# Matrix Computations

Matrix Computations

The fourth edition of Gene H. Golub and Charles F. Van Loan's classic is an essential reference for computational scientists and engineers in addition to researchers in the numerical linear algebra community. Les mer
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Matrix Computations

The fourth edition of Gene H. Golub and Charles F. Van Loan's classic is an essential reference for computational scientists and engineers in addition to researchers in the numerical linear algebra community. Anyone whose work requires the solution to a matrix problem and an appreciation of its mathematical properties will find this book to be an indispensible tool. This revision is a cover-to-cover expansion and renovation of the third edition. It now includes an introduction to tensor computations and brand new sections on: fast transforms; parallel LU; discrete Poisson solvers; pseudospectra; structured linear equation problems; structured eigenvalue problems; large-scale SVD methods; and, polynomial eigenvalue problems. Matrix Computations is packed with challenging problems, insightful derivations, and pointers to the literature-everything needed to become a matrix-savvy developer of numerical methods and software.

Preface
Global References
Other Books
Useful URLs
Common Notation
1. Matrix Multiplication
1.1. Basic Algorithms and Notation
1.2. Structure and Efficiency
1.3. Block Matrices and Algorithms
1.4. Fast Matrix-Vector Products
1.5. Vectorization and Locality
1.6. Parallel Matrix Multiplication
2. Matrix Analysis
2.1. Basic Ideas from Linear Algebra
2.2. Vector Norms
2.3. Matrix Norms
2.4. The Singular Value Decomposition
2.5. Subspace Metrics
2.6. The Sensitivity of Square Systems
2.7. Finite Precision Matrix Computations
3. General Linear Systems
3.1. Triangular Systems
3.2. The LU Factorization
3.3. Roundoff Error in Gaussian Elimination
3.4. Pivoting
3.5. Improving and Estimating Accuracy
3.6. Parallel LU
4. Special Linear Systems
4.1. Diagonal Dominance and Symmetry
4.2. Positive Definite Systems
4.3. Banded Systems
4.4. Symmetric Indefinite Systems
4.5. Block Tridiagonal Systems
4.6. Vandermonde Systems
4.7. Classical Methods for Toeplitz Systems
4.8. Circulant and Discrete Poisson Systems
5. Orthogonalization and Least Squares
5.1. Householder and Givens Transformations
5.2. The QR Factorization
5.3. The Full-Rank Least Squares Problem
5.4. Other Orthogonal Factorizations
5.5. The Rank-Deficient Least Squares Problem
5.6. Square and Underdetermined Systems
6. Modified Least Squares Problems and Methods
6.1. Weighting and Regularization
6.2. Constrained Least Squares
6.3. Total Least Squares
6.4. Subspace Computations with the SVD
6.5. Updating Matrix Factorizations
7. Unsymmetric Eigenvalue Problems
7.1. Properties and Decompositions
7.2. Perturbation Theory
7.3. Power Iterations
7.4. The Hessenberg and Real Schur Forms
7.5. The Practical QR Algorithm
7.6. Invariant Subspace Computations
7.7. The Generalized Eigenvalue Problem
7.8. Hamiltonian and Product Eigenvalue Problems
7.9. Pseudospectra
8. Symmetric Eigenvalue Problems
8.1. Properties and Decompositions
8.2. Power Iterations
8.3. The Symmetric QR Algorithm
8.4. More Methods for Tridiagonal Problems
8.5. Jacobi Methods
8.6. Computing the SVD
8.7. Generalized Eigenvalue Problems with Symmetry
9. Functions of Matrices
9.1. Eigenvalue Methods
9.2. Approximation Methods
9.3. The Matrix Exponential
9.4. The Sign, Square Root, and Log of a Matrix
10. Large Sparse Eigenvalue Problems
10.1. The Symmetric Lanczos Process
10.3. Practical Lanczos Procedures
10.4. Large Sparse SVD Frameworks
10.5. Krylov Methods for Unsymmetric Problems
10.6. Jacobi-Davidson and Related Methods
11. Large Sparse Linear System Problems
11.1. Direct Methods
11.2. The Classical Iterations