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# Matrix Computations

Gene H. Golub and Charles F. Van Loan

fourth edition

Publication Date

**A comprehensive treatment of numerical linear algebra from the standpoint of both theory and practice.**

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…

**A comprehensive treatment of numerical linear algebra from the standpoint of both theory and practice.**

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

• 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. The second most cited math book of 2012 according to MathSciNet, the book has placed in the top 10 for since 2005.

Reviews

Problems, solutions, and discussions of the formulas, methods and literature surrounding matrix computations make for a reference that is specific and well detailed: perfect for any college-level math collection appealing to engineers.

Written for scientists and engineers, *Matrix Computations *provides comprehensive coverage of numerical linear algebra. Anyone whose work requires the solution to a matrix problem and an appreciation of mathematical properties will find this book to be an indispensable tool.

A mine of insight and information and a provocation to thought; the annotated bibliographies are helpful to those wishing to explore further. One could not ask for more, and the book should be considered a resounding success.

About

Publication Date

Status

Available

Trim Size

7

x 10

Pages

784

ISBN

9781421407944

Illustration Description

31 line drawings

Table of Contents

Preface

Global References

Other Books

Useful URLs

Common Notation

Chapter 1. Matrix Multiplication

1.1. Basic Algorithms and Notation

1.2. Structure and Efficiency

1.3. Block Matrices and Algorithms

1.4. Fast

Preface

Global References

Other Books

Useful URLs

Common Notation

Chapter 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

Chapter 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

Chapter 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

Chapter 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

Chapter 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

Chapter 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

Chapter 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

Chapter 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

Chapter 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

Chapter 10. Large Sparse Eigenvalue Problems

10.1. The Symmetric Lanczos Process

10.2. Lanczos, Quadrature, and Approximation

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

Chapter 11. Large Sparse Linear System Problems

11.1. Direct Methods

11.2. The Classical Iterations

11.3. The Conjugate Gradient Method

11.4. Other Krylov Methods

11.5. Preconditioning

11.6. The Multigrid Framework

Chapter 12. Special Topics

12.1. Linear Systems with Displacement Structure

12.2. Structured-Rank Problems

12.3. Kronecker Product Computations

12.4. Tensor Unfoldings and Contractions

12.5. Tensor Decompositions and Iterations

Index

Resources

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