2014/09/17 by Anders Forsgren, Forsgren, Anders, Tove Odland +1 · 1 citation
Computer Science · Engineering · Physics and Astronomy · #Matrix Theory and Algorithms #Optical Network Technologies #Theoretical and Computational Physics
paper · pdf · doi:10.48550/arxiv.1409.4937
In an unnormalized Krylov subspace framework for solving symmetric systems of\nlinear equations, the orthogonal vectors that are generated by a Lanczos\nprocess are not necessarily on the form of gradients. Associating each\northogonal vector with a triple, and using only the three-term recurrences of\nthe triples, we give conditions on whether a symmetric system of linear\nequations is compatible or incompatible. In the compatible case, a solution is\ngiven and in the incompatible case, a certificate of incompatibility is\nobtained. In particular, the case when the matrix is singular is handled.\n We also derive a minimum-residual method based on this framework and show how\nthe iterates may be updated explicitly based on the triples, and in the\nincompatible case a minimum-residual solution of minimum Euclidean norm is\nobtained.\n