Isospectral flow method for nonnegative inverse eigenvalue problem with prescribed structure
type: Item
instance of: scholarly article
title: Isospectral flow method for nonnegative inverse eigenvalue problem with prescribed structure
zbMATH Open document ID: 1229.65067
author: Xi Chen
author: Da-li Liu
published in: Journal of Computational and Applied Mathematics
publication date: 2011-06-28T00:00:00Z
review text: The authors develop a numerical isospectral method including convergence results and error estimates for the solution of the following generalized inverse eigenvalue problem: Given a set of complex numbers \(\lambda = \{\lambda_j\} \in {\mathbb{C}, \, j = 1(1)n}\), a subset of index subscript pairs \({(i_{\nu}, j_{\nu})}_{\nu = 1}^l, \, 1 \leqq i_{\nu}, j_{\nu} \leqq n\), and a subset of prescribed values \(a_\nu \in \mathbb{R}, \, a_\nu \geqq 0, \, \nu = 1(1)l\). Find a nonnegative matrix \(X = \{x_{i,j}\} \in {\mathbb{R}}^{n \times n}\) such that \(\lambda (X) = \lambda\) and \(x_{i_\nu, j_\nu} = a_{\nu}, \, \nu = 1(1)l\). A corresponding problem is treated for symmetric nonnegative matrices. The described problems reduce to the common nonnegative inverse eigenvalue problems if the subset of prescribed entries \(\{a_\nu\}\) is empty. The here investigated problems have multiple solutions as the most inverse eigenvalue problems. Roughly speaking, the isospectral method for the reconstruction of the matrix \(X\) consists of the finding the intersection of a set determinated by the spectral constraints \(\{\lambda_j\}\) and a set depending on the structural constraints \(\{a_\nu\}\). The intersection is formulated as the distance between the two sets which is minimized using the steepest descent method. The first set is built of the isospectral surface \(V \Lambda V^{-1}, \, V \in {\mathbb{R}}^{n \times n}\), where \(V\) is nonsingular and \(\Lambda \in {\mathbb{R}}^{n \times n}\) denotes a diagonal block matrix (with blocks of order 1 or 2) with the given spectrum. The second set is constructed by the Hadamard product \(A \circ A\) of the matrix \(A\) built by the prescribed entries. The ill-posedness of nonnegative inverse eigenvalue problems (NIEP) caused by nonunique solutions is transferred to the dependence of solutions of ordinary differential equations on initial values, i.e., different initial values correspond with different solutions of the NIEP. The resul…
reviewed by: Georg Hebermehl
Mathematics Subject Classification ID: 65F18
Mathematics Subject Classification ID: 65K10
Mathematics Subject Classification ID: 15B48
zbMATH DE Number: 5914067
zbMATH Keywords: nonnegative inverse eigenvalue problem
zbMATH Keywords: nonnegative symmetric inverse eigenvalue problem
zbMATH Keywords: structured matrix
zbMATH Keywords: isospectral flow method
zbMATH Keywords: minimization problem
zbMATH Keywords: steepest descent method
zbMATH Keywords: numerical examples
MaRDI profile type: Publication
full work available at URL: j.cam.2011.02.005
OpenAlex ID: W1983654800
cites work: Inverse Eigenvalue Problems
cites work: Structured inverse eigenvalue problems
cites work: Q4326384
cites work: Q4039989
cites work: Q5811557
cites work: Matrices similar to a positive or essentially positive matrix
cites work: A note on eigenvalues of nonnegative matrices
cites work: Eigenvalues of nonnegative symmetric matrices
cites work: Constructing symmetric nonnegative matrices
cites work: An inverse eigenvalue problem for nonnegative matrices
cites work: Eigenvalues of nonnegative matrices
cites work: The nonnegative inverse eigenvalue problem.
cites work: Applications of a Brauer theorem in the nonnegative inverse eigenvalue problem
cites work: Construction of nonnegative symmetric matrices with given spectrum
cites work: Existence and construction of nonnegative matrices with prescribed spectrum
cites work: Q4315443
cites work: Constructing Symmetric Nonnegative Matrices with Prescribed Eigenvalues by Differential Equations
cites work: A Numerical Method for the Inverse Stochastic Spectrum Problem
cites work: Numerical Methods for Solving Inverse Eigenvalue Problems for Nonnegative Matrices
cites work: Inverse matrix eigenvalue problems
cites work: Gradient flow methods for matrix completion with prescribed eigenvalues.
cites work: A note on the construction of a positive oscillatory matrix with a prescribed spectrum
cites work: An introduction to the mathematical theory of inverse problems
cites work: Q3790737
DOI: 10.1016/J.CAM.2011.02.005
Recommended article: Isospectral gradient flows for non-symmetric eigenvalue problem
Recommended article: Q3765915
Recommended article: Isospectral flows and the inverse eigenvalue problem for Toeplitz matrices
Recommended article: Q4841315
Recommended article: Semi-explicit methods for isospectral flows
Recommended article: On solving an isospectral flow
Recommended article: Numerical methods for solving nonnegative inverse singular value problems with prescribed structure
Recommended article: Centrosymmetric isospectral flows and some inverse eigenvalue problems
Recommended article: Isospectral sets and inverse problems for vector-valued Sturm–Liouville equations
Recommended article: An Inverse Iteration Method for Eigenvalue Problems with Eigenvector Nonlinearities
description: scientific article; zbMATH DE number 5914067