特征值反问题的迭代解法

§1.引言 近年来,由于许多应用科学,如地球物理、海洋、地质、声学、光学、量子力学和识别等问题的需要,提出了特征值反问题和广义特征值反问题.这些问题形成一类区别于经典代数特征值问题的复杂非线性问题.这类问题中只有少量在理论上、数值上有一些求解的方法,前人的工作主要集中于sturm-Liouville反问题,见[1,2,3,4].本文讨论下列各种特征值反问题:     

Inverse unitary eigenproblems and related orthogonal functions

Summary. This paper explores the relationship between certain inverse unitary eigenvalue problems and orthogonal functions. In particular, the inverse eigenvalue problems for unitary Hessenberg matrices and for Schur parameter pencils are considered. The Szeg? recursion is known to be identical to the Arnoldi process and can be seen as an algorithm for solving an inverse unitary Hessenberg eigenvalue problem. Reformulation of this inverse unitary Hessenberg eigenvalue problem yields an inverse eigenvalue problem for Schur parameter pencils. It is shown that solving this inverse eigenvalue problem is equivalent to computing Laurent polynomials orthogonal on the unit circle. Efficient and reliable algorithms for solving the inverse unitary eigenvalue problems are given which require only O() arithmetic operations as compared with O() operations needed for algorithms that ignore the structure of the problem. Received April 3, 1995 / Revised version received August 29, 1996     

THE UNSOLVABILITY OF GENERALIZED INVERSE EIGENVALUE PROBLEMS ALMOST EVERYWHERE

In this paper the unsolvability of generalized inverse eigenvalue problems almost everywhere is discussed.We first give the definitions for the unsolvability of generalized inverse eigenvalue problems almost everywhere.Then adopting the method used in [14],we present some sufficient conditions such that the generalized inverse eigenvalue problems are unsohable almost everywhere.     

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．     

The solvability conditions for the inverse eigenvalue problems of reflexive matrices

This paper involves related inverse eigenvalue problems of reflexive matrices and their optimal approximation, the sufficient and necessary conditions under which the solvable problems of inverse eigenvalue, and the general provided form of the solution. Furthermore, the algorithm to compute the optimal approximate solution and some numerical experiments are given.     

Sufficient conditions for the solubility of inverse eigenvalue problems

We define an inverse eigenvalue problem, which contains as special cases the classical additive and multiplicative inverse eigenvalue problems. Using some results on the distance of eigenvalues from matrix diagonal elements and Brouwer's fixed-point theorem, we give sufficient conditions for the solubility of the problem.     

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.     

A geometric theory for preconditioned inverse iteration I: Extrema of the Rayleigh quotient

The discretization of eigenvalue problems for partial differential operators is a major source of matrix eigenvalue problems having very large dimensions, but only some of the smallest eigenvalues together with the eigenvectors are to be determined. Preconditioned inverse iteration (a “matrix-free” method) derives from the well-known inverse iteration procedure in such a way that the associated system of linear equations is solved approximately by using a (multigrid) preconditioner. A new convergence analysis for preconditioned inverse iteration is presented. The preconditioner is assumed to satisfy some bound for the spectral radius of the error propagation matrix resulting in a simple geometric setup. In this first part the case of poorest convergence depending on the choice of the preconditioner is analyzed. In the second part the dependence on all initial vectors having a fixed Rayleigh quotient is considered. The given theory provides sharp convergence estimates for the eigenvalue approximations showing that multigrid eigenvalue/vector computations can be done with comparable efficiency as known from multigrid methods for boundary value problems.     

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.     

Solvability and stability of the inverse Sturm–Liouville problem with analytical functions in the boundary condition

The paper deals with the Sturm–Liouville eigenvalue problem with the Dirichlet boundary condition at one end of the interval and with the boundary condition containing entire functions of the spectral parameter at the other end. We study the inverse problem, which consists in recovering the potential from a part of the spectrum. This inverse problem generalizes partial inverse problems on finite intervals and on graphs and also the inverse transmission eigenvalue problem. We obtain sufficient conditions for global solvability of the studied inverse problem, which prove its local solvability and stability. In addition, application of our main results to the partial inverse Sturm–Liouville problem on the star-shaped graph is provided.     

哈密顿系统的若干逆问题

1.引言 本文讨论一个能量守恒系统的若干逆问题以及逆特征值问题的关系.该系统的数学方程可用下列Hilbert空间H中的微分方程来描述:     

A Newton iteration process for inverse eigenvalue problems

Summary We suppose an inverse eigenvalue problem which includes the classical additive and multiplicative inverse eigenvalue problems as special cases. For the numerical solution of this problem we propose a Newton iteration process and compare it with a known method. Finally we apply it to a numerical example.     

Method for Solving the Sine-Gordon Equation in Laboratory Coordinates

By solving the direct and inverse scattering problems for a rather ‘unconventional’ eigenvalue problem, we can solve the initial value problem for the sine-Gordon equation entirely in laboratory coordinates. This allows us to extend and accurately define the class of initial value problems which can be solved by ‘inverse scattering transforms.’ Simple examples are given to illustrate both the direct and the inverse scattering methods.     

The Stability Analysis of the Solutions of Inverse Eigenvalue Problems

This paper gives perturbation bounds of some solutions of the classical additive and multiplicative inverse eigenvalue problems for real symmetric matrices.     

特征值问题的自适应反迭代有限元算法

1引言设Ω∈R~2为Lipschitz单连通的有界闭区域,X为定义在Ω的Sobolev空间,a(·,·)和b(·,·)为X×X→C的有界双线性或半双线性泛函,考虑变分特征值问题:求(λ,u≠0)∈C×X使得a(u,v)=λb(u,u),(?)u∈X,其中a(·,·)满足X上的"V-强制性"条件或者连续的inf-sup条件,设M_h为Q区域上的正则三角形剖分,X_h∈X为定义在M_h有限元子空间,上述变分问题对应的有限元离散问题为:求(λ_h,u_h)∈R×X,u_h≠0使得     

To solving inverse eigenvalue problems for parametric matrices

Some statements of inverse eigenvalue problems for one-parameter and multiparameter regular polynomial matrices with linear and nonlinear dependences on spectral parameters are considered. Methods for solving inverse eigenvalue problems based on rank factorization, exhaustion, and reduction to nonlinear equations are proposed. Bibliography: 12 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 334, 2006, pp. 174–192.     

Analyzing the convergence factor of residual inverse iteration

We will establish here a formula for the convergence factor of the method called residual inverse iteration, which is a method for nonlinear eigenvalue problems and a generalization of the well-known inverse iteration. The formula for the convergence factor is explicit and involves quantities associated with the eigenvalue to which the iteration converges, in particular the eigenvalue and eigenvector. Residual inverse iteration allows for some freedom in the choice of a vector w _k and we can use the formula for the convergence factor to analyze how it depends on the choice of w _k . We also use the formula to illustrate the convergence when the shift is close to the eigenvalue. Finally, we explain the slow convergence for double eigenvalues by showing that under generic conditions, the convergence factor is one, unless the eigenvalue is semisimple. If the eigenvalue is semisimple, it turns out that we can expect convergence similar to the simple case.     

A NOTE ON THE BACKWARD ERRORS FOR INVERSE EIGENVALUE PROBLEMS

In this note,we consider the backward errors for more general inverse eigenvalus prob-lems by extending Sun‘‘‘‘s approach.The optimal backward errors defined for diagonal-ization matrix inverse eigenvalue problem with respect to an approximate solution,and the upper and lower bounds are derived for the optimal backward errors.The results may be useful for testing the stability of practical algorithms.     

Inverse eigenvalue problems

The classical inverse additive and multiplicative inverse eigenvalue problems for matrices are studied. Using general results on the solvability of polynomial systems it is shown that in the complex case these problems are always solvable by a finite number of solutions. In case of real symmetric matrices the inverse problems are reformulated to have a real solution. An algorithm is given to obtain this solution.     

关于代数特征值反问题对称情况可解的充分条件

§1.引言 本文讨论下述特征值反问题的可解性: 问题 G.设A_0=(a_(ij)~((0)))和A_k=(a_(ij)~((k)))(k=1,…,n)是一组n+1个n×n实对称矩阵,λ_1,…,λ_n是n个不同的实数.求实数c_1,…,c_n使得矩阵A_0+sum from k-1 to n C_k·A_k的特征值为λ_1,…,λ_n. [1]和[2]曾给出此问题可解的充分条件.本文应用Rothe不动点定理[3]给出问题G可解的另外两个充分条件.本文的结果可判定[1]和[2]中定理所不能判定的某些问题