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