Inverse Eigenvalue Problems: Theory and Applications

Transcription

1 Inverse Eigenvalue Problems: Theory and Applications A Series of Lectures to be Presented at IRMA, CNR, Bari, Italy Moody T. Chu (Joint with Gene Golub) Department of Mathematics North Carolina State University June 27, 2001

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3 Preface An inverse eigenvalue problem concerns the reconstruction of a matrix from prescribed spectral data. The spectral data involved may consist of the complete or only partial information of eigenvalues or eigenvectors. The objective of an inverse eigenvalue problem is to construct a matrix that maintains a certain specific structure as well as that given spectral property. Inverse eigenvalue problems arise in a remarkable variety of applications, including system and control theory, geophysics, molecular spectroscopy, particle physics, structure analysis, and so on. Generally speaking, the basic goal of an inverse eigenvalue problem is to reconstruct the physical parameters of a certain system from the knowledge or desire of its dynamical behavior. Since the dynamical behavior often is governed by the underlying natural frequencies and/or normal modes, the spectral constraints are thus imposed. On the other hand, in order that the resulting model is physically realizable, additional structural constraints must also be imposed upon the matrix. Depending on the application, inverse eigenvalue problems appear in many different forms. Associated with any inverse eigenvalue problem are two fundamental questions the theoretic issue on solvability and the practical issue on computability. Solvability concerns obtaining a necessary or a sufficient condition under which an inverse eigenvalue problem has a solution. Computability concerns developing a procedure by which, knowing a priori that the given spectral data are feasible, a matrix can be constructed numerically. Both questions are difficult and challenging. In this note the emphasis is to provide an overview of the vast scope of this fascinating problem. The fundamental questions, some known results, many applications, mathematical properties, a variety of numerical techniques, as well as several open problems will be discussed. This research was supported in part by the National Science Foundation under the grants DMS and DMS The lectures are to be presented at the Istituto per Ricerche di Matematica Applicata (IRMA), Bari, Italy, from June 23 to July 20, 2001, upon the invitation by Fasma Diele. The visit to present this series of lectures is made possible by Professor Roberto Peluso at the IRMA and Professor Dario Bini at the Universita di Pisa with the support from the Il Consiglio Nazionale delle Ricerche (CNR) and the Gruppo Nazionale per il Calcolo Scientifico (GNCS) of the Istituto Nazionale di Alta Matematica (INDAM) under the project Algebra Lineare Numerica per Problemi con Struttura e Applicazioni. The warm kindness and encouragement received from these colleagues are greatly appreciated. MoodyT.Chu Raleigh, North Carolina iii

4 iv PREFACE May, 2001

5 Contents Preface iii 1 Introduction 1 Overview... 2 Inverse Eigenvalue Problem (IEP)... 3 Fundamental Questions... 4 Brief History... 5 Literature Review... 6 Applications... 8 An Example... 9 Classification Via Algebraic Characteristics Via Physical Characteristics A Glimpse of Some Major Issues Complex Solvability Real Solvability Numerical Methods Sensitivity Analysis Summary Applications 27 Pole Assignment Problem State Feedback Control Output Feedback Control Control of Vibration Undamped System Damped System Inverse Sturm-Liouville Problem Matrix Analogue Applied Physics Quantum Mechanics Geophysics Neutron Transport Theory v

6 vi CONTENTS Numerical Analysis Preconditioning High Order Stable Runge-Kutta Schemes Gauss Quadratures Low Rank Approximation Parameterized Inverse Eigenvalue Problems 51 Overview Generic Form Variations General Results Existence Theory for Linear PIEP Complex Solvability Real Solvability (n = m) Multiple Eigenvalue Sensitivity Analysis Forward Problem for General A(c) Inverse Problem for Linear Symmetric A(c) Numerical Methods Direct Method Iterative Methods Continuous Methods Additive Inverse Eigenvalue Problems Subvariations Solvability Issues Sensitivity Issues (for AIEP2) Numerical Methods Newton s Method (for AIEP2) Homotopy Method (for AIEP3) Multiplicative Inverse Eigenvalue Problems Subvariations Solvability Issues Optimal Conditioning by Diagonal Matrices Sensitivity Issues (for MIEP2) Numerical Methods Reformulate MIEP1 as Nonlinear Equations Newton s Method (for MIEP2) Structured Inverse Eigenvalue Problems 97 Jacobi Inverse Eigenvalue Problems Overview Subvariations Physical Interpretations...104

7 CONTENTS vii Existence Theory Sensitivity Issues Numerical Methods Lanczos Method (for SIEP6a) Orthogonal Reduction Method (for SIEP6a) Toeplitz Inverse Eigenvalue Problem Overview Spectral Properties of Centrosymmetric Matrices A 3 3 Example Inverse Problem for Centrosymmetric Matrices Existence Numerical Methods Continuous Method Refined Newton to Centrosymmetric Structure Numerical Experiment Conclusion Nonnegative Inverse Eigenvalue Problem Overview Some Existence Results Symmetric Nonnegative Inverse Eigenvalue Problem Numerical Method Stochastic Inverse Eigenvalue Problem General View Karpelevič s Theorem Relation to Nonnegative Matrices Basic Formulation Steepest Descent Flow ASVD flow Convergence Numerical Experiment Conclusion Unitary Inverse Eigenvalue Problem Overview Formulation Existence Theory Inverse Eigenvalue Problems with Prescribed Entries Overview Prescribed Entries along the Diagonal Schur-Horn Theorem Mirsky Theorem Sing-Thompson Theorem de Oliveira Theorem Prescribed Entries at Arbitrary Locations...188

8 viii CONTENTS Cardinality and Locations Numerical Methods Inverse Singular Value Problems IEP versus ISVP Existence Question A Continuous Approach An Iterative Approach for ISVP Inverse Singualar/Eigenvalue Problem Overview Weyl-Horn Theorem A Recursive Algorithm The 2 2 Case Ideas in Horn s Proof Key to the Algorithmic Success Outline of Proof A MATLAB Program Matrix Structure Correct the Mistake A Variation of Horn s Proof A Symbolic Example Numerical Experiment Rosser Test Wilkinson Test Conclusion Least Squares Inverse Eigenvalue Problems 249 Overview Formulation Least Squares Approximating the Spectrum Least Squares Approximating the Structure Main Theorem One Particular Case Geometric Sketch Algorithm Partially Described Inverse Eigenvalue Problems 259 Overview Generic Form PDIEP for Toeplitz Matrices...262

9 CONTENTS ix 7 Spectrally Constrained Approximation 265 Overview Spectrally Constrained Problem Singular-Value Constrained Problem Reformulation Feasible Set O(n) & Gradient of F The Projected Gradient An Isospectral Descent Flow The Second Order Derivative Least Squares Approximation Sorting Property Wielandt-Hoffman Theorem Toeplitz Inverse Eigenvalue Problem Toeplitz Annihilator Eigenvalue Computation Simultaneous Reduction Structured Low Rank Approximation 283 Overview Difficulties An Example of Existence Another Hidden Catch Low Rank Toeplitz Approximation A General Remark Algebraic Structure Constructing Lower Rank Toeplitz Matrices Lift and Project Algorithm Implicit Optimization Numerical Experiment Explicit Optimization An Illustration Using Existing Optimization Codes Conclusions Low Rank Circulant Approximation Basic Properties Elementary Spectral Properties (Inverse) Eigenvalue Problem Conjugate Evenness Low Rank Approximation Trivial O(n log n) TSVD Algorithm Data Matching Problem An Example using a Tree Structure Numerical Experiment...322

10 x CONTENTS Example 1: Symmetric Illustration Example 2: Complex Illustration Example 3: Perturbed Case Conclusion Lor Rank Covarance Approximation Euclidian Distance Matrix Approximation Approximate GCD...331