Mathematical background

The Cassini ovals are another method to determine a bound for the eigenvalues of a matrix. Let $A=((a_{ij}))$ and the Cassini ovals defined as the set of $z$ such that:

$$O_{jk} =\vert z-a_{jj}\vert\vert z-a{kk} \le (\sum_i \vert a_{ji}\vert, i \not= j)(\sum_i \vert a_{ki}\vert, i \not= k)$$

Column based Cassini ovals may also be defined. The roots of the characteristic polynomial are enclosed in the union of the row-based and column-based $O_{jk}$. Although more complicate to calculate the bounds obtained with the Cassini ovals are usually tighter than the bounds obtained with the Gerschgorin circles.