Eigenvalue Algorithms for Symmetric Hierarchical Matrices Thomas Mach MAX PLANCK INSTITUTE FOR DYNAMICS OF COMPLEX TECHNICAL SYSTEMS MAGDEBURG Eigenvalue Algorithms for Symmetric Hierarchical Matrices Dissertation submitted to Department of Mathematics at Chemnitz University of Technology in accordance with the requirements for the degree Dr. rer. nat. ν( 0.5) = 4 -6− -4 -2 -1 0 1 2 4 6 a0 = 6.5 b0 = 5.5 − µ = 0.5 1 − ν( 3.5) = 2 -6− -4 -2 -1 0 1 2 4 6 a1 = 6.5 b1 = 0.5 − µ = 3.5 − 2 − ν( 2) = 4 1, 2, 3, 4, 5, 6, 7, 8 -6 − -4 -2 -1 0 1 2 4 6 { } a2 = 3.5 b2 = 0.5 S∗(r) − µ = 2 − 3 − r = 5, 6, 7, 8 ν( 2.75) = 3 1, 2, 3, 4 { } -6− -4 -2 -1 0 1 2 4 6 { } S(r) a3 = 3.5 b3 = 2 −µ = 2.75 − 4 − 1, 2 3, 4 5, 6 7, 8 { } { } { } { } 1 2 3 4 5 6 7 8 { } { } { } { } { } { } { } { } φ1 φ2 φ3 φ4 φ5 φ6 φ7 φ8 τ = σ = supp (φ1, φ2) dist (τ, σ) supp (φ7, φ8) diam (τ) presented by: Dipl.-Math. techn. Thomas Mach Advisor: Prof. Dr. Peter Benner Reviewer: Prof. Dr. Steffen B¨orm March 13, 2012 ACKNOWLEDGMENTS Personal Thanks. I thank Peter Benner for giving me the time and the freedom to follow several ideas and for helping me with new ideas. I would also like to thank my colleagues in Peter Benner's groups in Chemnitz and Magdeburg. All of them helped me a lot during the last years. The following four will be highlighted, since they are most important for me. I would like to thank Jens Saak for explaining me countless things and for providing his LATEX framework that I used for this document. I stressed our compute cluster specialist, Martin K¨ohler,a lot with questions and request. I thank him for his help. Further, I am greatly indebted to Ulrike Baur, who explained me many details concerning hierarchical matrices. I want to thank Martin Stoll for pushing me forward during the final year. I also want to thank Jessica G¨ordesand Steffen B¨orm(CAU Kiel) for discussing eigen- value algorithms for -matrices, Marc Van Barel (KU Leuven) for pointing out the H` relations to semiseparable matrices, and Lars Grasedyck (RWTH Aachen) for a discus- sion on QR decompositions and the QR-like algorithms for -matrices. H I am indebted to Tom Barker for carefully proof reading this thesis. Finally this work would not be possible without adequate recreation by (extensive) sports activities. Therefore, I thank my friends Benjamin Schulz, Jens Lang, and Tobias Breiten for cycling and jogging with me. Last but not least I thank my sister for proof reading and my parents, who made all of this possible in the first place. Financial Support. This research was supported by a grant from the Free State of Saxony (S¨achsisches Landesstipendium) for two years. CONTENTS List of Figures xi List of Tables xiii List of Algorithms xv List of Acronyms xvii List of Symbols xix Publications xxi 1 Introduction1 1.1 Notation.....................................2 1.2 Structure of this Thesis............................3 2 Basics 5 2.1 Linear Algebra and Eigenvalues........................6 2.1.1 The Eigenvalue Problem........................7 2.1.2 Dense Matrix Algorithms.......................9 2.2 Integral Operators and Integral Equations.................. 14 2.2.1 Definitions............................... 14 2.2.2 Example - BEM............................ 16 2.3 Introduction to Hierarchical Arithmetic................... 17 2.3.1 Main Idea................................ 17 2.3.2 Definitions............................... 19 2.3.3 Hierarchical Arithmetic........................ 24 2.3.4 Simple Hierarchical Matrices ( -Matrices)............. 30 H` 2.4 Examples.................................... 33 2.4.1 FEM Example............................. 33 2.4.2 BEM Example............................. 36 2.4.3 Randomly Generated Examples.................... 37 viii Contents 2.4.4 Application Based Examples..................... 38 2.4.5 One-Dimensional Integral Equation.................. 38 2.5 Related Matrix Formats............................ 39 2.5.1 2-Matrices............................... 40 H 2.5.2 Diagonal plus Semiseparable Matrices................ 40 2.5.3 Hierarchically Semiseparable Matrices................ 42 2.6 Review of Existing Eigenvalue Algorithms.................. 44 2.6.1 Projection Method........................... 44 2.6.2 Divide-and-Conquer for (1)-Matrices............... 45 H` 2.6.3 Transforming Hierarchical into Semiseparable Matrices....... 46 2.7 Compute Cluster Otto............................. 47 3 QR Decomposition of Hierarchical Matrices 49 3.1 Introduction................................... 49 3.2 Review of Known QR Decompositions for -Matrices........... 50 H 3.2.1 Lintner's -QR Decomposition.................... 50 H 3.2.2 Bebendorf's -QR Decomposition.................. 52 H 3.3 A new Method for Computing the -QR Decomposition.......... 54 H 3.3.1 Leaf Block-Column........................... 54 3.3.2 Non-Leaf Block Column........................ 56 3.3.3 Complexity............................... 57 3.3.4 Orthogonality.............................. 60 3.3.5 Comparison to QR Decompositions for Sparse Matrices...... 61 3.4 Numerical Results............................... 62 3.4.1 Lintner's -QR decomposition.................... 62 H 3.4.2 Bebendorf's -QR decomposition.................. 66 H 3.4.3 The new -QR decomposition.................... 66 H 3.5 Conclusions................................... 67 4 QR-like Algorithms for Hierarchical Matrices 69 4.1 Introduction................................... 70 4.1.1 LR Cholesky Algorithm........................ 70 4.1.2 QR Algorithm............................. 70 4.1.3 Complexity............................... 71 4.2 LR Cholesky Algorithm for Hierarchical Matrices.............. 72 4.2.1 Algorithm................................ 72 4.2.2 Shift Strategy.............................. 72 4.2.3 Deflation................................ 73 4.2.4 Numerical Results........................... 73 4.3 LR Cholesky Algorithm for Diagonal plus Semiseparable Matrices.... 75 4.3.1 Theorem................................ 75 4.3.2 Application to Tridiagonal and Band Matrices........... 79 4.3.3 Application to Matrices with Rank Structure............ 79 4.3.4 Application to -Matrices....................... 80 H Contents ix 4.3.5 Application to -Matrices...................... 82 H` 4.3.6 Application to 2-Matrices...................... 83 H 4.4 Numerical Examples.............................. 84 4.5 The Unsymmetric Case............................ 84 4.6 Conclusions................................... 88 5 Slicing the Spectrum of Hierarchical Matrices 89 5.1 Introduction................................... 89 5.2 Slicing the Spectrum by LDLT Factorization................ 91 5.2.1 The Function ν(M µI)....................... 91 − 5.2.2 LDLT Factorization of -Matrices.................. 92 H` 5.2.3 Start-Interval [a; b]........................... 96 5.2.4 Complexity............................... 96 5.3 Numerical Results............................... 97 5.4 Possible Extensions............................... 100 5.4.1 LDLT Slicing Algorithm for HSS Matrices.............. 103 5.4.2 LDLT Slicing Algorithm for -Matrices............... 103 H 5.4.3 Parallelization............................. 105 5.4.4 Eigenvectors.............................. 107 5.5 Conclusions................................... 107 6 Computing Eigenvalues by Vector Iterations 109 6.1 Power Iteration................................. 109 6.1.1 Power Iteration for Hierarchical Matrices.............. 110 6.1.2 Inverse Iteration............................ 111 6.2 Preconditioned Inverse Iteration for Hierarchical Matrices......... 111 6.2.1 Preconditioned Inverse Iteration................... 113 6.2.2 The Approximate Inverse of an -Matrix.............. 115 H 6.2.3 The Approximate Cholesky Decomposition of an -Matrix.... 116 H 6.2.4 PINVIT for -Matrices........................ 117 H 6.2.5 The Interior of the Spectrum..................... 120 6.2.6 Numerical Results........................... 123 6.2.7 Conclusions............................... 130 7 Comparison of the Algorithms and Numerical Results 133 7.1 Theoretical Comparison............................ 133 7.2 Numerical Comparison............................. 135 8 Conclusions 141 Theses 143 Bibliography 145 Index 153 LIST OF FIGURES 2.1 Dense, sparse and data-sparse matrices.................... 11 1 2.2 Singular values of ∆2−;h............................. 18 1 2.3 Singular values of ∆2−;h............................. 18 2.4 -tree T ..................................... 20 H I 2.5 Example of basis functions........................... 20 2.6 -matrix based on T from Figure 2.4..................... 23 H I 2.7 -matrix structure diagram.......................... 30 H 2.8 Structure of an -matrix........................... 31 H3 2.9 Eigenvectors of FEM32............................. 35 2.10 Eigenvectors of FEM3D8............................ 36 2.11 Set diagram of different matrix formats.................... 39 2.12 Eigenvalues before and after projection.................... 45 2.13 Block structure condition............................ 46 2.14 Photos of Linux-cluster Otto.......................... 47 3.1 Explanation of Line5 and6 of Algorithm 3.5................. 58 3.2 Explanation of Line 16 and 17 of Algorithm 3.5............... 58 3.3 Computation time for the FEM example series................ 63 3.4 Accuracy and orthogonality for the FEM example series........... 63 3.5 Computation time for the BEM example series................ 64 3.6 Accuracy and orthogonality for the BEM example series........... 64 3.7 Computation time for the