## Constraint Preconditioning for Indefinite Linear Systems (2000)

Venue: | SIAM J. Matrix Anal. Appl |

Citations: | 81 - 12 self |

### BibTeX

@ARTICLE{Keller00constraintpreconditioning,

author = {Carsten Keller and Nicholas I. M. Gould and Andrew and Andrew J. Wathen},

title = {Constraint Preconditioning for Indefinite Linear Systems},

journal = {SIAM J. Matrix Anal. Appl},

year = {2000},

volume = {21},

pages = {1300--1317}

}

### Years of Citing Articles

### OpenURL

### Abstract

. The problem of nding good preconditioners for the numerical solution of indenite linear systems is considered. Special emphasis is put on preconditioners that have a 2 2 block structure and which incorporate the (1; 2) and (2; 1) blocks of the original matrix. Results concerning the spectrum and form of the eigenvectors of the preconditioned matrix and its minimum polynomial are given. The consequences of these results are considered for a variety of Krylov subspace methods. Numerical experiments validate these conclusions. Key words. preconditioning, indenite matrices, Krylov subspace methods AMS subject classications. 65F10, 65F15, 65F50 1. Introduction. In this paper, we are concerned with investigating a new class of preconditioners for indenite systems of linear equations of a sort which arise in constrained optimization as well as in least-squares, saddle-point and Stokes problems. We attempt to solve the indenite linear system A B T B 0 | {z } A x 1 x...

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Citation Context ... implies that Z T GZ is at least as good a preconditioner for Z T AZ. To show that the preconditionersZ T GZ can in fact be much better, consider the following example, taken from the CUTE collection =-=[3]-=-. Example 2.8. Consider the convex quadratic programming problem BLOWEYC which may be formulated as minimize u(s) T Au(s) + u(s) T w(s) v(s) T Au(s) 2:0v(s) T w(s) u(s) T v(s) subject to Aw(s) = u(s);... |

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Citation Context ...all call G a constraint preconditioner. A preconditioner of the form G has recently been used by Luksan and Vlcek [16] in the context of constrained non-linear programming problems|see also Coleman [7=-=-=-], Polyak [20] and Gould et al. [12]. Here, we derive arguments that conrm and extend some of the results in [16] and highlight the favourable features of a preconditioner of the form G. Note that Gol... |

4 |
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Citation Context ...and Gould et al. [12]. Here, we derive arguments that conrm and extend some of the results in [16] and highlight the favourable features of a preconditioner of the form G. Note that Golub and Wathen [=-=11-=-] recently considered a symmetric preconditioner of the form (1.4) for problems of the form (1.1) where A is non-symmetric. We comment that for certain partial dierential equation problems, which give... |

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Citation Context ...1 Ax = G 1 b: (1.5) Since these blocks are unchanged from the original system, we shall call G a constraint preconditioner. A preconditioner of the form G has recently been used by Luksan and Vlcek [1=-=6-=-] in the context of constrained non-linear programming problems|see also Coleman [7], Polyak [20] and Gould et al. [12]. Here, we derive arguments that conrm and extend some of the results in [16] and... |

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Citation Context ...te nature of matrix (5.1) suggests the use of MINRES in the unpreconditioned case. Of course, positive denite preconditioning could be employed with MINRES|see, for example, Murphy, Golub and Wathen [=-=18]-=-, or Silvester and Wathen [25], and Wathen and Silvester [26] in case of the Stokes Problem. However, this is not done here. We refer to Battermann and Heikenschloss [3] for some numerical results of ... |

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