有界灰矩阵的非奇异性与秩

本文重新论述有界灰方阵的非奇异性判别问题,提出“条件非奇异”与“最大非奇异子元值域”的新概念,指出现有结果的局限性,给出了一些实用判据,并对灰逆阵的存在性条件与定义域作了研究.此外,本文又提出有界灰矩阵的“灰秩”的新概念与算法,建立了有界灰矩阵恒满秩、恒不满秩、条件满秩的判据。     

关于非奇异性判别的Gudkov定理

本文主要研究矩阵的非奇异性判别,推广了Ｓｚｕｌｃ和李最近在该领域所取得的主要结果。     

对角占优矩阵奇异性研究

本研究了对角占化矩阵的奇异性．得到了此类矩阵非奇异的一个筒单判断法．改进了已有结论。     

上三角Toeplitz矩阵的一个结论

本文得出了上三角 Ｔｏｅｐｌｉｔｚ 矩阵关于矩阵乘法构成一交换群的结果,并给出其逆矩阵的计算方法．     

Toeplitz型矩阵的逆矩阵的快速三角分解算法

针对有关“型”矩阵的三角分解问题 ,提出了一种 Toeplitz型矩阵的逆矩阵的快速三角分解算法 .首先假设给定 n阶非奇异矩阵 A,利用一组线性方程组的解 ,得到 A- 1的一个递推关系式 ,进而利用该关系式得到 A- 1的一种三角分解表达式 ,然后从 Toeplitz型矩阵的特殊结构出发 ,利用上述定理的结论 ,给出了Toeplitz型矩阵的逆矩阵的一种快速三角分解算法 ,算法所需运算量为 O( mn2 ) .最后 ,数值计算表明该算法的可靠性 .     

上三角Toeplitz矩阵的一个结论

本得出了上三角Toeplitz矩阵关于矩阵乘法构成一交换群的结果,并给出其逆矩阵的计算方法．     

分块r-循环Toeplitz矩阵的特征值方程

给出了分块 r-循环 Toeplitz矩阵特征方程的一个求法 ,推广了文 [1 ]的结果 .     

分块r—循环Toeplitz矩阵的特征值方程

给出了分块ｒ－循环Ｔｏｅｐｌｉｔｚ矩阵特征方程的一个求法,推广了「１」的结果。     

一类Toeplitz矩阵的群逆

Toeplitz矩阵Tn=(ti-j)n/i·j=0在信号处理、系统理论、逼近论、正交多项式．积分方程数值解等许多领域常常遇到,易知,Toeplitz矩阵T．的逆矩阵一般不再是Toeplitz矩阵．1972年Gohberg和Semencul给出了一个名结果：如果将Toeplirz矩阵T。     

分块K—循环Toeplitz矩阵求逆的快速付氏变换法

1算法描述及推导 Toeplitz矩阵及Toeplitz系统的求解在谱分析、线性预测、误差控制码、自回归滤波器设计等领域内起着重要的作用~[1-3]，而分块Toeplitz矩阵在计算机的时序分析、自回归时序模型滤波中也经常出现~[4]。对一般Toeplitz矩阵求逆，其算术复杂性为O(n~2)~[5]-[6]，其中n为Toepleitz矩阵的阶，而K-循环Toeplitz矩阵的求逆，其算术复杂性可降为O(nlog_2n)，本文提供了mn附分块K-循环Toeplitz矩阵求逆的一种快速付氏变换算法，其算术复杂性为O（mnlog_2mn）.     

On the reconstruction of Toeplitz matrix inverses from columns

In this paper we discuss the problem whether and how the inverse of a Toeplitz matrix can be recovered from some of its columns or parts of columns under the requirement that only 2n−1 parameters are involved. The results generalize and strengthen earlier findings by Trench, Gohberg, Semencul, Krupnik, Ben-Artzi, Shalom, Labahn, Rodman and others. Special attention is paid to symmetric, skewsymmetric and hermitian Toeplitz matrix inverses and the question whether such a matrix can be retrieved from a single column.     

A mean value algorithm for Toeplitz matrix completion

In this paper, we propose a new mean value algorithm for the Toeplitz matrix completion based on the singular value thresholding (SVT) algorithm. The completion matrices generated by the new algorithm keep a feasible Toeplitz structure. Meanwhile, we prove the convergence of the new algorithm under some reasonal conditions. Finally, we show the new algorithm is much more effective than the ALM (augmented Lagrange multiplier) algorithm through numerical experiments and image inpainting.     

Noncirculant Toeplitz matrices all of whose powers are Toeplitz

Let a, b and c be fixed complex numbers. Let M _n (a, b, c) be the n × n Toeplitz matrix all of whose entries above the diagonal are a, all of whose entries below the diagonal are b, and all of whose entries on the diagonal are c. For 1 ⩽ k ⩽ n, each k × k principal minor of M _n (a, b, c) has the same value. We find explicit and recursive formulae for the principal minors and the characteristic polynomial of M _n (a, b, c). We also show that all complex polynomials in M _n (a, b, c) are Toeplitz matrices. In particular, the inverse of M _n (a, b, c) is a Toeplitz matrix when it exists.     

关于Toeplitz矩阵的某些注记

In this paper,we study real symmetric Toeplitz matrices commutable with tridi-agonal matrices, present more detailed results than those in [1], and extend them to non-symmetric Toeplitz matrices. Also, complex Toeplitz matrices, especially the corresponding matrices of lower order, are discussed.     

On the asymptotic spectrum of Hermitian block Toeplitz matrices with Toeplitz blocks

We study the asymptotic behaviour of the eigenvalues of Hermitian block Toeplitz matrices , with Toeplitz blocks. Such matrices are generated by the Fourier coefficients of an integrable bivariate function , and we study their eigenvalues for large and , relating their behaviour to some properties of as a function; in particular we show that, for any fixed , the first eigenvalues of tend to , while the last tend to , so extending to the block case a well-known result due to Szegö. In the case the 's are positive-definite, we study the asymptotic spectrum of , where is a block Toeplitz preconditioner for the conjugate gradient method, applied to solve the system , obtaining strict estimates, when and are fixed, and exact limit values, when and tend to infinity, for both the condition number and the conjugate gradient convergence factor of the previous matrices. Extensions to the case of a deeper nesting level of the block structure are also discussed.

     

A fast algorithm for solving a Toeplitz system

In this paper, we present a fast algorithm for solving Symmetric penta-diagonal systems. We give the feasibility and Stability analysis of the algorithm. Moreover, parallel computations can be implemented in the algorithm. The numerical examples verify the efficiency of the algorithm.     

Convergence of the Multigrid Method of Ill-conditioned Block Toeplitz Systems

We study the solutions of block Toeplitz systems A _mn u = b by the multigrid method (MGM). Here the block Toeplitz matrices A _mn are generated by a nonnegative function f (x,y) with zeros. Since the matrices A _mn are ill-conditioned, the convergence factor of classical iterative methods will approach 1 as the size of the matrices becomes large. These classical methods, therefore, are not applicable for solving ill-conditioned systems. The MGM is then proposed in this paper. For a class of block Toeplitz matrices, we show that the convergence factor of the two-grid method (TGM) is uniformly bounded below 1 independent of mn and the full MGM has convergence factor depending only on the number of levels. The cost per iteration for the MGM is of O(mn log mn) operations. Numerical results are given to explain the convergence rate.     

三对角与五对角Toeplitz矩阵求逆的算法

提出了一种求三对角与五对角Toeplitz矩阵逆的快速算法,其思想为先将Toeplitz矩阵扩展为循环矩阵,再快速求循环矩阵的逆,进而运用恰当矩阵分块求原Toeplitz矩阵的逆的算法.算法稳定性较好且复杂度较低.数值例子显示了算法的有效性和稳定性,并指出了算法的适用范围.     

Band Toeplitz preconditioners for block Toeplitz systems

We consider the solutions of block Toeplitz systems with Toeplitz blocks by the preconditioned conjugate gradient (PCG) method. Here the block Toeplitz matrices are generated by nonnegative functions f(x,y). We use band Toeplitz matrices as preconditioners. The generating functions g(x,y) of the preconditioners are trigonometric polynomials of fixed degree and are determined by minimizing (f − g)/f∞. We prove that the condition number of the preconditioned system is O(1). An a priori bound on the number of iterations for convergence is obtained.