A New Quotient and Iterative Detection Method in an Affine Krylov Subspace for Solving Eigenvalue Problems

Abstract

The paper solves the eigenvalues of a symmetric matrix by using three novel algorithms developed in the m-dimensional affine Krylov subspace. The n-dimensional eigenvector is superposed by a constant shifting vector and an m-vector. In the first algorithm, the m-vector is derived as a function of eigenvalue by maximizing the Rayleigh quotient to generate the first characteristic equation, which, however, is not easy to determine the eigenvalues since its roots are not of simple ones, exhibiting turning points, spikes, and even no intersecting point to the zero line. To overcome that difficulty by the first algorithm, we propose the second characteristic equation through a new quotient with the inner product of the shifting vector to the eigen-equation. The Newton method and the fictitious time integration method are convergent very fast due to the simple roots of the second characteristic equation. For both symmetric and nonsymmetric eigenvalue problems solved by the third algorithm, we develop a simple iterative detection method to maximize the Euclidean norm of the eigenvector in terms of the eigen-parameter, of which the peaks of the response curve correspond to the eigenvalues. Through a few finer tunings to the smaller intervals sequentially, a very accurate eigenvalue and eigenvector can be obtained. The efficiency and accuracy of the proposed iterative algorithms are verified and compared to the Lanczos algorithm and the Rayleigh quotient iteration method.