Spectral portrait for non-hermitian large sparse matrices

Abstract

The spectral portrait of a matrix is the picture of its ɛ-spectra forε ∈[ε 1,ε 2], where an ɛ-spectrum ofA is the union of all the eigenvalues of all the matricesA+Δ with ∥Δ∥2≤ε∥A∥2. The spectral portrait is, for example, useful to study the stability of various problems, or, as we illustrate in this paper, to visualize the condition number of an eigenvalue. Some methods to estimate the spectral portrait already exist, but only for small matrices. We propose here a new algorithm for non hermitian large sparse matrices.