Hamiltonian System Approach to Distributed Spectral Decomposition in Networks

Because of the significant increase in size and complexity of the networks, the distributed computation of eigenvalues and eigenvectors of graph matrices has become very challenging and yet it remains as important as before. In this paper we develop efficient distributed algorithms to detect, with higher resolution, closely situated eigenvalues and corresponding eigenvectors of symmetric graph matrices. We model the system of graph spectral computation as physical systems with Lagrangian and Hamiltonian dynamics. The spectrum of Laplacian matrix, in particular, is framed as a classical spring-mass system with Lagrangian dynamics. The spectrum of any general symmetric graph matrix turns out to have a simple connection with quantum systems and it can be thus formulated as a solution to a Schr\"odinger-type differential equation. Taking into account the higher resolution requirement in the spectrum computation and the related stability issues in the numerical solution of the underlying differential equation, we propose the application of symplectic integrators to the calculation of eigenspectrum. The effectiveness of the proposed techniques is demonstrated with numerical simulations on real-world networks of different sizes and complexities

A comparison of the LR and QR transformations for finding the eigenvalues for real nonsymmetric matrices

The LR and QR algorithms, two of the best available iterative methods for finding the eigenvalues of a nonsymmetric matrix associated with a system of linear homogeneous equations, are studied. These algorithms are discussed as they apply to the determination of the eigenvalues of real nonsymmetric matrices. A comparison of the speed and accuracy of these transformations is made. A detailed discussion of the criterion for convergence and the numerical difficulties which may occur in the computation of multiple and complex conjugate eigenvalues are included. The results of this study indicate that the QR algorithm is the more successful method for finding the eigenvalues of a real nonsymmetric matrix --Abstract, page ii

Fast computation of the matrix exponential for a Toeplitz matrix

The computation of the matrix exponential is a ubiquitous operation in numerical mathematics, and for a general, unstructured $n\times n$ matrix it can be computed in $\mathcal{O}(n^3)$ operations. An interesting problem arises if the input matrix is a Toeplitz matrix, for example as the result of discretizing integral equations with a time invariant kernel. In this case it is not obvious how to take advantage of the Toeplitz structure, as the exponential of a Toeplitz matrix is, in general, not a Toeplitz matrix itself. The main contribution of this work are fast algorithms for the computation of the Toeplitz matrix exponential. The algorithms have provable quadratic complexity if the spectrum is real, or sectorial, or more generally, if the imaginary parts of the rightmost eigenvalues do not vary too much. They may be efficient even outside these spectral constraints. They are based on the scaling and squaring framework, and their analysis connects classical results from rational approximation theory to matrices of low displacement rank. As an example, the developed methods are applied to Merton's jump-diffusion model for option pricing

Computational error bounds for multiple or nearly multiple eigenvalues

AbstractIn this paper bounds for clusters of eigenvalues of non-selfadjoint matrices are investigated. We describe a method for the computation of rigorous error bounds for multiple or nearly multiple eigenvalues, and for a basis of the corresponding invariant subspaces. The input matrix may be real or complex, dense or sparse. The method is based on a quadratically convergent Newton-like method; it includes the case of defective eigenvalues, uncertain input matrices and the generalized eigenvalue problem. Computational results show that verified bounds are still computed even if other eigenvalues or clusters are nearby the eigenvalues under consideration

A direct method for completing eigenproblem solutions on a parallel computer

AbstractThe computation of eigenvalues and eigenvectors of a real symmetric matrix A with distinct eigenvalues can be speeded up at the end of the Jacobi process when the off-diagonal elements have become sufficiently small for A to be regarded as a perturbation of a diagonal matrix. A leading-order approximation to the eigensolution is calculated by formulae particularly suitable for the distributed array processor (DAP). A single application of this direct method reduces A to diagonal form and is asymptotically equivalent to an entire sweep of the Jacobi method