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
- Index
1 - Direct Solution Methods
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
- Index
Summary
The need to solve large linear systems of algebraic equations arises in almost any mathematical model, as illustrated in the instances below. In particular, we go into some detail regarding electrical networks. The most common method used to solve such linear systems is based on factorization of the matrix in triangular factors, which is discussed and shown to be equivalent to the Gaussian elimination method (learned in almost any elementary course in algebra).
We present this method in a form that can be the basis for a computer algorithm. When one wants to solve very large systems of equations, it is important to know how computational complexity grows with the size of the problem. This topic is also addressed, including the case where the matrix has a special structure in the form of a bandmatrix. The solution of tridiagonal systems is considered in particular detail. In addition, some basic dimension theory for matrices, such as the relation between rank and the dimension of the nullspace, is discussed and derived using the factored form of the matrix.
- Type
- Chapter
- Information
- Iterative Solution Methods , pp. 1 - 45Publisher: Cambridge University PressPrint publication year: 1994