Eigenvalues Of Large Symmetric Matrices Research Articles

Iterative methods for calculations of extreme eigenvalues of large symmetric matrices

It is shown that the iterative methods for the calculation of the extreme eigenvalues and corresponding eigenvectors of the generalized symmetric matrix eigenvalue problem can be divided into two general classes that differ from each other in the method of combining the Krylov subspace with iterations. The paper demonstrates unused possibilities in the development of iterative methods. Correspondingly, many new iterative methods are presented. Difficult problems related to the classification of some methods are also considered.

Kron's method for symmetric eigenvalue problems

Kron's method has been used successfully by engineers for over 30 years to find eigenvalues of large symmetric matrices. These matrices arise from domain decomposition of nonoverlapping domains of self-adjoint partial differential operators. This paper discusses some theoretical aspects of Kron's method. Specifically, the poles and zeroes of eigenvalues of the Kron matrix and their relationships with the eigenvalues of the original matrix are given.

Computation of dominant eigenvalues and eigenvectors: A comparative study of algorithms.

We investigate two widely used recursive algorithms for the computation of eigenvectors with extreme eigenvalues of large symmetric matrices---the modified Lancz\os method and the conjugate-gradient method. The goal is to establish a connection between their underlying principles and to evaluate their performance in applications to Hamiltonian and transfer matrices of selected model systems of interest in condensed matter physics and statistical mechanics. The conjugate-gradient method is found to converge more rapidly for understandable reasons, while storage requirements are the same for both methods.

On the asymptotic distribution of the eigenvalues of random matrices

Abstract : A generalization of E. P. Wigner's semi-circle law concerning the distribution of the eigenvalues of large symmetric matrices A = (a sub ij) whose entries a sub ij are for i > or = j independent and identically distributed random variables. (Author)