Numerical analysis of Eigenvalue algorithms based on subspace iterations

Eigenvalue problems are important in many applications involving mathematical modelling. The development of hardware and software over the years has enlarged the class of problems that can be solved efficiently. In particular, the iterative algorithms that project the problem to low-dimensional subspaces are able to find eigenvalues of very large matrices. Several aspects of such algorithms are treated here from a theoretical point of view. After considering an upperbound of the errors in approximations of an eigenvector generated by the projection-procedure, the well-known Arnoldi algorithm is described as a function of its parameters in order to answer the question which parameters generate the same sequence of eigenvalue- approximations. Then some inexact subspace methods from three different classes are described and analysed. The inexactness is caused by the fact that the solutions of the systems of linear equations that occur in these algorithms, are not calculated exactly, but only approximated.