On the stable implementation of the generalized minimal error method

The paper reviews several implementations of the Generalized minimal error method (GMERR method) for solving nonsymmetric systems of linear equations that minimize the Euclidean norm of the error in the related generalized Krylov subspace. We show the relation to the methods in the symmetric indefinite case. A new variant of the GMERR method is proposed and the stable implementation based on the Householder transformations is discussed. Numerical stability of the most frequent implementations is analyzed and the theoretical results are illustrated by numerical examples

By How Much Can Residual Minimization Accelerate The Convergence Of Orthogonal Residual Methods?

We capitalize upon the known relationship between pairs of orthogonal and minimal residual methods (or, biorthogonal and quasi-minimal residual methods) in order to estimate how much smaller the residuals or quasi-residuals of the minimizing methods can be compared to the those of the corresponding Galerkin or Petrov-Galerkin method. Examples of such pairs are the conjugate gradient (CG) and the conjugate residual (CR) methods, the full orthogonalization method (FOM) and the generalized minimal residual (GMRes) method, the CGNE and CGNR versions of applying CG to the normal equations, as well as the biconjugate gradient (BiCG) and the quasi-minimal residual (QMR) methods. Also the pairs consisting of the (bi)conjugate gradient squared (CGS) and the transpose-free QMR (TFQMR) methods can be added to this list if the residuals at half-steps are included, and further examples can be created easily. The analysis is more generally applicable to the minimal residual (MR) and quasi-minimal residual (QMR) smoothing processes, which are known to provide the transition from the results of the first method of such a pair to those of the second one. By an interpretation of these smoothing processes in coordinate space we deepen the understanding of some of the underlying relationships and introduce a unifying framework for minimal residual and quasi-minimal residual smoothing. This framework includes the general notion of QMR-type methods

On the round-off error analysis of the Gram-Schmidt algorithm with reorthogonalization

In this paper we analyse the numerical behavior of the Gram-Schmidt orthogonalization process with reorthogonalization. Assuming numerical nonsingularity of the matrix we prove that two steps of iterative Gram-Schmidt process are enough for preserving the orthogonality of computed vectors close to the machine precision level. We give a rounding error analysis of classical reorthogonalization and modified Gram-Schmidt algorithm with (exactly one) reorthogonalization and relate our results to the approach used in the Kahan-Parlett "twice is enough" algorithm as well as to results shown by Abdelmalek, Daniel et al, Hoffmann and others

Numerical stability of the GMRES method

Available from STL Prague, CZ / NTK - National Technical LibrarySIGLECZCzech Republi

Preconditioning as a tool to improve diving safety

info:eu-repo/semantics/publishe

Functional comparison between critical flicker fusion frequency and simple cognitive tests in subjects breathing air or oxygen in normobaria.

Measurement of inert gas narcosis and its degree is difficult during operational circumstances, hence the need for a reliable, reproducible and adaptable tool. Although being an indirect measure of brain function, if reliable, critical flicker fusion frequency (CFFF) could address this need and be used for longitudinal studies on cortical arousal in humans.Comparative StudyJournal ArticleResearch Support, Non-U.S. Gov'tinfo:eu-repo/semantics/publishe