The saddle point problem and the Manteuffel algorithm
| Autor: | Apostolos Hadjidimos |
|---|---|
| Rok vydání: | 2016 |
| Předmět: |
Matrix coefficient
Rank (linear algebra) Computer Networks and Communications Iterative method Applied Mathematics Linear system 0211 other engineering and technologies Block (permutation group theory) 021107 urban & regional planning 010103 numerical & computational mathematics 02 engineering and technology Positive-definite matrix 01 natural sciences Combinatorics Computational Mathematics Conjugate gradient method Saddle point 0101 mathematics Algorithm Software Mathematics |
| Zdroj: | BIT Numerical Mathematics. 56:1281-1302 |
| ISSN: | 1572-9125 0006-3835 |
| DOI: | 10.1007/s10543-016-0617-x |
| Popis: | In the last two decades, the augmented linear systems and the saddle point problems have been solved by many researchers who have used the conjugate gradient method or the generalized SOR iterative method and variants of them. In the latter class of methods, when the block \(A \in {\mathrm{I\!R}}^{m\times m}\) of the matrix coefficient \(\mathcal {A} = \left[ \begin{array}{cc} A &{} B \\ -B^T &{} 0\end{array}\right] \in {\mathrm{I\!R}}^{(m+n) \times (m+n)}\), \(m \ge n,\) of the linear system to be solved, is symmetric positive definite and \({\mathrm{rank}}(B) =r \le n\), convergence regions and optimal values of the parameters involved have been determined. In this work, we consider the block A to be nonsymmetric positive definite, \({\mathrm{rank}}(B) =r < n \), and use a two-level stationary iterative method whose main step is the linear second-order stationary iterative method for the solution of this class of problems. This method leads to the singular Manteuffel algorithm and the determination of its optimal parameters. As a byproduct, the optimal parameters of the Generalized Modified SSOR method in a particular case are also determined. Numerical examples verify our theoretical findings. |
| Databáze: | OpenAIRE |
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