Numerical Methods for General and Structured Eigenvalue Problems (eBook)

Preface 6
Contents 9
1 The QR Algorithm 13
1.1 The Standard Eigenvalue Problem 14
1.2 Perturbation Analysis 15
1.3 The Basic QR Algorithm 30
1.4 Balancing 47
1.5 Block Algorithms 51
1.6 Advanced Deflation Techniques 65
1.7 Computation of Invariant Subspaces 69
1.8 Case Study: Solution of an Optimal Control Problem 75
2 The QZ Algorithm 79
2.1 The Generalized Eigenvalue Problem 80
2.2 Perturbation Analysis 82
2.3 The Basic QZ Algorithm 88
2.4 Balancing 103
2.5 Block Algorithms 105
2.6 Aggressive Early Deflation 117
2.7 Computation of Deflating Subspaces 120
3 The Krylov-Schur Algorithm 124
3.1 Basic Tools 125
3.2 Restarting and the Krylov-Schur Algorithm 130
3.3 Balancing Sparse Matrices 137
4 Structured Eigenvalue Problems 142
4.1 General Concepts 143
4.2 Products of Matrices 157
4.3 Skew-Hamiltonian and Hamiltonian Matrices 186
4.4 Skew-Hamiltonian Matrices 192
4.5 Hamiltonian matrices 202
4.6 A Bouquet of Other Structures 220
A Background in Control Theory 226
A.1 Basic Concepts 226
A.2 Balanced Truncation Model Reduction 230
A.3 Linear-Quadratic Optimal Control 231
A.4 Distance Problems 232
B Software 235
B.1 Computational Environment 235
B.2 Flop Counts 235
B.3 Software for Standard and Generalized Eigenvalue Problems 236
B.4 Software for Structured Eigenvalue Problems 238
References 242
Index 261