Inverse Eigenvalue Problems: Theory, Algorithms, and Applications

Transcription

1 Inverse Eigenvalue Problems: Theory, Algorithms, and Applications Moody T. Chu North Carolina State University Gene H. Golub Stanford University OXPORD UNIVERSITY PRESS

2 List of Acronyms List of Figures List of Tables XIV XV xvii 1 Introduction Direct problem Inverse problem Constraints Fundamental issues Nomenclature Summary Applications Overview Pole assignment problem State feedback control Output feedback control Applied mechanics A string with beads Quadratic eigenvalue problem Engineering applications Inverse Sturm-Liouville problem Applied physics Quantum mechanics Neuron transport theory Numerical analysis Preconditioning Numerical ODEs Quadrature rules Signal and data processing Signal processing Computer algebra Molecular structure modelling Principal component analysis, data mining and others Summary 28 Parameterized inverse eigenvalue problems Overview Generic form Variations ix

3 3.2 General results for linear PIEP Existence theory Sensitivity analysis Ideas of computation Newton's method (for LiPIEP2) Projected gradient method (for LiPIEP2) 3.3 Additive inverse eigenvalue problems Solvability Sensitivity and stability (for AIEP2) Numerical methods 3.4 Multiplicative inverse eigenvalue problems Solvability Sensitivity (for MIEP2) Numerical methods 3.5 Summary Structured inverse eigenvalue problems 4.1 Overview 4.2 Jacobi inverse eigenvalue problems Variations Physical interpretations Existence theory Sensitivity issues Numerical methods 4.3 Toeplitz inverse eigenvalue problems Symmetry and parity Existence Numerical methods 4.4 Nonnegative inverse eigenvalue problems Some existence results Symmetrie nonnegative inverse eigenvalue probl Minimum realizable spectral radius 4.5 Stochastic inverse eigenvalue problems Existence Numerical method 4.6 Unitary Hessenberg inverse eigenvalue problems 4.7 Inverse eigenvalue problems with prescribed entries Prescribed entries along the diagonal Prescribed entries at arbitrary locations Additive inverse eigenvalue problem revisit Cardinality and locations Numerical methods 4.8 Inverse Singular value problems Distinct Singular values

4 XI Multiple singular values Rank deficiency Inverse singular/eigenvalue problems The 2 x 2 building block Divide and conquer A symbolic example A numerical example Equality constrained inverse eigenvalue problems Existence and equivalence to PAPs Summary Partially described inverse eigenvalue problems Overview PDIEP for Toeplitz matrices An example General consideration PDIEP for quadratic pencils Recipe of construction Eigenstructure of Q(X) Numerical experiment Monic quadratic inverse eigenvalue problem Real linearly dependent eigenvectors Complex linearly dependent eigenvectors Numerical examples Summary 189 Least Squares inverse eigenvalue problems 6.1 Overview 6.2 ample of MIEP An ex Least Least Summary Squares LiPIEP2 Formulation Equivalence Lift and projection The Newton method Numerical experiment Squares PDIEP Spectrally constrained approximation Overview Spectral constraint Singular value constraint Constrained optimization 216

5 Xll CONTENTS 7.2 Central framework Projected gradient Projected Hessian Applications Approximation with fixed spectrum Toeplitz inverse eigenvalue problem revisit Jacobi-type eigenvalue computation Extensions Approximation with fixed singular values Jacobi-type singular value computation Simultaneous reduction Background review Orthogonal similarity transformation A nearest commuting pair problem Orthogonal equivalence transformation Closest normal matrix problem First-order optimality condition Second-order optimality condition Numerical methods Summary 245 Structured low rank approximation Overview Low rank Toeplitz approximation Theoretical considerations Tracking structured low rank matrices Numerical methods Summary Low rank circulant approximation Preliminaries Basic spectral properties Conjugate-even approximation Algorithm Numerical experiment An application to image reconstruction Summary Low rank covariance approximation Low dimensional random variable approximation TruncatedSVD Summary Euclidean distance matrix approximation Preliminaries Basic formulation, Analytic gradient and Hessian 291

6 Xlll Modification Numerical examples Summary Low rank approximation on unit sphere Linear model Fidelity of low rank approximation Compact form and Stiefel manifold Numerical examples Summary Low rank nonnegative factorization First-order optimality condition Numerical methods An air pollution and emission example Summary Group orbitally constrained approximation Overview A case study Discreteness versus continuousness Generalization General Framework Matrix group and actions Tangent space and projection Canonical form Objective functions Least Squares and projected gradient Systems for other objectives Generalization to non-group structures Summary 358 References 359 Index 381