PAT – a Reliable Path-Following Algorithm

Abstract

This paper presents a new technique for the reliable computation of the σ-pseudospectrum defined by Λσ(A)={z∈C : σmin(A−zI)≤σ} where σmin is the smallest singular value. The proposed algorithm builds an orbit of adjacent equilateral triangles to capture the level curve ϒσ(A)={z∈C : σmin(A−zI)=σ} and uses a bisection procedure on specific triangle vertices to compute a numerical approximation to ϒσ. The method is guaranteed to terminate, even in the presence of round-off errors.