\main" i i 2004/5/6 page i i i Numerical Methods and Software for General and Structured Eigenvalue Problems vorgelegt von Dipl.-Math. Daniel Kre¼ner von der FakultÄat II - Mathematik und Naturwissenschaften der Technischen UniversitÄat Berlin zur Erlangung des akademischen Grades Doktor der Naturwissenschaften - Dr. rer. nat. - genehmigte Dissertation Promotionsausschuss: Vorsitzender: Prof. Dr. GunÄ ter M. Ziegler Berichter: Prof. Dr. Volker Mehrmann Berichter: Prof. Dr. Ralph Byers Berichter: Prof. Dr. Bo KºagstrÄom Tag der wissenschaftlichen Aussprache: 14.05.2004 Berlin 2004 D 83 i i i i \main" i i 2004/5/6 page ii i i i i i i \main" i i 2004/5/6 page iii i i Zusammenfassung Die Dissertationsschrift beschÄaftigt sich mit numerischen Verfahren zur LÄosung allgemeiner und strukturierter Eigenwertprobleme. Weiterhin werden Software- Implementierungen der behandelten numerischen Verfahren vorgestellt. Die Arbeit enthÄalt zahlreiche BeitrÄage zu verschiedenen Aspekten des QR-Algorithmus, des QZ-Algorithmus, des periodischen QR-Algorithmus, verschiedener strukturerhal- tender Algorithmen furÄ (schief-)Hamiltonische Eigenwertprobleme und Arnoldi-Äahn- lichen Verfahren. Abstract In this thesis, we have investigated numerical methods for the solution of general and structured eigenvalue problems. Moreover, we have presented software imple- menting these methods. Contributions have been made to various aspects of the QR algorithm, the QZ algorithm, the periodic QR algorithm, structure-preserving methods for (skew-)Hamiltonian matrices, and the Krylov-Schur algorithm. i i i i \main" i i 2004/5/6 page iv i i i i i i \main" i i 2004/5/6 page v i i Contents Preface ix 1 The QR Algorithm 1 1 The Standard Eigenvalue Problem . 2 2 Perturbation Analysis . 4 2.1 Spectral Projectors and Separation . 4 2.2 Eigenvalues and Eigenvectors . 6 Eigenvalues. Eigenvectors. 2.3 Eigenvalue Clusters and Invariant Subspaces . 10 On the computation of sep. 2.4 Global Perturbation Bounds . 14 3 The Basic QR Algorithm . 17 3.1 Local Convergence . 17 Stationary shifts. Instationary shifts. 3.2 Hessenberg Form . 22 Reduction to Hessenberg form. 3.3 Implicit Shifted QR Iteration . 25 3.4 Deation . 27 3.5 The Overall Algorithm . 28 3.6 Failure of Global Converge . 30 4 Balancing . 32 4.1 Isolating Eigenvalues . 32 4.2 Scaling . 33 4.3 Merits of Balancing . 35 5 Block Algorithms . 36 5.1 Compact WY Representation . 36 5.2 Block Hessenberg Reduction . 37 5.3 Multishifts and Bulge Pairs . 39 5.4 Connection to Pole Assignment . 41 5.5 Tightly Coupled Tiny Bulges . 43 Introducing a chain of bulges. Chasing a chain of bulges. Getting rid o® a chain of bulges. Numerical results. 6 Advanced Deation Techniques . 48 6.1 Aggressive Early Deation . 48 Numerical results. 6.2 Aggressive Early Deation at Top Left Corner . 50 7 Computation of Invariant Subspaces . 52 7.1 Swapping Two Diagonal Blocks . 53 v i i i i \main" i i 2004/5/6 page vi i i vi Contents 7.2 Reordering . 54 7.3 Block Algorithm . 55 Numerical Results. 8 Case Study: Solution of an Optimal Control Problem . 57 2 The QZ Algorithm 61 1 The Generalized Eigenvalue Problem . 62 2 Perturbation Analysis . 63 2.1 Spectral Projectors and Dif . 64 2.2 Local Perturbation Bounds . 65 On the computation of dif. 2.3 Global Perturbation Bounds . 69 3 The Basic QZ Algorithm . 69 3.1 Hessenberg-Triangular Form . 70 3.2 Implicit Shifted QZ Iteration . 72 3.3 On the Use of Householder Matrices . 75 Early attempts. Opposite Householder matrices. The role of RQ decompositions. 3.4 Deation . 79 Deation of in¯nite eigenvalues. 3.5 The Overall Algorithm . 81 4 Balancing . 83 4.1 Isolating Eigenvalues . 84 4.2 Scaling . 84 5 Block Algorithms . 86 5.1 Reduction to Hessenberg-Triangular Form . 86 Stage 1. Stage 2. 5.2 Multishifts and Bulge Pairs . 92 5.3 Deation of In¯nite Eigenvalues Revisited . 93 5.4 Tightly Coupled Tiny Bulge Pairs . 94 Introducing a chain of bulge pairs. Chasing a chain of bulge pairs. Getting rid o® a chain of bulge pairs. Numerical results. 6 Aggressive Early Deation . 97 Numerical results. 3 The Periodic QR Algorithm 101 1 The Periodic Eigenvalue Problem . 102 2 Perturbation Theory . 104 2.1 Eigenvalues . 106 2.2 Invariant Subspaces . 108 On the computation of cA( ). 3 Derivation of the Periodic QR AlgorithmX . 110 3.1 The Perfect Shu²e . 110 The perfect shu²e version of the periodic Schur form. 3.2 Reduction to Hessenberg Form . 111 3.3 QR Iteration . 114 3.4 Deation . 115 3.5 Summary . 116 4 Computation of Invariant Subspaces . 116 5 Further Notes and References . 118 i i i i \main" i i 2004/5/6 page vii i i Contents vii 4 QR-Based Algorithms for (Skew-)Hamiltonian Matrices 121 1 Preliminaries . 123 1.1 Elementary Orthogonal Symplectic Matrices . 123 1.2 The Symplectic QR Decomposition . 125 2 The Skew-Hamiltonian Eigenvalue Problem . 127 2.1 Structured Decompositions . 127 2.2 Structured Condition Numbers for Eigenvalues . 130 2.3 Structured Condition Numbers for Invariant Subspaces . 132 Isotropic invariant subspaces and a quadratic matrix equation. Skew- symmetrizing the bottom part. Solving the decoupled linearized equation. Solving the coupled linearized equation. Solving the quadratic matrix equation. Perturbation bounds and a condition y number. On the computation of TW . Numerical Example. 2.4 An Algorithm for Computing thek Skew-Hamiltoniank Schur De- composition . 143 2.5 Computation of Invariant Subspaces . 145 2.6 Other Algorithms . 145 3 The Hamiltonian Eigenvalue Problem . 145 3.1 Structured Decompositions . 145 3.2 Structured Condition Numbers for Eigenvalues . 146 3.3 Structured Condition Numbers for Invariant Subspaces . 147 3.4 An Explicit Hamiltonian QR Algorithm . 148 3.5 Reordering a Hamiltonian Schur Decomposition . 149 3.6 Algorithms Based on H 2 . 150 3.7 Computation of Invariant Subspaces Based on H 2 . 154 3.8 Re¯nement of Stable Invariant Subspaces . 154 3.9 Other Algorithms . 155 4 Symplectic Balancing . 155 4.1 Isolating Eigenvalues . 156 4.2 Scaling . 157 5 Block Algorithms . 159 5.1 A WY-like representation for products of elementary orthogonal symplectic matrices . 159 5.2 Block Symplectic QR Decomposition . 161 5.3 Block Symplectic URV Decomposition . 162 5.4 Numerical Stability . 165 5.5 Numerical Results . 167 6 Numerical Experiments . 168 6.1 Accuracy of Eigenvalues . 169 6.2 Computing Times . 170 5 Krylov-Schur Algorithms 173 1 Basic Tools . 174 1.1 Krylov Subspaces . 174 1.2 The Arnoldi Method . 176 The Arnoldi Decomposition. The Basic Algorithm. 2 Restarting and the Krylov-Schur Algorithm . 179 2.1 Restarting an Arnoldi Decomposition . 179 2.2 The Krylov Decomposition . 180 2.3 Restarting a Krylov Decomposition . 182 i i i i \main" i i 2004/5/6 page viii i i viii Contents 2.4 Deating a Krylov Decomposition . 184 3 A Krylov-Schur Algorithm for Periodic Eigenproblems . 186 3.1 Periodic Arnoldi and Krylov Decompositions . 186 3.2 Restarting a Periodic Krylov Decomposition . 191 3.3 Deating a Periodic Krylov Decomposition . 192 Deation of singular factors. 4 Krylov-Schur Algorithms for (Skew-)Hamiltonian Eigenproblems . 195 4.1 SHIRA . 195 4.2 A Two-Sided Krylov-Schur Algorithm for Hamiltonian Matrices . 197 5 Balancing Sparse General Matrices . 198 5.1 Irreducible forms . 199 5.2 Krylov-Based Balancing . 200 6 Balancing Sparse Hamiltonian Matrices . 201 6.1 Irreducible forms . 202 6.2 Krylov-Based Balancing . ..