一类二次特征值反问题的中心对称解及其最佳逼近

1引言给定n阶实矩阵M,C和K,二次特征值问题:求数λ和非零向量x使得Q(λ)x=0, (1．1)其中Q(λ)=λ2M λC K称为二次束．数λ和相应的非零向量x分别称为二次束Q(λ)的特征值和特征向量．Tisseur和Meerbergen概述了二次特征值问题的各种应用、数学理论和数值方法．在工程技术,特别是结构动力模型修正技术领域经常遇到与二次特征值问题相反的问题(称之为二次特征值反问题)．对阻尼结构进行动力分析时,应用有限元方法可得到系统的质量矩阵M,阻尼矩阵C和刚度矩阵K,从而可求得二次特征值问题的特征值(频率)和特征向量(振型)．但是有限元模型毕竟是实际结构系统的离散化,并且     

解广义特征值反问题的同伦方法

1.引言 我们讨论下列广义特征值反问题: (G)已知B是n×n阶对称半正定矩阵,λ=(λ_1,…,λ_(2n-1))~T∈R~(2n-1),且{λ_i}~(n_3),和{λ_i}_(n+1)~(2n-1)严格交错。问题是欲求一个实对称三对角n×n阶矩阵A,使得λ_1…,λ_n是Ax=λBx的特征值,λ_(n+1),…,λ_(2n-1)是A_(n-1)x=λB_(n-1)x的特征值,其中A_(n-1),B_(n-1)分别是矩阵A,B的前n-1阶主子阵。     

实对称矩阵广义特征值反问题

本文研究如下实对称矩阵广义特征值反问题: 问题IGEP,给定X∈R~(n×m),1=diag(λ_II_k_I,…,λ_pI_k_p)∈R~(n×m),并且λ_I,…,λ_p互异,sum from i=1 to p(k_i=m,求K,M∈SR~(n×n),或K∈SR~(n×n),M∈SR_0~(n×m),或K,M∈SR_0~(n×n),或K∈SR~(n×n),M∈SR_+~(n×n),或K∈SR_0~(n×n),M∈SR_+~(n×n),或K,M∈SR_+~(n×m), (Ⅰ)使得 KX=MXA, (Ⅱ)使得 X~TMX=I_m,KX=MXA,其中SR~(n×n)={A∈R~(n×n)|A~T=A},SR_0~(n×n)={A∈SR~(n×n)|X~TAX≥0,X∈R~n},SR_+~(n×n)={A∈SR~(n×n)|X~TAX>0,X∈R~n,X≠0}. 利用矩阵X的奇异值分解和正交三角分解,我们给出了上述问题的解的表达式.     

加法与乘法逆特征值问题的可解性

1.引言 本文讨论如下代数特征值反问题可解的充分条件: 问题A(加法逆特征值问题)。给定一Hermite矩阵A=(a_(ij))_(n×n)及n个实数λ_1,…,λ_n,求一实对角阵D=diag(c_1…,c_n),使得A+D的特征值为λ_1,…,λ_n。 问题M(乘法逆特征值问题)。给定一正定Hermite矩阵A=(a_(ij))_(n×n)和n个正实数     

代数特征值反问题之解的稳定性分析

考虑如下代数特征值反问题: 问题 G(A;{A_k}_1~n;λ).设 A=(a_(ij)),A_k=(a_(ij)~((k))),k=1,…,n是n+1个n×n的实对称矩阵,λ=(λ_1,…,λ_n)是n维实向量且λ_i≠λ_j,i≠j.求n维实向量c=(c_1,…,c_n)~T,使矩阵A(c)=A+sum from k=1 to n (c_kA_k)的特征值是λ_1,…,λ_n. 这一问题是经典加法问题的推广.当A_k-e_ke_k~~T(e_k是n阶单位阵的第k列)时,     

实对称矩阵的两类逆特征值问题

§gi.两类逆特征值问题先说明一些记号.R~(m×n)是所有m×n实矩阵的全体,R~n=R~(n×1),R=R~1;SR~(n×n)是 所有n×n实对称矩阵的全体;OR~(n×n)是所有n×n实正交矩阵的全体;I~((n))是n阶单位矩阵;A~T是矩阵A的转置;A>0表示A是正定的实对称矩阵.?(A)是矩阵A的列空间;A~+是矩阵A的Moore-Penrose广义逆;P_A=AA~+表示到?(A)的正交投影.λ(A)是A的特征值的全体;λ(K,M)是广义特征值问题K_x=λM_x的特征值的     

QR分解与非线性特征值问题

考察m×n矩阵A(λ),其中元素a_(ij)(λ)均为复(实)变量λ的解析(至少有一阶导数)函数.称此类矩阵为泛函λ-矩阵。特别,当a_(ij)(λ)是λ的多项式时,A(λ)就是熟知的λ-矩阵.给定A(λ)∈C~(n×n)(m=n),有时需确定其非线性特征值及其相应的特征向量,即求满足     

计算对称带状矩阵广义特征值问题的并行分治多分法

1 引 言 本文研究了广义特征值问题 Ax=λBx (1)的并行计算。其中,A,B均为半带宽为r的n阶实对称带状矩阵且其中之一是正定的.本文总假设B是正定的.     

非齐次对称特征值问题

引言 用SR~(n×n)表示所有。n×n实对称矩阵的集合。R~n表示n维线性空间。||·||_2表示向量的Euclid范数或矩阵的谱范数。 本文研究如下问题: 问题ISEP 给定矩阵A∈SR~n×n和向量b∈R~n,求实数λ和向量X∈R~n使得 AX=λX+b, (1) ||X||_2=1. (2) 若b=0,则问题ISEP就是通常的实对称矩阵特征值问题,若b≠0,则问题ISEP称为非齐次对称特征值问题,使(1)和(2)式成立的数λ和向量X分别称为非齐次特征值和相应的非齐     

经典代数特征值反问题的一般提法

§1.引言 [3]曾提出两类Hermiie阵的代数特征值反问题,后来被人们称之为加法问题和乘法问题并推广到更一般的情形.到目前止,经典代数特征值反问题在数学上的最一般提法如下: 问题G.给定n+1个n阶实对称矩阵A,A_1,…,A_n和n个实数λ_1,…,λ_n,求n个实数x_1,…,x_n,使矩阵     

解非对称矩阵特征值问题的一种并行分治算法

１引言考虑矩阵特征值问题其中Ａ是非对称矩阵．通过正交变换（如Ｈｏｕｓｅｈｏｌｄｅｒ变换或Ｇｉｖｅｎｓ变换）,Ａ可化为上Ｈｅｓｓｅｎｂｅｒｇ形．因而,本文假设Ａ为上Ｈｅｓｓｅｎｂｅｒｇ矩阵,表示如下：不失一般性,进一步假设所有的（ｊ＝２,…,ｎ）,即认为Ａ是不可约的关于如何求解上述问题,人们进行了不懈的努力,提出了许多行之有效的算法［１－８］．其中分治算法因具有良好的并行性而引人注目．分治算法的典型代表是基于同伦连续的分治算法［２,３,４］和基于Ｎｅｗｔｏｎ迭代的分治算法［１］．本文提出一种新的分…     

A new type of full multigrid method for the elasticity eigenvalue problem

This paper introduces a new type of full multigrid method for the elasticity eigenvalue problem. The main idea is to avoid solving large scale elasticity eigenvalue problem directly by transforming the solution of the elasticity eigenvalue problem into a series of solutions of linear boundary value problems defined on a multilevel finite element space sequence and some small scale elasticity eigenvalue problems defined on the coarsest correction space. The involved linear boundary value problems will be solved by performing some multigrid iterations. Besides, some efficient techniques such as parallel computing and adaptive mesh refinement can also be absorbed in our algorithm. The efficiency and validity of the multigrid methods are verified by several numerical experiments.     

Inertia‐based spectrum slicing for symmetric quadratic eigenvalue problems

In the quadratic eigenvalue problem (QEP) with all coefficient matrices symmetric, there can be complex eigenvalues. However, some applications need to compute real eigenvalues only. We propose a Lanczos‐based method for computing all real eigenvalues contained in a given interval of large‐scale symmetric QEPs. The method uses matrix inertias of the quadratic polynomial evaluated at different shift values. In this way, for hyperbolic problems, it is possible to make sure that all eigenvalues in the interval have been computed. We also discuss the general nonhyperbolic case. Our implementation is memory‐efficient by representing the computed pseudo‐Lanczos basis in a compact tensor product representation. We show results of computational experiments with a parallel implementation in the SLEPc library.     

Domain decomposition approaches for accelerating contour integration eigenvalue solvers for symmetric eigenvalue problems

This paper discusses techniques for computing a few selected eigenvalue–eigenvector pairs of large and sparse symmetric matrices. A recently developed class of techniques to solve this type of problems is based on integrating the matrix resolvent operator along a complex contour that encloses the interval containing the eigenvalues of interest. This paper considers such contour integration techniques from a domain decomposition viewpoint and proposes two schemes. The first scheme can be seen as an extension of domain decomposition linear system solvers in the framework of contour integration methods for eigenvalue problems, such as FEAST. The second scheme focuses on integrating the resolvent operator primarily along the interface region defined by adjacent subdomains. A parallel implementation of the proposed schemes is described, and results on distributed computing environments are reported. These results show that domain decomposition approaches can lead to reduced run times and improved scalability.     

Calculating the minimal/maximal eigenvalue of symmetric parameterized matrices using projection

In applications of linear algebra including nuclear physics and structural dynamics, there is a need to deal with uncertainty in the matrices. We focus on matrices that depend on a set of parameters ω and we are interested in the minimal eigenvalue of a matrix pencil ( A , B ) with A , B symmetric and B positive definite. If ω can be interpreted as the realization of random variables, one may be interested in statistical moments of the minimal eigenvalue. In order to obtain statistical moments, we need a fast evaluation of the eigenvalue as a function of ω . Because this is costly for large matrices, we are looking for a small parameterized eigenvalue problem whose minimal eigenvalue makes a small error with the minimal eigenvalue of the large eigenvalue problem. The advantage, in comparison with a global polynomial approximation (on which, e.g., the polynomial chaos approximation relies), is that we do not suffer from the possible nonsmoothness of the minimal eigenvalue. The small‐scale eigenvalue problem is obtained by projection of the large‐scale problem. Our main contribution is that, for constructing the subspace, we use multiple eigenvectors and derivatives of eigenvectors. We provide theoretical results and document numerical experiments regarding the beneficial effect of adding multiple eigenvectors and derivatives.     

A note on “Methods for constructing distance matrices and the inverse eigenvalue problem”

In this paper, a symmetric nonnegative matrix with zero diagonal and given spectrum, where exactly one of the eigenvalues is positive, is constructed. This solves the symmetric nonnegative eigenvalue problem (SNIEP) for such a spectrum. The construction is based on the idea from the paper Hayden, Reams, Wells, “Methods for constructing distance matrices and the inverse eigenvalue problem”. Some results of this paper are enhanced. The construction is applied for the solution of the inverse eigenvalue problem for Euclidean distance matrices, under some assumptions on the eigenvalues.     

A property of eigenvalue bounds for a class of symmetric tridiagonal interval matrices

The eigenvalue bounds of interval matrices are often required in some mechanical and engineering fields. In this paper, we consider an interval eigenvalue problem with symmetric tridiagonal matrices. A theoretical result is obtained that under certain assumptions the upper and lower bounds of interval eigenvalues of the problem must be achieved just at some vertex matrices of the interval matrix. Then a sufficient condition is provided to guarantee the assumption to be satisfied. The conclusion is illustrated also by a numerical example. Copyright © 2010 John Wiley & Sons, Ltd.     

The constrained inverse eigenvalue problem and its approximation for normal matrices

In this paper, the constrained inverse eigenvalue problem and associated approximation problem for normal matrices are considered. The solvability conditions and general solutions of the constrained inverse eigenvalue problem are presented, and the expression of the solution for the optimal approximation problem is obtained.     

Efficient initials for computing maximal eigenpair

This paper introduces some efficient initials for a well-known algorithm (an inverse iteration) for computing the maximal eigenpair of a class of real matrices. The initials not only avoid the collapse of the algorithm but are also unexpectedly efficient. The initials presented here are based on our analytic estimates of the maximal eigenvalue and a mimic of its eigenvector for many years of accumulation in the study of stochastic stability speed. In parallel, the same problem for computing the next to the maximal eigenpair is also studied.     

INVERSE EIGENVALUE PROBLEM OF HERMITIAN GENERALIZED ANTI-HAMILTONIAN MATRICES

§1　IntroductionWe considerthe following inverse eigenvalue problem offinding an n-by-n matrix A∈S such thatAxi =λixi,i =1,2 ,...,m,where S is a given set of n-by-n matrices,x1 ,...,xm(m≤n) are given n-vectors andλ1 ,...,λmare given constants.Let X=(x1 ,...,xm) ,Λ=(λ1 ,λ2 ,...,λm) ,then the above inverse eigenvalue problemcan be written as followsProblem Given X∈Cn×m,Λ=(λ1 ,...,λm) ,find A∈S such thatAX =XΛ,where S is a given matrix set.We also discuss the so-called opti…