A filtering method for the interval eigenvalue problem
The real eigenvalue set \(\Lambda({\mathbf A})\) of an interval matrix \({\mathbf A}= [\underline A,\overline A]\) is defined by \(\Lambda({\mathbf A})= \{\lambda\in\mathbb{R}\mid Ax=\lambda x\), \(x\neq 0\), \(A\in{\mathbf A}\}\). A filtering method for \(\Lambda({\mathbf A})\) is a method which iteratively cuts off some parts from a given interval \({\mathbf a}\) that do not include any eigenvalue. Finally it delivers an interval \({\mathbf b}\subseteq{\mathbf a}\) such that \(({\mathbf a}\setminus{\mathbf b})\cap\Lambda({\mathbf A})=\emptyset\). The authors present such a filtering method based on a sufficient regularity condition for a shifted matrix of \({\mathbf A}\). They apply it also to symmetric interval matrices \({\mathbf A}^S= \{A\in{\mathbf A}\mid A= A^T\}\) in order to get enclosures of the corresponding eigenvalue set \(\Lambda({\mathbf A}^S)\subseteq\Lambda({\mathbf A})\). By three numerical examples they demonstrate the efficiency of their method.
scientific article; zbMATH DE number 5862460
scientific article