Linear algebra and power method combined with method of deflation applied to Toeplitz sinc matrices: A new approach for the computation of eigenvalues and eigenvectors

Abstract

A new robust method for the computation of the eigenvalues and eigenvectors of real symmetric sinc \(n \times n\)-matrices \(A(t)=A(t, n)\) with \(t\) in \(0<t<1\) is developed. If the eigenvalues \(\mu_j(t)=\mu_j(A(t))=\mu_j(A(t, n)), j=\) \(1, \ldots, n\) are arranged in descending order, one has \(\mu_1(t) \rightarrow n(t \rightarrow 0)\) and \(\mu_j(t) \rightarrow 0(t \rightarrow 0), j=2, \ldots, n\). The new approach consists in computing only a limited number \(i_q\) of the eigenvalues and to set the remaining very small eigenvalues equal to zero, of which the corresponding eigenvectors are reconstructed via the Gram-Schmidt process. This opens the way for conceiving a new method for the computation of the eigenvalues and associated eigenvectors of severely ill-conditioned real symmetric Toeplitz sinc matrices that delivers good results.