This extends a method by S. Friedland [Linear Algebra Appl. 17, 15- 51 (1977; Zbl 0358.15007)] for determining all diagonal matrices \(D\) such that \(A+D\) with given real symmetric \(A\) has a given real spectrum \(\omega\); the motivation comes from the correction of energy levels in atomic systems.
The authors discuss the case of real solution (obtained by an algorithm based on Newton’s method) and place particular emphasis on the case when no real solutions exist. For the latter case they give an algorithm (different from Friedland’s) for obtaining a real \(D^*\) such that \(\| \lambda^*-\omega\|\) is minimum, avoiding slow convergence that might otherwise occur. An error analysis and numerical results are given.