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.