Hermitian Toeplitz矩阵向量乘积的快速算法（英文）

众所周知,大规模Hermitian Toeplitz矩阵向量乘积Ax可由快速Fourier变换(FFT)进行计算.事实上,Hermitian Toeplitz矩阵在酉相似变换下可约化为一个实的Toeplitz矩阵与Hankel矩阵之和.基于此,本文利用DCT和DST,构造了一个更有效的方法,只需O(n)的复运算.     

由特征值唯一确定的实3×3Hankel矩阵(英文)

本文研究了由特征值唯一确定的3×3实Hankel矩阵.借助于M.Fielder[1]的结论并经过细致的讨论,得到3×3实Hankel矩阵由其特征值唯一确定的充分必要条件,刻画了3×3实Hankel矩阵的一种特征值性质.     

基于l_∞-模的Hankel矩阵填充的保结构阈值算法

文章基于l_∞-范数的性质及奇异值阈值方法,提出Hankel矩阵填充的一种算法.该算法保证每次迭代产生的填充矩阵是可行的Hankel矩阵,不仅减少了奇异值分解所用的时间,而且获得更精确的填充矩阵.同时,讨论了新算法的收敛性.最后通过数值实验以及简单的图像修复证明新算法比加速邻近梯度算法、阈值的增广Lagrange乘子算法以及基于F-模的Hankel矩阵填充的保结构阈值算法更有效.     

结构矩阵的三角表示及其应用

利用位移铁和交换Hessenberg矩阵代数给出结构矩阵的三角表示,并讨论在Toeplitz矩阵和Toeplitz Hankel矩阵方面的应用。     

对称r-循环矩阵的快速算法和并行算法

借助于快速付立叶变换(FFT),给出了一种判断对称r-循环线性系统是否有解的快速算法,并且在有解的情况下求出其解,该算法的计算复杂度为O(nlogn),且具有很好的并行性,若使用n台处理机并行处理该算法则只需要O(logn)步.当r=0时,对称r-循环矩阵变成一个上三角型Hankel矩阵,我们也给出了此类矩阵求逆的一种算法.最后将该算法推广到线性同余系统,其运算量仅为O(nlogn).     

加权Motzkin序列的Hankel行列式

基于经典的Motzkin路引入了一类新的加权Motzkin路的定义,用这种路给出了一类指数型Riordan矩阵的组合解释,得到了相应的Riordan矩阵第0列元素(加权Motzkin序列)的加法公式.作为应用,得到了一类加权Motzkin序列的Hankel行列式的计算方法.     

加权的最小二乘形式矩阵Padé-型逼近

当最小二乘形式矩阵Pade-型逼近(LSMPTA)中Hankel矩阵呈病态时,其逼近解往往很不稳定.通过引入适当的权因子矩阵,将LSMPTA转化为与之等价且稳定性较高的一种新的LSMPTA,即加权的最小二乘形式矩阵Pade-型逼近,并给出了最佳权因子的选择标准.最后,通过数值实例说明了该方法的有效性.     

本刊英文版Vol.35(2019),No.12论文摘要（英文）

Asymptotic and Partial Asymptotic Hankel Operators on H^2(D^n)Anuradha GUPTA Bhawna GUPTA Abstract In this paper,we generalize the concept of asymptotic Hankel operators on H^2(D)to the Hardy space H^2(D^n)(over polydisk)in terms of asymptotic Hankel and partial asymptotic Hankel operators and investigate some properties in case of its weak and strong convergence.     

Hankel算子乘积的两个问题

设f、u和g是单位圆周Hardy空间H~2中的函数,h是单位圆周上平方可积的函数,■和H_h都是从单位圆周Hardy空间H~2到其正交补空间(H~2)~⊥上有界的Hankel算子.本文得到了 3个Hankel算子的乘积等于一个Hankel算子(即■)成立的充分必要条件,以及■成立的充分必要条件.     

无限广义块Toeplitz和Hankel矩阵求逆的统一方法

利用 Sylvester位移方程的统一办法给出所谓的无限广义块Toeplitz和 Hankel矩阵的求逆公式 .     

Model order determination using the Hankel matrix of impulse responses

This letter studies identification problems of model orders using the Hankel matrix of impulse responses of a system and presents two order identification methods: one is based on the singularities or ratios of the Hankel matrix determinants and the other is based on the singular value decomposition of the Hankel matrix. A numerical example verifies the proposed methods.     

On two particular cases of solving the normal Hankel problem

The normal Hankel problem is one of characterizing all the complex matrices that are normal and Hankel at the same time. The matrix classes that can contain normal Hankel matrices admit a parameterization by real 2 × 2 matrices with determinant one. Here, the normal Hankel problem is solved in the case where the characteristic matrix of a given class is an order two Jordan block for the eigenvalue 1 or ?1.     

Block diagonalization and LU-equivalence of Hankel matrices

This article presents a new algorithm for obtaining a block diagonalization of Hankel matrices by means of truncated polynomial divisions, such that every block is a lower Hankel matrix. In fact, the algorithm generates a block LU-factorization of the matrix. Two applications of this algorithm are also presented. By the one hand, this algorithm yields an algebraic proof of Frobenius’ Theorem, which gives the signature of a real regular Hankel matrix by using the signs of its principal leading minors. On the other hand, the close relationship between Hankel matrices and linearly recurrent sequences leads to a comparison with the Berlekamp–Massey algorithm.     

Quasidirect decompositions of Hankel and Toeplitz matrices

Decompositions, over an algebraically closed field, of a Hankel matrix into a sum of Hankel matrices the sum of the ranks of which is equal to the rank of the original matrix, are completely described. Similar results hold for Toeplitz matrices.     

Computation of matrix-valued formally orthogonal polynomials and applications

We present a computational procedure for generating formally orthogonal polynomials associated with a given bilinear Hankel form with rectangular matrix-valued moments. Our approach covers the most general case of moments of any size and is not restricted to square moments. Moreover, our algorithm has a built-in deflation procedure to handle linearly dependent or almost linearly dependent columns and rows of the block Hankel matrix associated with the bilinear form. Possible singular or close-to-singular leading principal submatrices of the deflated block Hankel matrix are avoided by means of look-ahead techniques. Applications of the computational procedure to eigenvalue computations, reduced-order modeling, the solution of multiple linear systems, and the fast solution of block Hankel systems are also briefly described.     

调和Dirichlet空间上Toeplitz算子与小Hankel算子的交换性

本文研究了调和Dirichlet空间上调和符号的Toeplitz算子与小Hankel算子交换性的问题.利用算子矩阵表示的方法,获得了调和Dirichlet空间上调和符号的Toeplitz算子与小Hankel算子交换的充要条件,将Dirichlet空间上的相应结果推广到了调和Dirichlet空间上.     

Hankel matrices for system identification

The coefficients of a linear system, even if it is a part of a block-oriented nonlinear system, normally satisfy some linear algebraic equations via Hankel matrices composed of impulse responses or correlation functions. In order to determine or to estimate the coefficients of a linear system it is important to require the associated Hankel matrix be of row-full-rank. The paper first discusses the equivalent conditions for identifiability of the system. Then, it is shown that the row-full-rank of the Hankel matrix composed of impulse responses is equivalent to identifiability of the system. Finally, for the row-full-rank of the Hankel matrix composed of correlation functions, the necessary and sufficient conditions are presented, which appear slightly stronger than the identifiability condition. In comparison with existing results, here the minimum phase condition is no longer required for the case where the dimension of the system input and output is the same, though the paper does not make such a dimensional restriction.     

On the skew-symmetric part of the Toeplitz component in the real normal (<Emphasis Type="Italic">T</Emphasis> + <Emphasis Type="Italic">H</Emphasis>)-problem

The real normal Toeplitz-plus-Hankel problem is to characterize the matrices that can be represented as sums of two real matrices of which one is Toeplitz and the other Hankel. For a matrix of this type, relations are found between the skew-symmetric part of the Toeplitz component and the matrix obtained by reversing the order of columns in the Hankel component.