In the context of Krylov methods for solving systems of linear equations, expressions and bounds are derived for the norm of the minimal residual, like the one produced by GMRES or MINRES. It is shown that the minimal residual norm is large as long as the Krylov basis is well-conditioned. In the context of non-normal matrices, examples are given where the minimal residual norm is a function of ...

This paper starts oo with studying simple extrapolation methods for the classical iteration schemes such as Richardson, Jacobi and Gauss-Seidel iteration. The extrapolation procedures can be interpreted as approximate minimal residual methods in a Krylov subspace. It seems therefore logical to consider, conversely, classical methods as pre-processors for Krylov subspace methods, as was done by ...

The rational Krylov space is recognized as a powerful tool within Model Order Reduction techniques for linear dynamical systems. However, its success has been hindered by the lack of a parameter-free procedure, which would effectively generate the sequence of shifts used to build the space. In this talk we propose an adaptive computation of these shifts. The whole procedure only requires to inj...

By introducing the second order Krylov subspace, a method for the reduction of second order systems is proposed leading to a reduced system of the same structure. This generalization of Krylov subspace involves two matrices and some starting vectors and the reduced order model is found by applying a projection directly to the second order model without any conversion to state space. A numerical...

In this paper we introduce a two-sided Arnoldi method that can be used in the reduction of high order systems, based on a two-sided Krylov subspace approach. The presented method can find better results than the well known Arnoldi algorithm and leads to the same reduced models as the (numerically unstable and more complicated) Lanczos algorithm. The new algorithm can find two orthogonal bases f...

It is well known that the conjugate gradient method and a quasi-Newton method, using any well-defined update matrix from the one-parameter Broyden family of updates, produce the same iterates on a quadratic problem with positive-definite Hessian. This equivalence does not hold for any quasi-Newton method. We discuss more precisely the conditions on the update matrix that give rise to this behav...

In this note we exploit the knowledge embodied in innnitesi-mal generators of Markov processes to compute eeciently and economically the transient solution of continuous time Markov processes. We consider the Krylov subspace approximation method which has been analysed by Y. Saad for solving linear diierential equations. We place special emphasis on error bounds and stepsize control. We discuss...

Constrained least squares problems arise in a variety of applications, and many iterative methods are already available to compute their solutions. This paper proposes a new efficient approach to solve nonnegative linear least squares problems. The associated KKT conditions are leveraged to form an adaptively preconditioned linear system, which is then solved by a flexible Krylov subspace metho...