This report provides an introductory overview of the numerical solution of large scale algebraic eigenvalue problems. The main focus is on a class of methods called Krylov subspace projection methods. The Lanczos method is the premier member of this class and the Arnoldi method is a generalization to the nonsymmetric case. A recently developed and very promising variant of the Arnoldi/Lanczos s...

It is very natural to associate the accuracy of the eigenvector with this quantity from a geometric perspective. The indicator in the right-hand side of (1.1) is called (the norm of) the orthogonal complement of the projection of xi onto the space spanned by Q and it can be interpreted as the sine of the canonical angle between the Krylov subspace and an eigenvector. For the moment, we will onl...

In this paper we study numerical methods for solving Sylvester matrix equations of the form AX +XB +CD = 0. A new projection method is proposed. The union of Krylov subspaces in A and its inverse and the union of Krylov subspaces in B and its inverse are used as the right and left projection subspaces, respectively. The Arnoldi-like process for constructing the orthonormal basis of the projecti...

We propose an accelerating method for the restarted Arnoldi iteration to compute a number of eigenvalues of the standard eigenproblem Ax = x and discuss the dependence of the convergence rate of the accelerated iteration on the distribution of spectrum. The e ectiveness of the approach is proved by numerical results. We also propose a new parallelization technique for the nonsymmetric double sh...

The Arnoldi process is a well known technique for approximating a few eigenvalues and corresponding eigenvectors of a general square matrix. Numerical difficulties such as loss of orthogonality and assessment of the numerical quality of the approximations as well as a potential for unbounded growth in storage have limited the applicability of the method. These issues are addressed by fixing the...

In this report the Krylov subspace methods are reviewed and some applications in linear system theory and modern control theory are introduced. A modiication to the Arnoldi-based method to solve the Lyapunov matrix equation is also proposed.

The Explicitly Restarted Arnoldi Method (ERAM) allows to find a few eigenpairs of a large sparse matrix. The Multiple Explicitly Restarted Arnoldi Method (MERAM) is a technique based upon a multiple projection of ERAM and accelerates its convergence [3]. The MERAM allows to update the restarting vector of an ERAM by taking the interesting eigen-information obtained by the other ones into accoun...

The spectral transformation Lanczos method and the shift-invert Arnoldi method are probably the most popular methods for the solution of linear generalized eigenvalue problems originating from engineering applications, including structural and acoustic analyses and fluid dynamics. The orthogonalization of the Krylov vectors requires inner products. Often, one employs the standard inner product,...