Best pseudo-isolated Gerschgorin disks for Eigenvalues

Let A be an arbitrary n×n matrix, partitioned so that if A=A_ij], then all submatrices A_ii are square. If x is a positive vector, it is well-known that $\text{G(}\text{x}\text{) =∪}{}^{\mathrm{N}}{}_{\mathrm{i=1}}\text{G}{}_{\mathrm{i}}\text{(}\text{x}\text{)}$, where

$\text{G}{}_{\mathrm{i}}\text{(}\text{x}\text{) =}\text{z}\text{6(zI ? A}{}_{\mathrm{ii}}\text{)}{}^{\mathrm{?1}}\text{6}{}^{\mathrm{?1}}\text{?}\text{1}\text{x}{}_{\mathrm{i}}\text{∑}\text{j = 1}\text{j ≠ i}\text{N}\text{`6A}{}_{\mathrm{ij}}\text{6x}{}_{\mathrm{j}}$

, 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 G_i(x) is replaced by the union of circular disks centered at the eigenvalues of A_ii.