Miroslav Rozložník

Rounding error analysis of the classical Gram-Schmidt orthogonalization process

This paper provides two results on the numerical behavior of the classical Gram-Schmidt algorithm. The first result states that, provided the normal equations associated with the initial vectors are numerically nonsingular, the loss of orthogonality of the vectors computed by the classical Gram-Schmidt algorithm depends quadratically on the condition number of the initial vectors. The second result states that, provided the initial set of vectors has numerical full rank, two iterations of the classical Gram-Schmidt algorithm are enough for ensuring the orthogonality of the computed vectors to be close to the unit roundoff level.

NUMERICAL STABILITY OF GMRES

AbstractThe Generalized Minimal Residual Method (GMRES) is one of the significant methods for solving linear algebraic systems with nonsymmetric matrices. It minimizes the norm of the residual on the linear variety determined by the initial residual and then-th Krylov residual subspace and is therefore optimal, with respect to the size of the residual, in the class of Krylov subspace methods. One possible way of computing the GMRES approximations is based on constructing the orthonormal basis of the Krylov subspaces (Arnoldi basis) and then solving the transformed least squares problem. This paper studies the numerical stability of such formulations of GMRES. Our approach is based on the Arnoldi recurrence for the actually, i.e., in finite precision arithmetic, computed quantities. We consider the Householder (HHA), iterated modified Gram-Schmidt (IMGSA), and iterated classical Gram-Schmidt (ICGSA) implementations. Under the obvious assumption on the numerical nonsingularity of the system matrix, the HHA implementation of GMRES is proved backward stable in the normwise sense. That is, the backward error

Mixed-hybrid finite element approximation of the potential fluid flow problem

||b - A\tilde x||/(||A||{\text{ ||}}\tilde x|| + ||b||)

Partitioned Triangular Tridiagonalization

for the approximation

On the stable implementation of the generalized minimal error method

\tilde x

Approximate inverse preconditioners with adaptive dropping

is proportional to machine precision ε. Additionally, it is shown that in most cases the norm of the residual computed from the transformed least squares problem (Arnoldi residual) gives a good estimate of the true residual norm, until the true residual norm has reached the level ε‖A‖ ‖x‖.