We consider the problem of how to compute eigenvalues of a self-adjoint operator when a direct application of the Galerkin (finite-section) method is unreliable. The last two decades have seen the development of the so-called quadratic methods for ad
dressing this problem. Recently a new perturbation approach has emerged, the idea being to perturb eigenvalues off the real line and, consequently, away from regions where the Galerkin method fails. We propose a simplified perturbation method which requires no a priori information and for which we provide a rigorous convergence analysis. The latter shows that, in general, our approach will significantly outperform the quadratic methods. We also present a new spectral enclosure for operators of the form $A+iB$ where $A$ is self-adjoint, $B$ is self-adjoint and bounded. This enables us to control, very precisely, how eigenvalues are perturbed from the real line. The main results are demonstrated with examples including magnetohydrodynamics, Schrodinger and Dirac operators.
Although coherent large-scale structures such as filaments and walls are apparent to the eye in galaxy redshift surveys, they have so far proven difficult to characterize with computer algorithms. This paper presents a procedure that uses the eigenva
lues and eigenvectors of the Hessian matrix of the galaxy density field to characterize the morphology of large-scale structure. By analysing the smoothed density field and its Hessian matrix, we can determine the types of structure - walls, filaments, or clumps - that dominate the large-scale distribution of galaxies as a function of scale. We have run the algorithm on mock galaxy distributions in a LCDM cosmological N-body simulation and the observed galaxy distributions in the Sloan Digital Sky Survey. The morphology of structure is similar between the two catalogues, both being filament-dominated on 10-20 h^{-1} Mpc smoothing scales and clump-dominated on 5 h^{-1} Mpc scales. There is evidence for walls in both distributions, but walls are not the dominant structures on scales smaller than ~25 h^{-1} Mpc. Analysis of the simulation suggests that, on a given comoving smoothing scale, structures evolve with time from walls to filaments to clumps, where those found on smaller smoothing scales are further in this progression at a given time.
