THE STABILITY OF A KIND OF DISCRETE-TIME HOPFIELD NEURAL NETWORKS WITH ASYMPTOTICAL WEIGHTED MATRICES

In this paper, the authors analyze the stability of a kind of discrete-time Hopfield neural network with asymptotical weighted matrix, which can be expressed as the product of a positive definite diagonal matrix and a symmetric matrix. We obtain that it has asymptotically stable equilibriums if the network is updated asynchronously, and asymptotically stable equilibriums or vibrating final stage with 2 period if updated synchronously. To prove these, Lassale's invariance principle in difference equation is applied.     

关于非线性矩阵方程$X^s-A^*X^{-t}A=Q$的Hermitian正定解及其扰动估计

In this paper, the Hermitian positive definite solutions of the nonlinear matrix equation X^s - A^＊X^-tA = Q are studied, where Q is a Hermitian positive definite matrix, s and t are positive integers. The existence of a Hermitian positive definite solution is proved. A sufficient condition for the equation to have a unique Hermitian positive definite solution is given. Some estimates of the Hermitian positive definite solutions are obtained. Moreover, two perturbation bounds for the Hermitian positive definite solutions are derived and the results are illustrated by some numerical examples.     

CONDITION NUMBERS OF A MULTIPLE EIGENVALUE

This paper studies the symmetric eigenvalue problem Ax = λBx, where A,B are real symmetric matrices, and B is positive definite. Condition numbers of a multiple eigenvalue are derived from the first order perturbation expansions.     

ON THE BEST APPROXIMATION MATRIX PROBLEM AND MATRIX FOURIER SERIES

In this paper the concept of positive definite bilinear matrix moment functional. acting on the space of all the matrix valued continuous functions defined on a bounded interval [a,b], is introduced. The best approximation matrix problem with respect to such a functional is solved in terms of matrix Fourier series. Basic properties of matrix Fourier series such as the Kiemann -Lebesgue matrix property and the bessel-parseval matrix inequality are proved. The concept of total set vjith respect to a positive definite matrix functional is introduced , and the totallity of an orthonormal sequence of matrix polynomials with respect to the functional, is established.     

Bi-block positive semidefiniteness of bi-block symmetric tensors

The positive definiteness of elasticity tensors plays an important role in the elasticity theory.In this paper,we consider the bi-block symmetric tensors,which contain elasticity tensors as a subclass.First,we define the bi-block M-eigenvalue of a bi-block symmetric tensor,and show that a bi-block symmetric tensor is bi-block positive(semi)definite if and only if its smallest bi-block M-eigenvalue is(nonnegative)positive.Then,we discuss the distribution of bi-block M-eigenvalues,by which we get a sufficient condition for judging bi-block positive(semi)definiteness of the bi-block symmetric tensor involved.Particularly,we show that several classes of bi-block symmetric tensors are bi-block positive definite or bi-block positive semidefinite,including bi-block(strictly)diagonally dominant symmetric tensors and bi-block symmetric(B)B0-tensors.These give easily checkable sufficient conditions for judging bi-block positive(semi)definiteness of a bi-block symmetric tensor.As a byproduct,we also obtain two easily checkable sufficient conditions for the strong ellipticity of elasticity tensors.     

THE MINIMAX DIRECTION FOR THE DIRECT PRODUCT OF A CONVEX CONE WITH ITS APPLICATION TO TESTING PROBLEMS

Let C be a closed convex cone in R~k and let C~p be the p-th direct product of C.Thispaper gives some results of the minimax direction with respect to C~p and an innerproduct based on(?),where is a k×k diagonal matrix with positive diagonalelements,is a p×p positive definite matrix and is the Kronecker product ofand.It is also shown that the results may be applied to test the homogeneity of knormal mean vectors where the mean vectors are restricted by a given partial order.     

Solving Large Sparse Linear and Quadratic Generalized Eigenvalue Problems

We study the methods for solving the following large order eigenvelue problems occurring in the analysis of structural vibration: (1) (2) and (3) where M and C are both symmetric matrices, while (?) is skew symmetric. Moreover, M is positive definite, and the matrix K in (2) and (3) is also assumed to be symmetric positive definite. 1 The eigenvalue problem for normal matrices in generalized inner prduct Let B be an n×n Hermitian positive definite matrix. Then     

Hypersurfaces with isotropic para-Blaschke tensor

Let Mnbe an n-dimensional submanifold without umbilical points in the(n + 1)-dimensional unit sphere Sn+1.Four basic invariants of Mnunder the Moebius transformation group of Sn+1are a 1-form Φ called moebius form,a symmetric(0,2) tensor A called Blaschke tensor,a symmetric(0,2) tensor B called Moebius second fundamental form and a positive definite(0,2) tensor g called Moebius metric.A symmetric(0,2) tensor D = A + μB called para-Blaschke tensor,where μ is constant,is also an Moebius invariant.We call the para-Blaschke tensor is isotropic if there exists a function λ such that D = λg.One of the basic questions in Moebius geometry is to classify the hypersurfaces with isotropic para-Blaschke tensor.When λ is not constant,all hypersurfaces with isotropic para-Blaschke tensor are explicitly expressed in this paper.     

A PROCESS FOR SOLVING A FEW EXTREME EIGENPAIRS OF LARGE SPARSE POSITIVE DEFINITE GENERALIZED EIGENVALUE PROBLEM

1. IntroductionWe are concerned in this work with finding a few extreme eigenvalues and theircorresponding eigenvectors of a generalized large scale eigenvalue problem in which thematrices are sparse and symmetric positive definite.Although finding a few extreme eigenpairs is of interest both in theory and practice,there are only few usable and efficient methods up to now. Reinsch and Baner ([12]),suggested a oR algorithm with Newton shift for the standard eigenproblem which included an ingen…     

ON HERMITIAN POSITIVE DEFINITE SOLUTION OF NONLINEAR MATRIX EQUATION X＋A^＊X^-2A=Q

Based on the fixed-point theory, we study the existence and the uniqueness of the maximal Hermitian positive definite solution of the nonlinear matrix equation X＋A^＊X^-2A=Q, where Q is a square Hermitian positive definite matrix and A＊ is the conjugate transpose of the matrix A. We also demonstrate some essential properties and analyze the sensitivity of this solution. In addition, we derive computable error bounds about the approximations to the maximal Hermitian positive definite solution of the nonlinear matrix equation X＋A^＊X^-2A=Q. At last, we further generalize these results to the nonlinear matrix equation X＋A^＊X^-nA=Q, where n≥2 is a given positive integer.     

A note on Inverse Iteration

Inverse iteration, if applied to a symmetric positive definite matrix, is shown to generate a sequence of iterates with monotonously decreasing Rayleigh quotients. We present sharp bounds from above and from below which highlight inverse iteration as a descent scheme for the Rayleigh quotient. Such estimates provide the background for the analysis of the behaviour of the Rayleigh quotient in certain approximate variants of inverse iteration. Copyright © 2004 John Wiley & Sons, Ltd.     

A geometric theory for preconditioned inverse iteration applied to a subspace

The aim of this paper is to provide a convergence analysis for a preconditioned subspace iteration, which is designated to determine a modest number of the smallest eigenvalues and its corresponding invariant subspace of eigenvectors of a large, symmetric positive definite matrix. The algorithm is built upon a subspace implementation of preconditioned inverse iteration, i.e., the well-known inverse iteration procedure, where the associated system of linear equations is solved approximately by using a preconditioner. This step is followed by a Rayleigh-Ritz projection so that preconditioned inverse iteration is always applied to the Ritz vectors of the actual subspace of approximate eigenvectors. The given theory provides sharp convergence estimates for the Ritz values and is mainly built on arguments exploiting the geometry underlying preconditioned inverse iteration.     

求解一类分块二阶线性方程组的QHSS迭代方法

本文将QHSS迭代方法运用于求解一类分块二阶线性方程组.通过适当地放宽QHSS迭代方法的收敛性条件,我们给出了用QHSS迭代方法求解一类分块二阶线性方程组的具体迭代格式,并证明了当系数矩阵中的(1,1)块对称半正定时该QHSS迭代方法的收敛性.我们还用数值实验验证了QHSS迭代方法的可行性和有效性.     

Modulus-Based GSTS Iteration Method for Linear Complementarity Problems

In this paper, a modulus-based generalized skew-Hermitian triangular splitting (MGSTS) iteration method is present for solving a class of linear complementarity problems with the system matrix either being an $H_+$-matrix with non-positive off-diagonal entries or a symmetric positive definite matrix. The convergence of the MGSTS iteration method is studied in detail. By choosing different parameters, a series of existing and new iterative methods are derived, including the modulus-based Jacobi (MJ) and the modulus-based Gauss-Seidel (MGS) iteration methods and so on. Experimental results are given to show the effectiveness and feasibility of the new method when it is employed for solving this class of linear complementarity problems.     

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.     

A CLASS OF NEW PARALLEL HYBRID ALGEBRAIC MULTILEVEL ITERATIONS

1. IntroductionConsider the large sparse system of linear equationsAx = b, (1.1)where, for a fixed positive integer cr, A e L(R") is a symmetric positive definite (SPD) matrir,having the bloCked formx,b E R" are the uDknwn and the known vectors, respectively, having the correspondingblocked formsni(ni S n, i = 1, 2,', a) are a given positthe integers, satisfying Z ni = n. This systemi= 1of linear equations often arises in sultable finite element discretizations of many secondorderseifad…     

Modulus‐based matrix splitting iteration methods for linear complementarity problems

For the large sparse linear complementarity problems, by reformulating them as implicit fixed‐point equations based on splittings of the system matrices, we establish a class of modulus‐based matrix splitting iteration methods and prove their convergence when the system matrices are positive‐definite matrices and H₊‐matrices. These results naturally present convergence conditions for the symmetric positive‐definite matrices and the M‐matrices. Numerical results show that the modulus‐based relaxation methods are superior to the projected relaxation methods as well as the modified modulus method in computing efficiency. Copyright © 2009 John Wiley & Sons, Ltd.     

ON HERMITIAN AND SKEW-HERMITIAN SPLITTING ITERATION METHODS FOR CONTINUOUS SYLVESTER EQUATIONS

We present a Hermitian and skew-Hermitian splitting (HSS) iteration method for solving large sparse continuous Sylvester equations with non-Hermitian and positive definite/semi-definite matrices. The unconditional convergence of the HSS iteration method is proved and an upper bound on the convergence rate is derived. Moreover, to reduce the computing cost, we establish an inexact variant of the HSS iteration method and analyze its convergence property in detail. Numerical results show that the HSS iteration method and its inexact variant are efficient and robust solvers for this class of continuous Sylvester equations.     

On inexact alternating direction implicit iteration for continuous Sylvester equations

In this paper, we study the alternating direction implicit (ADI) iteration for solving the continuous Sylvester equation AX + XB = C , where the coefficient matrices A and B are assumed to be positive semi‐definite matrices (not necessarily Hermitian), and at least one of them to be positive definite. We first analyze the convergence of the ADI iteration for solving such a class of Sylvester equations, then derive an upper bound for the contraction factor of this ADI iteration. To reduce its computational complexity, we further propose an inexact variant of the ADI iteration, which employs some Krylov subspace methods as its inner iteration processes at each step of the outer ADI iteration. The convergence is also analyzed in detail. The numerical experiments are given to illustrate the effectiveness of both ADI and inexact ADI iterations.