The authors consider the following problem (called the additive inverse eigenvalue problem): Given \(n + 1\) real symmetric \(n \times n\) matrices \(A_ i\), \(i = 0, 1, \dots, n\), and given \(n\) real numbers \(\lambda_ 1 < \lambda_ 2 < \dots < \lambda_ n\), prove that there are \(n\) real numbers \(c^*_ i\), \(i = 1, \dots, n\), such that the matrix \(A(c) : = A_ 0 + \sum^ n_{i=1} c_ i A_ i\), \(c : = (c_ i) \in \mathbb{R}^ n\), has for \(c_ i = c^*_ i\), \(i = 1,\dots,n\), the prescribed numbers \(\lambda_ i\) as eigenvalues. The authors solve this problem by means of interval computation. The proposed algorithm is based on Newton's method using a new criterion for terminating the iteration, in order to get an approximation \(\widetilde c\) of \(c^* = (c^*_ i)\). The existence of \(c^*\) using one step of the Krawczyk method is verified. Two numerical examples are reported.