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
7 - Variational and Krylov Subspace Methods
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 prove the equivalence of the solution to a linear system of equations with an HPD matrix to the problem of quadratic minimization. With the help of this equivalence, we study the minimization of a quadratic energy and introduce gradient descent methods with exact and approximate line search, we study the preconditioned steepest descent method. We introduce the conjugate gradient method, with preconditioning, as a Galerkin approximation over Krylov subspaces and show its convergence. For systems with non HPD matrices we discuss the GMRES method.
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- Classical Numerical AnalysisA Comprehensive Course, pp. 156 - 196Publisher: Cambridge University PressPrint publication year: 2022