Mathematical background

Let a system $F(X)=0$, $J$ the Jacobian matrix of this system and $X0$ a solution of the system. The purpose of the inflation method is to build a box that will contain only this solution. Let $B(X0)$ be a ball centered at $X0$: if for any point in $B$ $J$ 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 $H$ be the matrix $J(X0)^{-1}J(B)$ whose components are intervals. Let $u$ be the diagonal element of H having the lowest absolute value, let $v_i$ be the maximum of the absolute value of the sum of the elements at row $i$ of $H$, discarding the diagonal element of the row and let $v$ be the maximum of the $v_i$'s. If $u>v$, then the matrix is denoted diagonally dominant and all the matrices $J(B)$ are regular [19].

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

$$\begin{eqnarray*} &&B_0=X0\\ &&B_n=[B_{n-1}-\epsilon,B_{n-1}+\epsilon] \end{eqnarray*}$$

that will be repeated until $B_n$ is no more diagonally dominant. $\epsilon$ Note that in some cases (see for example section 2.16) it is possible to calculate directly the largest possible $\epsilon$ so that all the matrices in $J([B_0-\epsilon,B_0+\epsilon])$ are regular without relying on the iterative scheme.