MINRES-QLP: A Krylov Subspace Method for Indefinite or Singular Symmetric Systems
SIAM Journal on Scientific Computing
Preconditioning for Allen-Cahn variational inequalities with non-local constraints
Journal of Computational Physics
An optimal linear solver for the Jacobian system of the extreme type-II Ginzburg-Landau problem
Journal of Computational Physics
Preconditioning for large scale micro finite element analyses of 3d poroelasticity
PARA'12 Proceedings of the 11th international conference on Applied Parallel and Scientific Computing
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The three-term Lanczos process for a symmetric matrix leads to bases for Krylov subspaces of increasing dimension. The Lanczos basis, together with the recurrence coefficients, can be used for the solution of symmetric indefinite linear systems, by solving a reduced system in one way or another. This leads to well-known methods: MINRES (minimal residual), GMRES (generalized minimal residual), and SYMMLQ (symmetric LQ). We will discuss in what way and to what extent these approaches differ in their sensitivity to rounding errors.In our analysis we will assume that the Lanczos basis is generated in exactly the same way for the different methods, and we will not consider the errors in the Lanczos process itself. We will show that the method of solution may lead, under certain circumstances, to large additional errors, which are not corrected by continuing the iteration process.Our findings are supported and illustrated by numerical examples.