On nonsymmetric saddle point matrices that allow conjugate gradient iterations |
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Authors: | Jörg Liesen Beresford N Parlett |
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Institution: | 1. Institute of Mathematics, Technical University of Berlin, Stra?e des 17. Juni 136, 10623, Berlin, Germany 2. Department of Mathematics, University of California, Berkeley, CA, 94720-3840, USA
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Abstract: | Linear systems in saddle point form are usually highly indefinite,which often slows down iterative solvers such as Krylov subspace methods. It has been noted by several authors that negating the second block row of a symmetric indefinite saddle point matrix leads to a nonsymmetric matrix ${{\mathcal A}}Linear systems in saddle point form are usually highly indefinite,which often slows down iterative solvers such as Krylov
subspace methods. It has been noted by several authors that negating the second block row of a symmetric indefinite saddle
point matrix leads to a nonsymmetric matrix whose spectrum is entirely contained in the right half plane. In this paper we study conditions so that is diagonalizable with a real and positive spectrum. These conditions are based on necessary and sufficient conditions for
positive definiteness of a certain bilinear form,with respect to which is symmetric. In case the latter conditions are satisfied, there exists a well defined conjugate gradient (CG) method for
solving linear systems with . We give an efficient implementation of this method, discuss practical issues such as error bounds, and present numerical
experiments.
In memory of Gene Golub (1932–2007), our wonderful friend and colleague, who had a great interest in the conjugate gradient
method and the numerical solution of saddle point problems.
The work of J?rg Liesen was supported by the Emmy Noether-Program and the Heisenberg-Program of the Deutsche Forschungsgemeinschaft. |
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Keywords: | 65F15 65N22 65F50 |
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