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

On the Efficiency of Superscalar and Vector Computer for Some Problems in Scientific Computing

. Some details of arithmetic of two representatives of computers (a superscalar workstation and a vector uniprocessor) available in the Czech Republic for scientific computing are described. Consequently, their efficiency and precision on a set of linear algebraic tasks solved by different solvers is compared. 1 Introduction One very important question in scientific computing is how to guarantee efficient performance of a given computer on a set of fundamental operations. On the other hand, efficiency on a set of basic, often linear algebra operations, plays usually the key role in the choice of computer for scientific computations. Although a lot of effort has been devoted to developing benchmarks for scientific computing [5], [9], [2] there is still a lack of comparison for computers avilable using a broader spectrum of the important basic algorithms. We devote our attention to some properties of computer arithmetic of two computers which present different philosophy of performing a..

Mixed-Hybrid Finite Element Approximation Of The Potential Fluid Flow Problem

: In the paper a mixed-hybrid approximation of the potential fluid flow problem based on prismatic discretization of the domain is presented. Trilateral prismatic elements with vertical faces and nonparallel bases suitable for the modelling of real geological circumstances are considered. The set of linearly independent vector basis functions is defined and existence and uniqueness of the approximate solution from the resulting symmetric indefinite system are examined. Possible approaches to the solution of the discretized system are discussed. Keywords: potential flow problem in porous media, mixed-hybrid formulation, general prismatic elements, symmetric indefinite linear systems. AMS classification: 65N30, 65K10, 73C99 1 Introduction Solution of the underground water flow problem in real conditions must reflect complex geological structure of sedimented minerals. Layers of stratified rocks with substantially different physical properties must be modelled using an appropriate disc..

The loss of orthogonality in the Gram-Schmidt orthogonalization process

AbstractIn this paper, we study numerical behavior of several computational variants of the Gram-Schmidt orthogonalization process. We focus on the orthogonality of computed vectors which may be significantly lost in the classical or modified Gram-Schmidt algorithm, while the Gram-Schmidt algorithm with reorthogonalization has been shown to compute vectors which are orthogonal to machine precision level. The implications for practical implementation and its impact on the efficiency in the parallel computer environment are considered