The need to determine a few eigenvalues of a large sparse generalised eigenvalue problem Ax = Bx with semi-positive deenite B arises in many physical situations, for example, in a stability analysis of the discretised Navier-Stokes equation. A common technique is to apply Arnoldi's method to the shift-invert transformation, but this can suuer from numerical instabilities as is illustrated by a ...

For the solution of full-rank ill-posed linear systems a new approach based on the Arnoldi algorithm is presented. Working with regularized systems, the method theoretically reconstructs the true solution by means of the computation of a suitable function of matrix. In this sense the method can be referred to as an iterative refinement process. Numerical experiments arising from integral equati...

Even if both A and B are real-valued, it is likely that λ and x are complexvalued. Finding the solution of eigensystems is a fairly complicated procedure. It is at least as difficult as finding the roots of polynomials. Therefore, any numerical method for solving eigenvalue problems is expected to be iterative in nature. Algorithms for solving eigenvalue problems include the power method, subsp...

We consider the special case of the restarted Arnoldi method for approximating the product of a function of a Hermitian matrix with a vector which results when the restart length is set to one. When applied to the solution of a linear system of equations, this approach coincides with the method of steepest descent. We show that the method is equivalent with an interpolation process in which the...

The need to determine a few eigenvalues of a large sparse generalised eigenvalue problem Ax = λBx with positive semidefinite B arises in many physical situations, for example, in a stability analysis of the discretised Navier-Stokes equation. A common technique is to apply Arnoldi’s method to the shift-invert transformation, but this can suffer from numerical instabilities as is illustrated by ...

We discuss the computational problems when analyzing general, non-hermitian matrices and in particular the un-modified Wilson lattice Dirac operator. We report on our experiences with the Implicitly Restarted Arnoldi Method. The eigenstates of the Wilson-Dirac operator which have real eigenvalues and correspond to zero modes in the continuum are analyzed by correlating the size of the eigenvalu...

The construction of a closed-loop observer for linear control systems and the associated eigenvalue assignment problem are classical tasks in Control Theory. The present paper describes an algorithm for the solution of this eigenvalue assignment problem. The algorithm is based on the implicitly restarted Arnoldi method, and is well suited for large-scale problems, that require the (re)assignmen...