The goal of our report is to compare a number of algorithms for computing a large number of eigenvectors of the generalized eigenvalue problem arising from a modal analysis of elastic structures using preconditioned iterative methods. The shift-invert Lanczos algorithm has emerged as the workhorse for the solution of this generalized eigenvalue problem; however a sparse direct method is require...

In this paper we develop descriptor techniques for modeling swirling hydrodynamic structures. Using the descriptor notation we obtain the generalized eigenvalue formulation for differential-algebraic equations describing the spatial stability of the fluid flow. We describe the general framework for spatial stability investigation of vortex structures in both viscous and inviscid cases. The disp...

The paper deals with eigenvalue estimates for block incomplete factorization methods for symmetric matrices. First, some previous results on upper bounds for the maximum eigenvalue of preconditioned matrices are generalized to each eigenvalue. Second, upper bounds for the maximum eigenvalue of the preconditioned matrix are further estimated, which presents a substantial improvement of earlier r...

We present an iterative method to compute the Maxwell’s transmission eigenvalue problem which has importance in non-destructive testing of anisotropic materials. The transmission eigenvalue problem is first written as a quad-curl eigenvalue problem. Then we show that the real transmission eigenvalues are the roots of a non-linear function whose value is the generalized eigenvalue of a related s...

Feedback design for a second order control system leads to an eigenstructure assignment problem for a quadratic matrix polynomial. It is desirable that the feedback controller not only assigns specified eigenvalues to the second order closed loop system, but also that the system is robust, or insensitive to perturbations. We show that robustness of the quadratic inverse eigenvalue problem can b...

We present an extremely fast graph drawing algorithm for very large graphs, which we term ACE (for Algebraic multigrid Computation of Eigenvectors). ACE exhibits an improvement of something like two orders of magnitude over the fastest algorithms we are aware of; it draws graphs of millions of nodes in less than a minute. ACE finds an optimal drawing by minimizing a quadratic energy function. T...

We analyze the systematic errors made when using the generalized eigenvalue problem to extract energies and matrix elements in lattice gauge theory. Effective theories such as HQET are also discussed. Numerical results are shown for the extraction of ground-state and excited B-meson masses and the ground-state decay constant in the static approximation.

Let G be a connected reductive subgroup of a complex connected reductive group Ĝ. Fix maximal tori and Borel subgroups of G and Ĝ. Consider the cone LR(G, Ĝ) generated by the pairs (ν, ν̂) of dominant characters such that Vν is a submodule of Vν̂ (with usual notation). Here we give a minimal set of inequalities describing LR(G, Ĝ) as a part of the dominant chamber. In way, we obtain results about...

In this short note we highlight the fact that linear blind source separation can be formulated as a generalized eigenvalue decomposition under the assumptions of non-Gaussian, non-stationary, or non-white independent sources. The solution for the unmixing matrix is given by the generalized eigenvectors that simultaneously diagonalize the covariance matrix of the observations and an additional s...

When a symmetric, positive, isomorphism between a reeexive Banach space (that is densely and compactly embedded in a Hilbert space) and its dual varies smoothly over a Banach space, its eigenvalues vary in a Lipschitz manner. We calculate the generalized gradient of the extreme eigenvalues at an arbitrary crossing. We apply this to the generalized gradient, with respect to a coeecient in an ell...