“杨辉三角”中某些矩阵及其行列式的讨论

“杨辉三角”中某些矩阵及其行列式的讨论张之正，刘麦学（河南洛阳师范专科学校数学系４７１０２２）近来，许多作者对杨辉三角中的矩阵和行列式进行了讨论（如［１］─［５］），文［６］还给出了一类矩阵的逆矩阵的求法．本短文对这些矩阵及行列式做进一步的探讨．杨辉...     

求解线性矩阵方程的初等变换法

求解线性矩阵方程的初等变换法杨兴东（南京气象学院基科系，南京２１００４４）杨兴洲（南京大学成人教育学院，南京２１００９３）文［１］给出了线性矩阵方程ＡＸＢ＝Ｃ有解的简单判别法则，本文则应用初等变换，给出矩阵方程ＡＸＢ＝Ｃ（１）的简便解法．引理１［２］...     

矩阵的算术平均等于几何平均的特征

矩阵的算术平均等于几何平均的特征杨忠鹏（吉林师范学院，吉林市，１３２０１３）为第二届中国矩阵论及其应用学术会（１９９６，８．１２－１６，吉林市）的召开，“国际线性代数学会”赶印了其会刊［１］，在［１］的“ＩｍａｇｅＰｒｏｂｌｅｍＣｏｒｎｅｒ”一栏提出...     

（r）循环矩阵的Kronecker积

（ｒ）循环矩阵的Ｋｒｏｎｅｃｋｅｒ积韩瑞珠（东南大学）循环矩阵是一类很重要的矩阵，它有着广泛的应用［１］，并且有许多独特的性质。例如，循环阵的广义道［１］、逆矩阵［２］、伴随阵［３］仍为循环阵；循环阵的乘积仍为循环阵。自然地，作为矩阵论中有极其重要的...     

关于对广义正定矩阵的进一步研究

文［１］定义了广义正定矩阵集合Ｐ（Ｉ）．文［２］定义了较Ｐ（Ｉ）更广泛的另一个广义正定矩阵集合Ｐ（Ｓ＋）．本文把Ｐ（Ｉ）中矩阵的某些性质，推广到Ｐ（Ｓ＋）中从而丰富了Ｐ（Ｓ＋）矩阵集合的结果。     

广义严格对角占优矩阵与非奇M矩阵的判定

１引言Ｍ矩阵是计算数学中应给极其广泛的矩阵类,它出现于经济价值模型矩阵和反网络系统分析的系数矩阵及解某类确定微分方程问题的数值解法中．由于Ｍ矩阵的重要性,讨论Ｍ矩阵及相关的广义对角占优矩阵的判定及性质有着十分重要的意义．本文则是在文［１］～［３］基础上,给出了广义严格对角占优矩阵与非奇Ｍ矩阵几则新的充分条件．拓广了文［１］～［３］的相关结果．２主要结果定义１设Ａ＝（ａｉｊ）,如果存在正对角阵Ｄ,使得ＡＤ为严格对角占优阵,则称Ａ为广义严格对角占优阵．定义２设Ａ＝,Ｍ（Ａ）＝（Ｍｉｊ）,其中,则称Ｓ…     

关于一种循环类预条件方程组的快速求解

１引言考虑下列Ｎ阶线性方程组其中Ｃ１＝,Ｃ２＝０≤ｉ,ｊ≤Ｎ－１,是Ｎ阶循环矩阵,Ｊ１＝（Ｊ）是Ｎ阶置换矩阵,其元素分别满足１９９３年,Ｔ,Ｋ．Ｋｕ,Ｃ．Ｃ．Ｊ．Ｋｕｏ在［１］中取Ｃ１,Ｃ２为实对称循环矩阵,而Ｃ１＋Ｊ１Ｃ２作为预条件矩阵来求解在数字信号处理中有一定应用的Ｔｏｅｐｌｉｔｚ加Ｈａｎｋｅｌ线性方程组［２］,得到了一种高效的预处理其轭梯度算法．当Ｔｏｅｅｌｉｔｚ与Ｈａｎｋｅｌ矩阵之和为正定矩阵且条件数适中时,所需运算量可达到０（Ｎｌｏｇ２Ｎ）,比原有算法［２,３,４］的运算量０（Ｎ２）…     

线性方程组反问题的推广

线性方程组反问题的推广王卿文（山东昌潍师专数学系２６１０４３）自文［１］提出线性方程组Ａｘ＝ｂ的反问题以来，此反问题即成为人们研究的热门课题之一，文［１—７］分别给出了其正定对称矩阵解与对称矩阵解的某些解法及解集合的结构．最近，文［８］又提出了线性方...     

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…     

任意体上的矩阵方程［X_（nn）A_（ns），X_（nn）B_（nt）］＝［A_

本文研究了任意体上的矩阵方程［Ｘ（ｎｎ）Ａ（ｎｓ），Ｘ（ｎｎ）Ｂ（ｎｔ）］＝［Ａ（ｎｓ），０］（１）给出了（１）相容的充要条件、通解的表达式、解的性质及其实用解法．     

A Class of Nested Iteration Schemes for Linear Systems with a Coefficient Matrix with a Dominant Positive Definite Symmetric Part

We present a class of nested iteration schemes for solving large sparse systems of linear equations with a coefficient matrix with a dominant symmetric positive definite part. These new schemes are actually inner/outer iterations, which employ the classical conjugate gradient method as inner iteration to approximate each outer iterate, while each outer iteration is induced by a convergent and symmetric positive definite splitting of the coefficient matrix. Convergence properties of the new schemes are studied in depth, possible choices of the inner iteration steps are discussed in detail, and numerical examples from the finite-difference discretization of a second-order partial differential equation are used to further examine the effectiveness and robustness of the new schemes over GMRES and its preconditioned variant. Also, we show that the new schemes are, at least, comparable to the variable-step generalized conjugate gradient method and its preconditioned variant.     

求解带Toeplitz矩阵的线性互补问题的一类预处理模系矩阵分裂迭代法

针对系数矩阵为对称正定Toeplitz矩阵的线性互补问题，本文提出了一类预处理模系矩阵分裂迭代方法.先通过变量替换将线性互补问题转化为一类非线性方程组，然后选取Strang或T.Chan循环矩阵作为预优矩阵，利用共轭梯度法进行求解.我们分析了该方法的收敛性.数值实验表明，该方法是高效可行的.     

几乎各向同性的高维空间分数阶扩散方程的分块快速正则Hermite分裂预处理方法

一类空间分数阶扩散方程经过有限差分离散后所得到的离散线性方程组的系数矩阵是两个对角矩阵与Toeplitz型矩阵的乘积之和.在本文中,对于几乎各向同性的二维或三维空间分数阶扩散方程的离散线性方程组,采用预处理Krylov子空间迭代方法,我们利用其系数矩阵的特殊结构和具体性质构造了一类分块快速正则Hermite分裂预处理子.通过理论分析,我们证明了所对应的预处理矩阵的特征值大部分都聚集于1的附近.数值实验也表明,这类分块快速正则Hermite分裂预处理子可以明显地加快广义极小残量(GMRES)方法和稳定化的双共轭梯度(BiCGSTAB)方法等Krylov子空间迭代方法的收敛速度.     

A splitting preconditioner for saddle point problems

For large sparse systems of linear equations iterative techniques are attractive. In this paper, we study a splitting method for an important class of symmetric and indefinite system. Theoretical analyses show that this method converges to the unique solution of the system of linear equations for all t>0 (t is the parameter). Moreover, all the eigenvalues of the iteration matrix are real and nonnegative and the spectral radius of the iteration matrix is decreasing with respect to the parameter t. Besides, a preconditioning strategy based on the splitting of the symmetric and indefinite coefficient matrices is proposed. The eigensolution of the preconditioned matrix is described and an upper bound of the degree of the minimal polynomials for the preconditioned matrix is obtained. Numerical experiments of a model Stokes problem and a least‐squares problem with linear constraints presented to illustrate the effectiveness of the method. Copyright © 2011 John Wiley & Sons, Ltd.     

Right preconditioned MINRES for singular systems†

We consider solving large sparse symmetric singular linear systems. We first introduce an algorithm for right preconditioned minimum residual (MINRES) and prove that its iterates converge to the preconditioner weighted least squares solution without breakdown for an arbitrary right‐hand‐side vector and an arbitrary initial vector even if the linear system is singular and inconsistent. For the special case when the system is consistent, we prove that the iterates converge to a min‐norm solution with respect to the preconditioner if the initial vector is in the range space of the right preconditioned coefficient matrix. Furthermore, we propose a right preconditioned MINRES using symmetric successive over‐relaxation (SSOR) with Eisenstat's trick. Some numerical experiments on semidefinite systems in electromagnetic analysis and so forth indicate that the method is efficient and robust. Finally, we show that the residual norm can be further reduced by restarting the iterations.     

Minimum residual methods for augmented systems

For large systems of linear equations, iterative methods provide attractive solution techniques. We describe the applicability and convergence of iterative methods of Krylov subspace type for an important class of symmetric and indefinite matrix problems, namely augmented (or KKT) systems. Specifically, we consider preconditioned minimum residual methods and discuss indefinite versus positive definite preconditioning. For a natural choice of starting vector we prove that when the definite and indenfinite preconditioners are related in the obvious way, MINRES (which is applicable in the case of positive definite preconditioning) and full GMRES (which is applicable in the case of indefinite preconditioning) give residual vectors with identical Euclidean norm at each iteration. Moreover, we show that the convergence of both methods is related to a system of normal equations for which the LSQR algorithm can be employed. As a side result, we give a rare example of a non-trivial normal(1) matrix where the corresponding inner product is explicitly known: a conjugate gradient method therefore exists and can be employed in this case. This work was supported by British Council/German Academic Exchange Service Research Collaboration Project 465 and NATO Collaborative Research Grant CRG 960782     

On a Chebyshev accelerated splitting iteration method with application to two‐by‐two block linear systems

The Chebyshev accelerated preconditioned modified Hermitian and skew‐Hermitian splitting (CAPMHSS) iteration method is presented for solving the linear systems of equations, which have two‐by‐two block coefficient matrices. We derive an iteration error bound to show that the new method is convergent as long as the eigenvalue bounds are not underestimated. Even when the spectral information is lacking, the CAPMHSS iteration method could be considered as an exponentially converging iterative scheme for certain choices of the method parameters. In this case, the convergence rate is independent of the parameters. Besides, the linear subsystems in each iteration can be solved inexactly, which leads to the inexact CAPMHSS iteration method. The iteration error bound of the inexact method is derived also. We discuss in detail the implementation of CAPMHSS for solving two models arising from the Galerkin finite‐element discretizations of distributed control problems and complex symmetric linear systems. The numerical results show the robustness and the efficiency of the new methods.     

Nested splitting CG-like iterative method for solving the continuous Sylvester equation and preconditioning

We present a nested splitting conjugate gradient iteration method for solving large sparse continuous Sylvester equation, in which both coefficient matrices are (non-Hermitian) positive semi-definite, and at least one of them is positive definite. This method is actually inner/outer iterations, which employs the Sylvester conjugate gradient method as inner iteration to approximate each outer iterate, while each outer iteration is induced by a convergent and Hermitian positive definite splitting of the coefficient matrices. Convergence conditions of this method are studied and numerical experiments show the efficiency of this method. In addition, we show that the quasi-Hermitian splitting can induce accurate, robust and effective preconditioned Krylov subspace methods.     

嵌套简单ILU分解代数预处理方法

In this paper, we propose a nested simple incomplete LU decomposition (NSILU) method for preconditioning iterative methods for solving largely scale and sparse ill-conditioned hnear systems. NSILU consists of some numerical techniques such as simple modification of Schur complement, compression of ill-condition structure by permutation, nested simple ILU, and inner-outer iteration. We give detailed error analysis of NSILU and estimations of condition number of the preconditioned coefficient matrix, together with numerical comparisons. We also show an analysis of inner accuracy strategies for the inner-outer iteration approach. Our new approach NSILU is very efficient for linear systems from a kind of two-dimensional nonlinear energy equations with three different temperature variables, where most of the calculations centered around solving large number of discretized and illconditioned linear systems in large scale. Many numerical experiments are given and compared in costs of flops, CPU times, and storages to show the efficiency and effectiveness of the NSILU preconditioning method. Numerical examples include middle-scale real matrices of size n = 3180 or n = 6360, a real apphcation of solving about 755418 linear systems of size n = 6360, and a simulation of order n=814080 with structures and properties similar as the real ones.     

An inexact parallel splitting augmented Lagrangian method for large system of linear equations

Parallel iterative methods are powerful in solving large systems of linear equations (LEs). The existing parallel computing research results focus mainly on sparse systems or others with particular structure. Most are based on parallel implementation of the classical relaxation methods such as Gauss-Seidel, SOR, and AOR methods which can be efficiently carried out on multiprocessor system. In this paper, we propose a novel parallel splitting operator method in which we divide the coefficient matrix into two or three parts. Then we convert the original problem (LEs) into a monotone (linear) variational inequality problem (VI) with separable structure. Finally, an inexact parallel splitting augmented Lagrangian method is proposed to solve the variational inequality problem (VI). To avoid dealing with the matrix inverse operator, we introduce proper inexact terms in subproblems such that the complexity of each iteration of the proposed method is O(n²). In addition, the proposed method does not require any special structure of system of LEs under consideration. Convergence of the proposed methods in dealing with two and three separable operators respectively, is proved. Numerical computations are provided to show the applicability and robustness of the proposed methods.