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On solving symmetric systems of linear equations in an unnormalized Krylov subspace framework

Anders Forsgren (andersf***at***kth.se)
Tove Odland (odland***at***kth.se)

Abstract: In an unnormalized Krylov subspace framework for solving symmetric systems of linear equations, the orthogonal vectors that are generated by a Lanczos process are not necessarily on the form of gradients. Associating each orthogonal vector with a triple, and using only the three-term recurrences of the triples, we give conditions on whether a symmetric system of linear equations is compatible or incompatible. In the compatible case, a solution is given and in the incompatible case, a certificate of incompatibility is obtained. In particular, the case when the matrix is singular is handled. We also derive a minimum-residual method based on this framework and show how the iterates may be updated explicitly based on the triples, and in the incompatible case a minimum-residual solution of minimum Euclidean norm is obtained.

Keywords: Krylov subspace method, symmetric system of linear equations, unnormalized Lanczos vectors, minimum-residual method

Category 1: Nonlinear Optimization (Unconstrained Optimization )

Category 2: Nonlinear Optimization (Quadratic Programming )

Citation: arXiv:1409.4937 [math.OC]

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Entry Submitted: 03/14/2014
Entry Accepted: 03/14/2014
Entry Last Modified: 09/18/2014

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