Book contents
- Frontmatter
- Contents
- Preface
- Acknowledgments
- 1 Direct Solution Methods
- 2 Theory of Matrix Eigenvalues
- 3 Positive Definite Matrices, Schur Complements, and Generalized Eigenvalue Problems
- 4 Reducible and Irreducible Matrices and the Perron-Frobenius Theory for Nonnegative Matrices
- 5 Basic Iterative Methods and Their Rates of Convergence
- 6 M-Matrices, Convergent Splittings, and the SOR Method
- 7 Incomplete Factorization Preconditioning Methods
- 8 Approximate Matrix Inverses and Corresponding Preconditioning Methods
- 9 Block Diagonal and Schur Complement Preconditionings
- 10 Estimates of Eigenvalues and Condition Numbers for Preconditioned Matrices
- 11 Conjugate Gradient and Lanczos-Type Methods
- 12 Generalized Conjugate Gradient Methods
- 13 The Rate of Convergence of the Conjugate Gradient Method
- Appendices
- A Matrix Norms, Inherent Errors, and Computation of Eigenvalues
- B Chebyshev Polynomials
- C Some Inequalities for Functions of Matrices
- Index
A - Matrix Norms, Inherent Errors, and Computation of Eigenvalues
Published online by Cambridge University Press: 05 August 2012
- Frontmatter
- Contents
- Preface
- Acknowledgments
- 1 Direct Solution Methods
- 2 Theory of Matrix Eigenvalues
- 3 Positive Definite Matrices, Schur Complements, and Generalized Eigenvalue Problems
- 4 Reducible and Irreducible Matrices and the Perron-Frobenius Theory for Nonnegative Matrices
- 5 Basic Iterative Methods and Their Rates of Convergence
- 6 M-Matrices, Convergent Splittings, and the SOR Method
- 7 Incomplete Factorization Preconditioning Methods
- 8 Approximate Matrix Inverses and Corresponding Preconditioning Methods
- 9 Block Diagonal and Schur Complement Preconditionings
- 10 Estimates of Eigenvalues and Condition Numbers for Preconditioned Matrices
- 11 Conjugate Gradient and Lanczos-Type Methods
- 12 Generalized Conjugate Gradient Methods
- 13 The Rate of Convergence of the Conjugate Gradient Method
- Appendices
- A Matrix Norms, Inherent Errors, and Computation of Eigenvalues
- B Chebyshev Polynomials
- C Some Inequalities for Functions of Matrices
- Index
Summary
For various applications, we need to quantify errors and distances—i.e., we need to measure the size of a vector and a matrix. We shall find that it is not only the Euclidian measure of a vector that is appropriate in practice. Accordingly, we introduce vector and matrix norms, which are real valued functions of vectors and matrices, respectively. It is shown how to calculate matrix norms associated with certain vector norms and apply them in the estimation of eigenvalues. We shall find that properties of positive definite matrices play a crucial role in the calculation of the norm associated with the Euclidian vector norm. We show also that matrix norms are useful when estimating inherent errors—i.e., errors caused by errors in given data—in solutions of systems of linear algebraic equations. Finally, it is shown how to estimate the errors in eigenvalue computations by a posteriori estimates and how to compute sequences of approximations of them by matrix power methods. The following definitions are introduced in this appendix.
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- Information
- Iterative Solution Methods , pp. 595 - 638Publisher: Cambridge University PressPrint publication year: 1994