Numerical enclosure for each eigenvalue in generalized eigenvalue problem (Q765307)

The paper presents an algorithm for enclosing all eigenvalues in the generalized eigenvalue problem \[ Ax=\lambda Bx,\;A,B\in {\mathbb C}^{n\times n},\;\lambda\in{\mathbb C},\;x\in{\mathbb C}^n\tag{1} \] where \(\lambda\) is the eigenvalue and \(x\neq 0\) is an eigenvector corresponding to \(\lambda.\) This algorithm is applicable even if \(A\in {\mathbb C}^{n\times n}\) is not Hermitian and/or \(B\in{\mathbb C}^{n\times n}\) is not Hermitian positive definite, and supplies \textit{n error bounds} \(r_1,\dots,r_n\) such that the all eigenvalues are included in the set \(\bigcup_{i=1}^{n}\{z\in{\mathbb C}:|z-\overline\lambda_i|\leq r_i\}\) when \(\overline D\in{\mathbb C}^{n\times n}\) is a diagonal matrix (\(\lambda_i:=\overline D_{ii},\; i=1,\dots,n\)) and \(\overline X\in{\mathbb C}^{n\times n}\) such that \(A\overline X=B\overline X\overline D\) are given. The first section is an introductory one. The second section establishes the theory for computing \(r_1,\dots,r_n.\) The third section proposes an algorithm for enclosing all eigenvalues in ({1}). The efficiency of the proposed algorithm is proved through four numerical examples presented in the fourth section. The main conclusions are exposed in the last section.