\input zb-basic
\input zb-ioport
\iteman{io-port 06015764}
\itemau{Miyajima, Shinya}
\itemti{Numerical enclosure for each eigenvalue in generalized eigenvalue problem.}
\itemso{J. Comput. Appl. Math. 236, No. 9, 2545-2552 (2012).}
\itemab
The paper presents an algorithm for enclosing all eigenvalues in the generalized eigenvalue problem $$Ax=\lambda Bx,\;A,B\in {\Bbb C}^{n\times n},\;\lambda\in{\Bbb C},\;x\in{\Bbb C}^n\tag{1}$$ where $\lambda$ is the eigenvalue and $x\neq 0$ is an eigenvector corresponding to $\lambda.$ This algorithm is applicable even if $A\in {\Bbb C}^{n\times n}$ is not Hermitian and/or $B\in{\Bbb C}^{n\times n}$ is not Hermitian positive definite, and supplies {\it n error bounds} $r_1,\dots,r_n$ such that the all eigenvalues are included in the set $\bigcup_{i=1}^{n}\{z\in{\Bbb C}:|z-\overline\lambda_i|\leq r_i\}$ when $\overline D\in{\Bbb C}^{n\times n}$ is a diagonal matrix ($\lambda_i:=\overline D_{ii},\; i=1,\dots,n$) and $\overline X\in{\Bbb C}^{n\times n}$ 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.
\itemrv{R. Militaru (Craiova)}
\itemcc{}
\itemut{generalized eigenvalue problem; numerical enclosure; non-Hermitian matrices; eigenvector; numerical examples}
\itemli{doi:10.1016/j.cam.2011.12.013}
\end