Inverse Eigenvalue Problems Theory, Algorithms, and Applications

by Chu, Moody T.; Golub, Gene H.

• ISBN13: 9780198566649
• ISBN10: 0198566646
• Format: Hardcover
• Copyright: 2005-09-02
• Publisher: Oxford University Press

Summary

Inverse eigenvalue problems arise in a remarkable variety of applications and associated with any inverse eigenvalue problem are two fundamental questions-the theoretic issue on solvability and the practical issue on computability. Both questions are difficult and challenging. In this text, the authors discuss the fundamental questions, some known results, many applications, mathematical properties, a variety of numerical techniques as well as several open problems. This is the first book in the authoritative Numerical Mathematics and Scientific Computation series to cover numerical linear algebra, a broad area of numerical analysis. Authored by two world-renowned researchers, this book is aimed at graduates and researchers in applied mathematics, engineering and computer science and makes an ideal graduate text.

Author Biography

Moody Chu is currently an editor of the SIAM Journal on Matrix Analysis and Applications. He is a dedicated educators and has won outstanding teaching awards. Gene Golub has been and continues to be the editor of several important journals (Numerische Mathematik, Linear Algebra and Its Applications, Acta Numerica, etc.) in the field. He is a member of the National Academy of Engineering (1990) and of the National Academy of Sciences. He holds more than a dozen of Honorary Degrees or Honorary Fellows from around the world. He is a dedicated educators and has won outstanding teaching awards.

Table of Contents

List of Acronyms	p. xiv
List of Figures	p. xv
List of Tables	p. xvii
Introduction	p. 1
Direct problem	p. 2
Inverse problem	p. 4
Constraints	p. 4
Fundamental issues	p. 5
Nomenclature	p. 6
Summary	p. 9
Applications	p. 10
Overview	p. 10
Pole assignment problem	p. 11
State feedback control	p. 11
Output feedback control	p. 12
Applied mechanics	p. 13
A string with beads	p. 13
Quadratic eigenvalue problem	p. 15
Engineering applications	p. 17
Inverse Sturm-Liouville problem	p. 18
Applied physics	p. 19
Quantum mechanics	p. 19
Neuron transport theory	p. 20
Numerical analysis	p. 21
Preconditioning	p. 21
Numerical ODEs	p. 22
Quadrature rules	p. 23
Signal and data processing	p. 25
Signal processing	p. 25
Computer algebra	p. 25
Molecular structure modelling	p. 27
Principal component analysis, data mining and others	p. 27
Summary	p. 28
Parameterized inverse eigenvalue problems	p. 29
Overview	p. 29
Generic form	p. 30
Variations	p. 31
General results for linear PIEP	p. 34
Existence theory	p. 34
Sensitivity analysis	p. 39
Ideas of computation	p. 45
Newton's method (for LiPIEP2)	p. 47
Projected gradient method (for LiPIEP2)	p. 52
Additive inverse eigenvalue problems	p. 54
Solvability	p. 56
Sensitivity and stability (for AIEP2)	p. 59
Numerical methods	p. 60
Multiplicative inverse eigenvalue problems	p. 63
Solvability	p. 65
Sensitivity (for MIEP2)	p. 67
Numerical methods	p. 68
Summary	p. 70
Structured inverse eigenvalue problems	p. 71
Overview	p. 71
Jacobi inverse eigenvalue problems	p. 72
Variations	p. 73
Physical interpretations	p. 77
Existence theory	p. 79
Sensitivity issues	p. 81
Numerical methods	p. 82
Toeplitz inverse eigenvalue problems	p. 85
Symmetry and parity	p. 87
Existence	p. 89
Numerical methods	p. 89
Nonnegative inverse eigenvalue problems	p. 93
Some existence results	p. 94
Symmetric nonnegative inverse eigenvalue problem	p. 95
Minimum realizable spectral radius	p. 97
Stochastic inverse eigenvalue problems	p. 103
Existence	p. 104
Numerical method	p. 106
Unitary Hessenberg inverse eigenvalue problems	p. 110
Inverse eigenvalue problems with prescribed entries	p. 112
Prescribed entries along the diagonal	p. 112
Prescribed entries at arbitrary locations	p. 116
Additive inverse eigenvalue problem revisit	p. 117
Cardinality and locations	p. 118
Numerical methods	p. 119
Inverse singular value problems	p. 128
Distinct singular values	p. 129
Multiple singular values	p. 132
Rank deficiency	p. 134
Inverse singular/eigenvalue problems	p. 134
The 2 x 2 building block	p. 136
Divide and conquer	p. 136
A symbolic example	p. 141
A numerical example	p. 142
Equality constrained inverse eigenvalue problems	p. 144
Existence and equivalence to PAPs	p. 144
Summary	p. 145
Partially described inverse eigenvalue problems	p. 146
Overview	p. 146
PDIEP for Toeplitz matrices	p. 147
An example	p. 149
General consideration	p. 150
PDIEP for quadratic pencils	p. 160
Recipe of construction	p. 164
Eigenstructure of Q([lambda])	p. 167
Numerical experiment	p. 173
Monic quadratic inverse eigenvalue problem	p. 178
Real linearly dependent eigenvectors	p. 179
Complex linearly dependent eigenvectors	p. 181
Numerical examples	p. 185
Summary	p. 189
Least squares inverse eigenvalue problems	p. 192
Overview	p. 192
An example of MIEP	p. 193
Least Squares LiPIEP2	p. 194
Formulation	p. 195
Equivalence	p. 197
Lift and projection	p. 199
The Newton method	p. 201
Numerical experiment	p. 203
Least squares PDIEP	p. 209
Summary	p. 211
Spectrally constrained approximation	p. 212
Overview	p. 212
Spectral constraint	p. 212
Singular value constraint	p. 215
Constrained optimization	p. 216
Central framework	p. 217
Projected gradient	p. 219
Projected Hessian	p. 220
Applications	p. 220
Approximation with fixed spectrum	p. 221
Toeplitz inverse eigenvalue problem revisit	p. 223
Jacobi-type eigenvalue computation	p. 225
Extensions	p. 226
Approximation with fixed singular values	p. 226
Jacobi-type singular value computation	p. 229
Simultaneous reduction	p. 229
Background review	p. 230
Orthogonal similarity transformation	p. 234
A nearest commuting pair problem	p. 238
Orthogonal equivalence transformation	p. 239
Closest normal matrix problem	p. 241
First-order optimality condition	p. 242
Second-order optimality condition	p. 243
Numerical methods	p. 244
Summary	p. 245
Structured low rank approximation	p. 246
Overview	p. 246
Low rank Toeplitz approximation	p. 248
Theoretical considerations	p. 248
Tracking structured low rank matrices	p. 254
Numerical methods	p. 257
Summary	p. 263
Low rank circulant approximation	p. 264
Preliminaries	p. 264
Basic spectral properties	p. 266
Conjugate-even approximation	p. 267
Algorithm	p. 273
Numerical experiment	p. 275
An application to image reconstruction	p. 278
Summary	p. 279
Low rank covariance approximation	p. 279
Low dimensional random variable approximation	p. 280
Truncated SVD	p. 285
Summary	p. 286
Euclidean distance matrix approximation	p. 286
Preliminaries	p. 287
Basic formulation	p. 291
Analytic gradient and Hessian	p. 291
Modification	p. 294
Numerical examples	p. 295
Summary	p. 300
Low rank approximation on unit sphere	p. 300
Linear model	p. 302
Fidelity of low rank approximation	p. 306
Compact form and Stiefel manifold	p. 313
Numerical examples	p. 315
Summary	p. 320
Low rank nonnegative factorization	p. 320
First-order optimality condition	p. 322
Numerical methods	p. 324
An air pollution and emission example	p. 332
Summary	p. 337
Group orbitally constrained approximation	p. 339
Overview	p. 339
A case study	p. 341
Discreteness versus continuousness	p. 341
Generalization	p. 343
General Framework	p. 344
Matrix group and actions	p. 344
Tangent space and projection	p. 347
Canonical form	p. 350
Objective functions	p. 352
Least squares and projected gradient	p. 352
Systems for other objectives	p. 354
Generalization to non-group structures	p. 356
Summary	p. 358
References	p. 359
Index	p. 381
Table of Contents provided by Ingram. All Rights Reserved.