Let a system , the Jacobian matrix of this system and a solution of the system. The purpose of the inflation method is to build a box that will contain only this solution. Let be a ball centered at : if for any point in is not singular, then the ball contains only one solution of the system.

The problem now is to determine a ball such for any point in the ball
the Jacobian is regular. Let be the matrix
whose
components are intervals. Let be the diagonal element of H having
the lowest absolute value, let be the maximum of the absolute
value of the sum of the elements at row of , discarding the
diagonal element of the row and let be the maximum of the
's. If , then the matrix is denoted *diagonally dominant*
and all the matrices are regular [19].

Let be a small constant: we will build incrementally the ball by using an iterative scheme defined as:

that will be repeated until is no more diagonally dominant. Note that in some cases (see for example section 2.16) it is possible to calculate directly the largest possible so that all the matrices in are regular without relying on the iterative scheme.