Book contents
- Frontmatter
- Contents
- Preface
- Acknowledgements
- Symbols
- Part I Numerical Linear Algebra
- 1 Linear Operators and Matrices
- 2 The Singular Value Decomposition
- 3 Systems of Linear Equations
- 4 Norms and Matrix Conditioning
- 5 Linear Least Squares Problem
- 6 Linear Iterative Methods
- 7 Variational and Krylov Subspace Methods
- 8 Eigenvalue Problems
- Part II Constructive Approximation Theory
- Part III Nonlinear Equations and Optimization
- Part IV Initial Value Problems for Ordinary Differential Equations
- Part V Boundary and Initial Boundary Value Problems
- Appendix A Linear Algebra Review
- Appendix B Basic Analysis Review
- Appendix C Banach Fixed Point Theorem
- Appendix D A (Petting) Zoo of Function Spaces
- References
- Index
5 - Linear Least Squares Problem
from Part I - Numerical Linear Algebra
Published online by Cambridge University Press: 29 September 2022
Book contents
- Frontmatter
- Contents
- Preface
- Acknowledgements
- Symbols
- Part I Numerical Linear Algebra
- 1 Linear Operators and Matrices
- 2 The Singular Value Decomposition
- 3 Systems of Linear Equations
- 4 Norms and Matrix Conditioning
- 5 Linear Least Squares Problem
- 6 Linear Iterative Methods
- 7 Variational and Krylov Subspace Methods
- 8 Eigenvalue Problems
- Part II Constructive Approximation Theory
- Part III Nonlinear Equations and Optimization
- Part IV Initial Value Problems for Ordinary Differential Equations
- Part V Boundary and Initial Boundary Value Problems
- Appendix A Linear Algebra Review
- Appendix B Basic Analysis Review
- Appendix C Banach Fixed Point Theorem
- Appendix D A (Petting) Zoo of Function Spaces
- References
- Index
Summary
We study the solution of overdetermined systems of equations. Introduce weak, and in particular least squares solutions. For full rank systems, we show existence and uniqueness via the normal equations. We introduce projection matrices and the QR factorization. We discuss the computation of the QR factorization with the help of Householder reflectors. For rank defficient systems we prove the existence and uniqueness of a minimal norm least squares solution. We introduce the Moore-Penrose pseudoinverse, show how it relates to the SVD, and how it can be used to solve rank defficient systems.
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- Information
- Classical Numerical AnalysisA Comprehensive Course, pp. 88 - 120Publisher: Cambridge University PressPrint publication year: 2022