

本文建立了一类广义异步矩阵多分裂向前向后松驰算法，并在系数矩阵是Ｈ－矩阵的条件下，证明了这类算法的收敛性。     

不定常并行多分裂SOR方法的收敛性

不定常并行多分裂方法是关于解线性方程组ＡＸ＝ｂ的新的并行方法，如果引入某种松驰，这些方法的收敛性能期望得以改善。本研究了不定常并行多分裂ＳＯＲ方法及其推广。如果Ａ是一个Ｈ－矩阵，且松驰参数满足０〈ωｊ〈ω０，ｊ＝１，…，ｋ，ω０〉１。     

广义异步矩阵多分裂向前向后松弛算法

本文建立了一类广义异步矩阵多分裂向前向后松弛算法，并在系数矩阵是Ｈ－矩阵的条件下，证明了这类算法的收敛性．     

矩阵DA的特征值及在系统稳定性分析中的应用

本文分析了矩阵ＤＡ的特征值，并提出了一种新的矩阵稳定性问题：Ｄ－－稳定性和块Ｄ－－稳定性．这里Ｄ是一个正对角阵，Ａ是给定的矩阵．应用劳斯判据及李雅普诺夫方法等，详细地分析了此类稳定性问题，提出了一些易于验证的结果，可用于分析多变量奇异摄动系统稳定问题．     

K—对角占优矩阵

提出Ｋ－对角占优矩阵，它是对角占优矩阵及某些Ｈ－矩阵判别法的推广讨论了其基本性质以及与Ｈ－矩阵的关系，并给出其一些应用。     

按环路α-连对角占优阵及应用

１．引言与记号 利用矩阵的对角占优性研究矩阵的特征值分布和非奇Ｈ矩阵的判定,是数值代数的重要课题．［１］－［４］给出了利用 Ｏｓｔｒｏｗｓｋｉ定理及连对角占优性判定非奇 Ｈ－矩阵的最新成果．本文引入按环路α－连对角占优概念,给出了非奇Ｈ－矩阵的判定条件及等价表征,简化了计算,改进与推广了［１］－［９］的相应结果． 设Ａ＝．Γ（Ａ）表 Ａ的方向图,其顶点集及弧集分别记作 Ｖ（Ａ）及 Ｅ（Ａ）,ｅｉｊ表从顶点ｉ到顶点 ｊ的弧, Ｃ（Ａ）表 Γ（Ａ）中非平凡环路集合．对任意固定 α Ｅ［０,１］还记＊ｋ伪行、列足…     

M-矩阵线性互补问题模系多分裂迭代方法的收敛性

首先证明了M-矩阵的H-相容分裂都是正则分裂,反之不成立.这表明对于M-矩阵而言,其正则分裂包含H-相容分裂.然后针对系数矩阵为M-矩阵的线性互补问题,建立了两个收敛定理:一是模系多分裂迭代方法关于正则分裂的收敛定理;二是模系二级多分裂迭代方法关于外迭代为正则分裂和内迭代为弱正则分裂的收敛定理.     

On a type of matrix splitting preconditioners for a class of block two-by-two linear systems

Recently, Bai et al. (2013) proposed an effective and efficient matrix splitting iterative method, called preconditioned modified Hermitian/skew-Hermitian splitting (PMHSS) iteration method, for two-by-two block linear systems of equations. The eigenvalue distribution of the iterative matrix suggests that the splitting matrix could be advantageously used as a preconditioner. In this study, the CGNR method is utilized for solving the PMHSS preconditioned linear systems, and the performance of the method is considered by estimating the condition number of the normal equations. Furthermore, the proposed method is compared with other PMHSS preconditioned Krylov subspace methods by solving linear systems arising in complex partial differential equations and a distributed control problem. The numerical results demonstrate the difference in the performance of the methods under consideration.     

Iterative processes based on block <Emphasis Type="Italic">H</Emphasis>-splittings

Block H-splittings of block square matrices (which, in general, have complex entries) are examined. It is shown that block H-matrices are the only ones that admit this type of splittings. Iterative processes corresponding to these splittings are proved to be convergent. The concept of a simple splitting of a block matrix is introduced, and the convergence of iterative processes related to simple splittings of block H-matrices is investigated. Multisplitting and nonstationary iterative processes based on block H-splittings are considered. Sufficient conditions for their convergence are derived, and some estimates for the asymptotic convergence rate are given.     

块α-对角占优矩阵的讨论

应用矩阵块对角占优理论,讨论了块α-对角占优矩阵之间的蕴含关系,并得到了条件最弱的块严格α1-双对角占优的两个等价表征,并作为应用给出了块H矩阵新的判定准则,最后用数值例子说明结果的有效性.     

A class of block alternating splitting implicit iteration methods for double saddle point linear systems

In the framework of a special block alternating splitting implicit (BASI) iteration scheme for generalized saddle point problems, we establish some new iteration methods for solving double saddle point problems by means of a suitable partitioning strategy. Convergence analysis of the corresponding BASI iteration methods indicates that they are convergent unconditionally under certain weak requirements for the related matrix splittings, which are satisfied directly for our specific application to double saddle point problems. Numerical examples for liquid crystal director and time-harmonic eddy current models are presented to demonstrate the efficiency of the proposed BASI preconditioners to accelerate the GMRES method.     

两类组合设计及其在认证码中的应用

W．Ogata等定义了两种新的组合设计：外差族（EDF）与外平衡不完全区组设计（E-BIBD）．本文首先用有限域中的分圆类给出EDF的一个构造；接着用EBIBD构造出具有完善保密性的最优分裂A-码，然后证明了由满足一定条件的两个EBIBD通过上述方法构造出的两个认证码是同构的.     

Splitting iterative methods for fuzzy system of linear equations

A class of splitting iterative methods is considered for solving fuzzy system of linear equations, which cover Jacobi, Gauss–Seidel, SOR, SSOR, and their block variants proposed by others before. We give a convergence theorem for a regular splitting, where the corresponding iterative methods converge to the strong fuzzy solution for any initial vector and fuzzy right-hand vector. Two schemes of splitting are given to illustrate the theorem. Numerical experiments further show the efficiency of the splitting iterative methods.     

ChebStaBlkCG: A block variant of ChebFilterCG

The behavior of ChebFilterCG (an algorithm that combines the Chebyshev filter and Conjugate Gradient) applied to systems with unfavorable eigenvalue distribution is examined. To improve the convergence, a hybrid approach combining a stabilized version of the block conjugated gradient with Chebyshev polynomials as preconditioners (ChebStaBlkCG) is proposed. The performance of ChebStaBlkCG is illustrated and validated on a set of linear systems. It is shown how ChebStaBlkCG can be used to accelerate the block Cimmino method and to solve linear systems with multiple right‐hand sides.     

剖分Petersen图得到的新测地块

通过剖分Petersen图我们得到了一类新的测地块．由此成功构造了给定直径为6和围长为11的测地块，解决了近二十年来悬而未决的一个存在性和构造性问题．