特征值反问题的唯一性

证明了由特征值及特征向量反求矩阵时,特征值在对角矩阵中的排序可以是任意的,只须将对应特征向量作相应排序,所得矩阵唯一。对于重特征值的线性无关的特征向量可任意选取,所得矩阵唯一。     

Inverse Problems on the Least Eigenvalue

Some uniqueness theorems on the least eigenvalue are provided for wide classes of self-adjoint operators: differential operators with operator-valued potentials, higher-order partial differential operators and the p-Laplacian.     

Algorithms for Inverse Eigenvalue Problems

Two new algorithms based on QR decompositions (QRDs) (with column pivoting) are proposed for solving inverse eigenvalue problems, and under some non-singularity assumptions they are both locally quadratically convergent. Several numerical tests are presented to illustrate their convergence behavior.     

On the General Algebraic Inverse Eigenvalue Problems

A number of new results on sufficient conditions for the solvability and numerical algorithms of the following general algebraic inverse eigenvalue problem are obtained: Given $n+1$ real $n\times n$ matrices $A=(a_{ij}),A_k=(a_{ij}^{(k)})(k=1,2,\cdots,n)$ and $n$ distinct real numbers $\lambda_1,\lambda_2,\cdots,\lambda_n,$ find $n$ real number $c_1,c_2,\cdots,c_n$ such that the matrix $A(c)=A+\sum\limits_{k=1}^{n}c_k A_k$ has eigenvalues $\lambda_1,\lambda_2,\cdots,\lambda_n.$     

Inverse Eigenvalue Problems for Smooth Potential

We consider the inverse eigenvalue problem of the onedimensional Schrödinger operator for finite intervals. We give sufficient conditions for finitely many partially known spectra and partial information on the potential to determine the Schrodinger operator on the whole interval.     

Inverse Eigenvalue Problems on Directed Graphs

The differential operators and are constructed on certain finite directed weighted graphs. Two types of inverse spectral problems are considered. First, information about the graph weights and boundary conditions is extracted from the spectrum of . Second, the compactness of isospectral sets for is established by computation of the residues of the zeta function.

     

The Unsolvability of Inverse Algebraic Eigenvalue Problems Almost Everywhere

This paper revises the definition for the unsolvability of inverse algebraic eigenvalue problems almost everywhere (a.e.) given by Shapiro [5], and gives some sufficient and necessary conditions such that the inverse algebraic eigenvalue problems are unsolvable a.e.     

Inverse Eigenvalue Problems in Structural Dynamics

We consider the numerical solution of inverse eigenvalue problems (IEP). Two different formulations are used. The methods are applied to a certain mechanical system. Due to the special structure of the mass and stiffness matrix we benefit from a secular equation. The roots of that equation are the eigenvalues of the system. (© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

The Unsolvability of Multiplicative Inverse Eigenvalue Problems Almost Everywhere

The idea and technique used in [7] are applied to the multiplicative inverse eigenvalue problems as well. Some sufficient and necessary conditions that the multiplicative inverse eigenvalue problems be unsolvable almost everywhere are given. The results are similar to those of [7], but the proofs are more complicated.     

二阶微分方程的逆特征值问题

运用渐近分析的方法及Rayleigh商原理,将Sturm-Liouville问题的Ambarzumyan定理推广到具有Neumann边界条件或拟周期边界条件的二阶微分方程情形.同时,获得了二阶向量微分方程的有关Ambarzumyan型结果.     

二阶微分方程的逆特征值问题

运用渐近分析的方法及Rayleigh商原理,将Sturm-Liouville问题的Ambarzumyan定理推广到具有Neumann边界条件或拟周期边界条件的二阶微分方程情形.同时,获得了二阶向量微分方程的有关Ambarzumyan型结果.     

Nonlinear Eigenvalue Problems in the Stability Analysis of Morphogen Gradients

This paper is concerned with several eigenvalue problems in the linear stability analysis of steady state morphogen gradients for several models of Drosophila wing imaginal discs including one not previously considered. These problems share several common difficulties including the following: (a) The steady state solution which appears in the coefficients of the relevant differential equations of the stability analysis is only known qualitatively and numerically. (b) Though the governing differential equations are linear, the eigenvalue parameter appears nonlinearly after reduction to a problem for one unknown. (c) The eigenvalues are determined not only as solutions of a homogeneous boundary value problem with homogeneous Dirichlet boundary conditions, but also by an alternative auxiliary condition to one of the Dirichlet conditions allowed by a boundary condition of the original problem. Regarding the stability of the steady state morphogen gradients, we prove that the eigenvalues must all be positive and hence the steady state morphogen gradients are asymptotically stable. The other principal finding is a novel result pertaining to the smallest (positive) eigenvalue that determines the slowest decay rate of transients and the time needed to reach steady state. Here we prove that the smallest eigenvalue does not come from the nonlinear Dirichlet eigenvalue problem but from the complementary auxiliary condition requiring only to find the smallest zero of a rational function. Keeping in mind that even the steady state solution needed for the stability analysis is only known numerically, not having to solve the nonlinear Dirichlet eigenvalue problem is both an attractive theoretical outcome and a significant computational simplification.     

Inexact Inverse Iteration for Generalized Eigenvalue Problems

In this paper, we study an inexact inverse iteration with inner-outer iterations for solving the generalized eigenvalu problem Ax = Bx, and analyze how the accuracy in the inner iterations affects the convergence of the outer iterations. By considering a special stopping criterion depending on a threshold parameter, we show that the outer iteration converges linearly with the inner threshold parameter as the convergence rate. We also discuss the total amount of work and asymptotic equivalence between this stopping criterion and a more standard one. Numerical examples are given to illustrate the theoretical results.     

Direct and Inverse Eigenvalue Problems for Diagonal-Plus-Semiseparable Matrices

In this paper we study both direct and inverse eigenvalue problems for diagonal-plus-semiseparable (dpss) matrices. In particular, we show that the computation of the eigenvalues of a symmetric dpss matrix can be reduced by a congruence transformation to solving a generalized symmetric definite tridiagonal eigenproblem. Using this reduction, we devise a set of recurrence relations for evaluating the characteristic polynomial of a dpss matrix in a stable way at a linear time. This in turn allows us to apply divide-and-conquer eigenvalue solvers based on functional iterations directly to dpss matrices without performing any preliminary reduction into a tridiagonal form. In the second part of the paper, we exploit the structural properties of dpss matrices to solve the inverse eigenvalue problem of reconstructing a symmetric dpss matrix from its spectrum and some other informations. Finally, applications of our results to the computation of a QR factorization of a Cauchy matrix with real nodes are provided.     

对称双随机矩阵逆特征值问题

主要研究随机矩阵逆特征值问题．特别是对称双随机矩阵和列随机矩阵逆特征值问题．对参考文献[1]与[2]的结论作了一些推广．并给出了—个数值例子．     

带比例关系的实带状矩阵特征值反问题

研究了通过矩阵A的顺序主子矩阵A_((k))=(aij)_(i,j=1)^(n-k+1)的特征值{λ_i(n-k+1)的特征值{λ_i^((k)))}_(i=1)((k)))}_(i=1)^(n-k+1)k=1,2,…,r+1来构造一个带比例关系的实带状矩阵的特征值反问题.对当特征值{λ_i(n-k+1)k=1,2,…,r+1来构造一个带比例关系的实带状矩阵的特征值反问题.对当特征值{λ_i^((k))}_(i=1)((k))}_(i=1)^(n-k+1)中有多重特征值出现时,应当如何来构造这类矩阵进行了讨论,并给出了问题的具体算法及数值例子.     

一类广义特征值反问题

本文提出了一个实对称带状矩阵的广义特征值反问题,并且证明了对于Jacobi矩阵和一般对称矩阵,问题的存在性.     

结构动力学逆特征值问题的同伦解法

将结构动力学反问题视为拟乘法逆特征值问题,利用求解非线性方程组的同伦方法来解决结构动力学逆特征值问题,这种方法由于沿同伦路径求解,对初值的选取没有本质的要求,算例说明了这种方法是可行的．     

An Approach to Solving Inverse Eigenvalue Problems for Matrices

The paper considers different formulations of inverse eigenvalue problems for matrices whose entries either polynomially or rationally depend on unknown parameters. An approach to solving inverse problems together with numerical algorithms is suggested. The solution of inverse problems is reduced to the problem of finding the so-called discrete solutions of nonlinear algebraic systems. The corresponding systems are constructed using the method of traces, and their discrete roots are found by applying the algorithms for solving nonlinear algebraic systems in several variables previously suggested by the author. Bibliography: 30 titles.