INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING Int. J. Numer. Meth. Engng 2006; 68:401–424 Published online 30 March 2006 in Wiley InterScience (www.interscience.wiley.com). DOI: 10.1002/nme.1712 Iterative solution of the random eigenvalue problem with application to spectral stochastic finite element systems C. V. Verhoosel, M. A. Gutiérrez ∗, † and S. J. Hulshoff Faculty of Aerospace Engineering, Delft University of Technology, 2600 GB Delft, The Netherlands SUMMARY A new algorithm for the computation of the spectral expansion of the eigenvalues and eigenvectors of a random non-symmetric matrix is proposed. The algorithm extends the deterministic inverse power method using a spectral discretization approach. The convergence and accuracy of the algorithm is studied for both symmetric and non-symmetric matrices. The method turns out to be efficient and robust compared to existing methods for the computation of the spectral expansion of random eigenvalues and eigenvectors. Copyright  2006 John Wiley & Sons, Ltd. KEY WORDS: random eigenvalue problem; stochastic finite elements; inverse power method 1. INTRODUCTION Algebraic eigenvalue problems play an important role in a variety of fields. In structural mechanics, eigenvalue problems commonly appear in the context of, e.g. vibrations and buckling. Currently, the computation of eigenvalues and eigenvectors is well understood for deterministic problems [1, 2]. In many practical cases, however, physical characteristics are not deterministic. For example, the stiffness of a plate can locally be reduced by material imperfections, or the velocity of a flow can be influenced by turbulence. The traditional approach of dealing with these kind of uncertainties in structures or loading conditions is to use safety factors. Since the influence of uncertainties is in general unknown, the safety factors that are used need to be conservative, leading to overdimensioned structures and decreased economical efficiency. This overdimensioning can be reduced by describing the uncertain problem characteristics more realistically using random variables, since more insight in the effect of uncertainties is then obtained. The effect of describing the input parameters of a physical problem as random variables is that also the desired output will be random. The methods to compute these random results are ∗ Correspondence to: M. A. Gutiérrez, Faculty of Aerospace Engineering, Delft University of Technology, P.O. Box 5058, 2600 GB, Delft, The Netherlands. † E-mail: m.gutierrez.tudelft@gmail.com Received 19 October 2005 Revised 9 February 2006 Copyright  2006 John Wiley & Sons, Ltd. Accepted 9 February 2006