Preface to the Second Edition
Preface to the First Edition
1 Background in Linear Algebra
1.1 Matrices
1.2 Square Matrices and Eigenvalues
1.3 Types of Matrices
1.4 Vector Inner Products and Norms
1.5 Matrix Norms
1.6 Subspaces, Range, and Kernel
1.7 Orthogonal Vectors and Subspaces
1.8 Canonical Forms of Matrices
1.8.1 Reduction to the Diagonal Form
1.8.2 The Jordan Canonical Form
1.8.3 The Schur Canonical Form
1.8.4 Application to Powers of Matrices
1.9 Normal and Hermitian Matrices
1.9.1 Normal Matrices
1.9.2 Hermitian Matrices
1.10 Nonnegative Matrices, M-Matrices
1.11 Positive Definite Matrices
1.12 Projection Operators
1.12.1 Range and Null Space of a Projector
1.12.2 Matrix Representations
1.12.3 Orthogonal and Oblique Projectors
1.12.4 Properties of Orthogonal Projectors
1.13 Basic Concepts in Linear Systems
1.13.1 Existence of a Solution
1.13.2 Perturbation Analysis
Exercises
Notes and References
2 Discretization of Partial Differential Equations
2.1 Partial Differential Equations
2.1.1 Elliptic Operators
2.1.2 The Convection Diffusion Equation
2.2 Finite Difference Methods
2.2.1 Basic Approximations
2.2.2 Difference Schemes for the Laplacian Operator
2.2.3 Finite Differences for One-Dimensional Problerr
2.2.4 Upwind Schemes
2.2.5 Finite Differences for Two-Dimensional Problerr
2.2.6 Fast Poisson Solvers
2.3 The Finite Element Method
2.4 Mesh Generation and Refinement
2.5 Finite Volume Method
Exercises
Notes and References
3 Sparse Matrices
3.1 Introduction
3.2 Graph Representations
3.2.1 Graphs and Adjacency Graphs
3.2.2 Graphs of PDE Matrices
3.3 Permutations and Reorderings
3.3.1 Basic Concepts
3.3.2 Relations with the Adjacency Graph
3.3.3 Common Reorderings
3.3.4 Irreducibility
3.4 Storage Schemes
3.5 Basic Sparse Matrix Operations
3.6 Sparse Direct Solution Methods
3.6.1 MD Ordering
3.6.2 ND Ordering
3.7 Test Problems
Exercises
Notes and References
4 Basic Iterative Methods
4.1 Jacobi, Gauss-Seidel, and Successive Overrelaxation
4.1.1 Block Relaxation Schemes
4.1.2 Iteration Matrices and Preconditioning
4.2 Convergence
4.2.1 General Convergence Result
4.2.2 Regular Splittings
4.2.3 Diagonally Dominant Matrices
4.2.4 Symmetric Positive Definite Matrices
4.2.5 Property A and Consistent Orderings
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5 Projection Methods
6 Krylov Subspace Methods, Part Ⅰ
7 Krylov Subspace Methods, Part Ⅱ
8 Methods Related to the Normal Equations
9 Preconditioned Iterations
10 Preconditioning Techniques
11 Parallel Implementations
12 Parallel Preconditioners
13 Multigrid Methods
14 Domain Decomposition Methods
Bibliography
Index