DCT- and DST-based splitting methods for Toeplitz systems

New splitting iterative methods for Toeplitz systems are proposed by means of recently developed matrix splittings based on discrete sine and cosine transforms due to Kailath and Olshevsky [Displacement structure approach to discrete-trigonometric transform-based preconditioners of G. Strang type and of T. Chan type, SIAM J. Matrix Anal. Appl. 26 (2005), pp. 706–734]. Theoretical analysis shows that new splitting iterative methods converge to the unique solution of a symmetric Toeplitz linear system. Moreover, an upper bound of the contraction factor of our new splitting iterations is derived. Numerical examples are reported to illustrate the effectiveness of new splitting iterative methods.     

On the extreme eigenvalues of Toeplitz and real Hankel interval matrices

The purpose of this paper is to evaluate the extreme eigenvalues of a Hermitian Toeplitz interval matrix and a real Hankel interval matrix. A (n×n)-dimensional Hermitian Toeplitz (HT) matrix is determined by the elements of its first row, sayr. If the elements ofr lie in complex intervals (i.e., rectangles of the complex plane), we call the resulting set of matrices an HT interval (HTI) matrix. An HTI matrix can model real world HT matrices where the elements of the vectorr have finite precision (e.g., because of quantization, or imprecise measurement devices). In this paper we prove that the extreme eigenvalues of a given HTI matrix can be easily obtained from the 2^2(n–1) vertex HT matrices where the first element ofr is set to zero. Similarly, as a special case we obtain that the extreme eigenvalues of a real symmetric Toeplitz interval (RSTI) matrix can be obtained from 2 ^n–1 vertex matrices. Based on the above results we provide boxlike bounds for the eigenvalues on non-Hermitian complex and real Toeplitz interval matrices. Finally, we consider a real Hankel interval matrix. We prove that the maximal eigenvalue of a (n×n)-dimensional real Hankel interval matrix can be obtained from a subset of the vertex Hankel matrices containing 2^2n–3 vertex matrices, whereas the minimal eigenvalue can be obtained from another such subset also containing 2^2n–3 vertex matrices.     

用耦合模型法进行水质随机模拟的研究

基于水质变化同时存在的两种规律－确定性规律和随机性规律（前者反映事物必然的内在联系，后者反映外在的随机性因素作用）。建立了一种能同时体现这两种作用的确定性与随机性相耦合的水质模型，并进行了水质变化过程及其概率分布的随机模拟。经实际计算，表明是切实可行的。     

ASYMPTOTIC EFFICIENCY OF THE SAMPLE COVARIANCES IN A GAUSSIAN STATIONARY PROCESS

Abstract. This paper deals with the asymptotic efficiency of the sample autocovariances of a Gaussian stationary process. The asymptotic variance of the sample autocovariances and the Cramer–Rao bound are expressed as the integrals of the spectral density and its derivative. We say that the sample autocovariances are asymptotically efficient if the asymptotic variance and the Cramer–Rao bound are identical. In terms of the spectral density we give a necessary and sufficient condition that they are asymptotically efficient. This condition is easy to check for various spectra.     

主元凝集用于K──树法求有源网络的符号函数

本文利用主元凝集法逐步删减网络节点，从而简化了网络拓扑分析，拓宽了ｋ一树法求有源网络符号函数的范围。     

关于Vandermonde行列式的推广

本文定义了一类推广的Ｖａｎｄｅｒｍｏｎｄｅ行列式并给出了它的计算公式。Ｖａｎｄｅｒｍｏｎｄｅ行列式及其它已知的若干著名行列式是它的特例。     

一种门限可追踪的匿名签密方案

为解决匿名签密算法中签密者身份的追踪问题,提出了一种门限可追踪的匿名签密方案.该方案借助范德蒙行列式生成成员的公钥和私钥,通过在匿名签密过程中附加一些与签密者身份相关的额外信息,实现签密者身份的门限追踪.在不可分模型下,证明了该方案满足匿名性、门限可追踪性、不可伪造性和不相关性.     

Cramer法则的另一应用

求高阶行列式的值一直是数学算法中的一个难题.通过实例对Cramer法则重新进行了探讨,给出了该法则的一种新的应用,并得出了通过构造一个含有n个未知数n个方程的线性方程组计算n阶行列式值的方法.     

A new version of Kovarik’s approximate orthogonalization algorithm without matrix inversion

In 1970 Kovarik proposed approximate orthogonalization algorithms. One of them (algorithm B) has quadratic convergence but requires at each iteration the inversion of a matrix of similar dimension to the initial one. An attempt to overcome this difficulty was made by replacing the inverse with a finite Neumann series expansion involving the original matrix and its adjoint. Unfortunately, this new algorithm loses the quadratic convergence and requires a large number of terms in the Neumann series which results in a dramatic increase in the computational effort per iteration. In this paper we propose a much simpler algorithm which, by using only the first two terms in a different series expansion, gives us the desired result with linear convergence. Systematic numerical experiments for collocation and Toeplitz matrices are also described.     

Trigonometric transform splitting methods for real symmetric Toeplitz systems

In this paper we study efficient iterative methods for real symmetric Toeplitz systems based on the trigonometric transformation splitting (TTS) of the real symmetric Toeplitz matrix $A$. Theoretical analyses show that if the generating function $f$ of the $n\times n$ Toeplitz matrix $A$ is a real positive even function, then the TTS iterative methods converge to the unique solution of the linear system of equations for sufficient large $n$. Moreover, we derive an upper bound of the contraction factor of the TTS iteration which is dependent solely on the spectra of the two TTS matrices involved.Different from the CSCS iterative method in Ng (2003) in which all operations counts concern complex operations when the DFTs are employed, even if the Toeplitz matrix $A$ is real and symmetric, our method only involves real arithmetics when the DCTs and DSTs are used. The numerical experiments show that our method works better than CSCS iterative method and much better than the positive definite and skew-symmetric splitting (PSS) iterative method in Bai et al. (2005) and the symmetric Gauss–Seidel (SGS) iterative method.