Fast enclosure for all eigenvalues in generalized eigenvalue problems

A fast method for enclosing all eigenvalues in generalized eigenvalue problems Ax=λBxAx=λBx is proposed. Firstly a theorem for enclosing all eigenvalues, which is applicable even if AA is not Hermitian and/or BB is not Hermitian positive definite, is presented. Next a theorem for accelerating the enclosure is presented. The proposed method is established based on these theorems. Numerical examples show the performance and property of the proposed method. As an application of the proposed method, an efficient method for enclosing all eigenvalues in polynomial eigenvalue problems is also sketched.