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.     

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.     

Analytical inversion of general periodic tridiagonal matrices

In this paper we present an analytical forms for the inversion of general periodic tridiagonal matrices, and provide some very simple analytical forms which immediately lead to closed formulae for some special cases such as symmetric or perturbed Toeplitz for both periodic and non-periodic tridiagonal matrices. An efficient computational algorithm for finding the inverse of any general periodic tridiagonal matrices from the analytical form is given, it is suited for implementation using Computer Algebra systems such as MAPLE, MATLAB, MACSYMA, and MATHEMATICA. An example is also given to illustrate the algorithm.     

A fast algorithm for solving Toeplitz penta-diagonal systems

In this paper a fast algorithm for solving a large system with a symmetric Toeplitz penta-diagonal coefficient matrix is presented. This efficient method is based on the idea of a system perturbation followed by corrections and is competitive with standard methods. The error analysis is also given.     

Modified SMS method for computing outer inverses of Toeplitz matrices

We introduce a new algorithm based on the successive matrix squaring (SMS) method. This algorithm uses the strategy of ε-displacement rank in order to find various outer inverses with prescribed ranges and null spaces of a square Toeplitz matrix. Using the idea of displacement theory which decreases the memory space requirements as well as the computational cost, our method tends to be very effective for Toeplitz matrices.     

基于均值的Toeplitz矩阵填充的子空间算法

本文提出一种基于均值的Toeplitz矩阵填充的子空间算法.通过在左奇异向量空间中对已知元素的最小二乘逼近,形成了新的可行矩阵;并利用对角线上的均值化使得迭代后的矩阵保持Toeplitz结构,从而减少了奇异向量空间的分解时间.理论上,证明了在一定条件下该算法收敛于一个低秩的Toeplitz矩阵.通过不同已知率的矩阵填充数值实验展示了Toeplitz矩阵填充的新算法比阈值增广Lagrange乘子算法在时间上和精度上更有效.     

Efficient computation of the extreme solutions of

We propose a new quadratically convergent algorithm, having a low computational cost per step and good numerical stability properties, which allows the simultaneous approximation of the extreme solutions of the matrix equations and . The algorithm is based on the cyclic reduction method.

     

A fast algorithm for computing the smallest eigenvalue of a symmetric positive‐definite Toeplitz matrix

Recent progress in signal processing and estimation has generated considerable interest in the problem of computing the smallest eigenvalue of a symmetric positive‐definite (SPD) Toeplitz matrix. An algorithm for computing upper and lower bounds to the smallest eigenvalue of a SPD Toeplitz matrix has been recently derived (Linear Algebra Appl. 2007; DOI: 10.1016/j.laa.2007.05.008 ). The algorithm relies on the computation of the R factor of the QR factorization of the Toeplitz matrix and the inverse of R. The simultaneous computation of R and R^?1 is efficiently accomplished by the generalized Schur algorithm. In this paper, exploiting the properties of the latter algorithm, a numerical method to compute the smallest eigenvalue and the corresponding eigenvector of SPD Toeplitz matrices in an accurate way is proposed. Copyright © 2008 John Wiley & Sons, Ltd.     

Kernel structure of Toeplitz-plus-Hankel matrices

The structure of the kernel of block Toeplitz-plus-Hankel matrices R=[a_j−k+b_j+k], where a_j and b_j are the given p×q blocks with entries from a given field, is investigated. It is shown that R corresponds to two systems of at most p+q vector polynomials from which a basis of the kernel of R and all other Toeplitz-plus-Hankel matrices with the same parameters a_j and b_j can be built. The main result is an analogue of a known kernel structure theorem for block Toeplitz and block Hankel matrices.     

Fast solution of unsymmetric banded Toeplitz systems by means of spectral factorizations and Woodbury's formula

A fast algorithm for solving systems of linear equations with banded Toeplitz matrices is studied. An important step in the algorithm is a novel method for the spectral factorization of the generating function associated with the Toeplitz matrix. The spectral factorization is extracted from the right deflating subspaces corresponding to the eigenvalues inside and outside the open unit disk of a companion matrix pencil constructed from the coefficients of the generating function. The factorization is followed by the Woodbury inversion formula and solution of several banded triangular systems. Stability of the algorithm is discussed and its performance is demonstrated by numerical experiments. Copyright © 2012 John Wiley & Sons, Ltd.     

A fast method to block-diagonalize a Hankel matrix

In this paper, we consider an approximate block diagonalization algorithm of an n×n real Hankel matrix in which the successive transformation matrices are upper triangular Toeplitz matrices, and propose a new fast approach to compute the factorization in O(n ²) operations. This method consists on using the revised Bini method (Lin et al., Theor Comp Sci 315: 511–523, 2004). To motivate our approach, we also propose an approximate factorization variant of the customary fast method based on Schur complementation adapted to the n×n real Hankel matrix. All algorithms have been implemented in Matlab and numerical results are included to illustrate the effectiveness of our approach.     

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.     

关于Toeplitz矩阵的非奇异性

本讨论了Toeplitz矩阵的非奇异性，给出了Toeplitz矩阵非奇异的的一些判别条件。     

INERTIA SETS OF SYMMETRIC SIGN PATTERN MATRICES

1　IntroductionIn qualitative and combinatorial matrix theory,we study properties ofa matrix basedon combinatorial information,such as the signs of entries in the matrix.A matrix whoseentries are from the set{ + ,-,0 } is called a sign pattern matrix ( or sign pattern,or pat-tern) .We denote the setof all n× n sign pattern matrices by Qn.For a real matrix B,sgn( B) is the sign pattern matrix obtained by replacing each positive( respectively,negative,zero) entry of B by+ ( respectively,-,0 )…     

A sufficient condition for a non-negative matrix to have non-negative determinant

It is shown that a matrix with non-negative entries has non-negative determinant if in each row the elements decrease, by steadily smaller amounts, as one proceeds (in either direction) away from the main diagonal. This condition suffices to establish non- negativity of the determinant for certain matrices to which the familiar Minkowski- Hadamard-Ostrowski dominance conditions do not apply. In the symmetric case it provides a sufficient condition for non-negative definiteness. This may be applied to establish the positive definiteness of certain real symmetric Toeplitz matrices.     

一类Toeplitz线性代数方程组的预处理GMRES方法

本文研究一类来源于分数阶特征值问题的Toeplitz线性代数方程组的求解.构造Strang循环矩阵作为预处理矩阵来求解该Toeplitz线性代数方程组,分析了预处理后系数矩阵的特征值性质.提出求解该线性代数方程组的预处理广义极小残量法（PGMRES）,并给出该算法的计算量.数值算例表明了该方法的有效性.     

A Second-Order Method for the Electromagnetic Scattering from a Large Cavity

In this paper,we study the electromagnetic scattering from a two dimen- sional large rectangular open cavity embedded in an infinite ground plane,which is modelled by Helmholtz equations.By introducing nonlocal transparent boundary con- ditions,the problem in the open cavity is reduced to a bounded domain problem.A hypersingular integral operator and a weakly singular integral operator are involved in the TM and TE cases,respectively.A new second-order Toeplitz type approximation and a second-order finite difference scheme are proposed for approximating the hyper- singular integral operator on the aperture and the Helmholtz in the cavity,respectively. The existence and uniqueness of the numerical solution in the TE case are established for arbitrary wavenumbers.A fast algorithm for the second-order approximation is pro- posed for solving the cavity model with layered media.Numerical results show the second-order accuracy and efficiency of the fast algorithm.More important is that the algorithm is easy to implement as a preconditioner for cavity models with more general media.     

Eigenvector sensitivity under general and structured perturbations of tridiagonal Toeplitz‐type matrices

The sensitivity of eigenvalues of structured matrices under general or structured perturbations of the matrix entries has been thoroughly studied in the literature. Error bounds are available, and the pseudospectrum can be computed to gain insight. Few investigations have focused on analyzing the sensitivity of eigenvectors under general or structured perturbations. This paper discusses this sensitivity for tridiagonal Toeplitz and Toeplitz‐type matrices.     

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.