Hierarchical Matrices: Algorithms and Analysis

This self-contained monograph presents matrix algorithms and their analysis. The new technique enables not only the solution of linear systems but also the approximation of matrix functions, e.g., the matrix exponential. Other applications include the solution of matrix equations, e.g., the Lyapunov or Riccati equation. The required mathematical background can be found in the appendix.

The numerical treatment of fully populated large-scale matrices is usually rather costly. However, the technique of hierarchical matrices makes it possible to store matrices and to perform matrix operations approximately with almost linear cost and a controllable degree of approximation error. For important classes of matrices, the computational cost increases only logarithmically with the approximation error. The operations provided include the matrix inversion and LU decomposition.

Since large-scale linear algebra problems are standard in scientific computing, the subject of hierarchical matrices is of interest to scientists in computational mathematics, physics, chemistry and engineering.

Table of Contents

Part I Introductory and Preparatory Topics
Chapter 1 Introduction
Chapter 2 Rank-r Matrices
Chapter 3 Introductory Example
Chapter 4 Separable Expansions and Low-Rank Matrices
Chapter 5 Matrix Partition

Part II H-Matrices and Their Arithmetic
Chapter 6 Definition and Properties of Hierarchical Matrices
Chapter 7 Formatted Matrix Operations for Hierarchical Matrices
Chapter 8 H2-Matrices
Chapter 9 Miscellaneous Supplements

Part III Applications
Chapter 10 Applications to Discretised Integral Operators
Chapter 11 Applications to Finite Element Matrices
Chapter 12 Inversion with Partial Evaluation
Chapter 13 Eigenvalue Problems
Chapter 14 Matrix Functions
Chapter 15 Matrix Equations
Chapter 16 Tensor Spaces

Part IV Appendices
Appendix A Graphs and Trees
Appendix B Polynomials
Appendix C Linear Algebra and Functional Analysis
Appendix D Sinc Functions and Exponential Sums
Appendix E Asymptotically Smooth Functions