| Introduction | p. 1 |
| Low-Rank Matrices and Matrix Partitioning | p. 9 |
| Low-Rank Matrices | p. 9 |
| Efficient Representation | p. 10 |
| Adding and Multiplying Low-Rank Matrices | p. 12 |
| Approximation by Low-Rank Matrices | p. 13 |
| Singular Value Decomposition of Low-Rank Matrices | p. 15 |
| Approximate Addition of Low-Rank Matrices | p. 16 |
| Agglomerating Low-Rank Blocks | p. 18 |
| Structured Low-Rank Matrices | p. 19 |
| Admissible Partitions | p. 21 |
| Tensor vs. Hierarchical Partitions | p. 25 |
| Cluster Trees | p. 29 |
| Construction of Cluster Trees | p. 33 |
| Example: An Easily Analyzed Partition | p. 41 |
| Block Cluster Trees | p. 43 |
| Hierarchical Matrices | p. 49 |
| The Set of Hierarchical Matrices | p. 50 |
| Matrix-Vector Multiplication | p. 52 |
| Parallel Matrix-Vector Multiplication | p. 53 |
| Parallelization for Usual Matrices | p. 54 |
| Non-Uniform Block Distributions | p. 55 |
| Numerical Experiments | p. 60 |
| Blockwise and Global Norms | p. 63 |
| Adding H-Matrices | p. 65 |
| Preserving Positivity | p. 66 |
| Coarsening H-Matrices | p. 69 |
| Multiplying H-Matrices | p. 74 |
| Product Block Cluster Tree | p. 74 |
| Preserving the Original Block Structure | p. 77 |
| Rounded Multiplication | p. 79 |
| Multiplication of Hierarchical and Semi-Separable Matrices | p. 80 |
| Hierarchical Inversion | p. 83 |
| Computing the H-Matrix LU Decomposition | p. 85 |
| Hierarchical QR Decomposition | p. 87 |
| H[superscript 2]-Matrices and Fast Multipole Methods | p. 90 |
| Using Hierarchical Matrices for Preconditioning | p. 92 |
| Hermitian Positive Definite Coefficient Matrices | p. 94 |
| Non-Hermitian Coefficient Matrices | p. 97 |
| Approximation of Discrete Integral Operators | p. 99 |
| Boundary Integral Formulations | p. 104 |
| Asymptotic Smoothness of Kernel Functions | p. 109 |
| The Biharmonic Equation | p. 114 |
| Approximation by Degenerate Kernels | p. 116 |
| Degenerate Kernels through Taylor Expansion | p. 121 |
| Degenerate Kernels through Interpolation | p. 123 |
| Another Kind of Approximation | p. 128 |
| Matrix Approximation Error | p. 134 |
| Adaptive Cross Approximation (ACA) | p. 139 |
| The Algorithm | p. 140 |
| Error Analysis | p. 142 |
| The Right Choice of Rows | p. 148 |
| Overall Complexity | p. 152 |
| Numerical Experiments | p. 153 |
| Parallelization of ACA | p. 163 |
| A Recompression Technique for ACA (RACA) | p. 169 |
| Approximation Using Chebyshev Polynomials | p. 172 |
| Evaluation of the Approximation | p. 173 |
| Least Squares Approximation | p. 175 |
| Numerical Results | p. 178 |
| Preconditioning with Low-Accuracy Approximations | p. 180 |
| Dirichlet Problem | p. 183 |
| Neumann Problem | p. 184 |
| Mixed Boundary Value Problems | p. 187 |
| Application to Finite Element Discretizations | p. 193 |
| Approximating FE Matrix Inverses | p. 199 |
| Inverses of Banded Matrices | p. 200 |
| Approximating the Inverse Mass Matrix | p. 204 |
| An Algebraic Approach to the Approximation of the Inverse | p. 206 |
| Degenerate Approximation of the Green's Function | p. 209 |
| Approximation of Discrete Operators | p. 218 |
| Numerical Experiments | p. 222 |
| Schur Complements | p. 229 |
| Hierarchical LU Decomposition | p. 232 |
| Approximating Schur Complements Hierarchically | p. 234 |
| Constructing the Factors L[subscript H] and U[subscript H] | p. 236 |
| Numerical Experiments with the H-Matrix LU Factorization | p. 238 |
| Two-Dimensional Diffusion | p. 239 |
| Convection-Diffusion Problems | p. 244 |
| Three-Dimensional Diffusion | p. 245 |
| Nested Dissection LU Factorization | p. 248 |
| Matrix Partitioning | p. 249 |
| Approximation of the Factors of the LU Decomposition | p. 250 |
| Numerical Results | p. 253 |
| Parallel Approximate LU Factorization | p. 253 |
| Solving Nonlinear Problems with Broyden Updates | p. 257 |
| Broyden Updates | p. 260 |
| An Update Method for the LU Decomposition | p. 261 |
| The Influence of Truncation Errors | p. 264 |
| Numerical Experiments | p. 265 |
| References | p. 269 |
| Appendix | p. 281 |
| Index | p. 289 |
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