Best pseudo-isolated Gerschgorin disks for Eigenvalues

AbstractLet A be an arbitrary n×n matrix, partitioned so that if A=[Aij], then all submatrices Aii are square. If x is a positive vector, it is well-known that G(x) =∪Ni=1Gi(x), where Gi(x) = z‖(zI − Aii)−1‖−1 ⩽ 1xi∑j = 1j ≠ iN`‖Aij‖xj, contains all the eigenvalues of A. The purpose of this paper is to give a new definition of the concept of an isolated subregion of G(x). An algorithm is given for obtaining the best such isolated subregion in a certain sense, and examples are given to show that tighter bounds for some eigenvalues of A may be obtained than with previous algorithms. For ease of computation, each subregion Gi(x) is replaced by the union of circular disks centered at the eigenvalues of Aii