4 research outputs found
Implicit QR algorithms for palindromic and even eigenvalue problems
In the spirit of the Hamiltonian QR algorithm and other bidirectional chasing algorithms, a structure-preserving variant of the implicit QR algorithm for palindromic eigenvalue problems is proposed. This new palindromic QR algorithm is strongly backward stable and requires less operations than the standard QZ algorithm, but is restricted to matrix classes where a preliminary reduction to structured Hessenberg form can be performed. By an extension of the implicit Q theorem, the palindromic QR algorithm is shown to be equivalent to a previously developed explicit version. Also, the classical convergence theory for the QR algorithm can be extended to prove local quadratic convergence. We briefly demonstrate how even eigenvalue problems can be addressed by similar techniques. © 2008 Springer Science+Business Media, LLC
Spectral properties of kernel matrices in the flat limit
Kernel matrices are of central importance to many applied fields. In this
manuscript, we focus on spectral properties of kernel matrices in the so-called
"flat limit", which occurs when points are close together relative to the scale
of the kernel. We establish asymptotic expressions for the determinants of the
kernel matrices, which we then leverage to obtain asymptotic expressions for
the main terms of the eigenvalues. Analyticity of the eigenprojectors yields
expressions for limiting eigenvectors, which are strongly tied to discrete
orthogonal polynomials. Both smooth and finitely smooth kernels are covered,
with stronger results available in the finite smoothness case.Comment: 40 pages, 8 page