On the inverse eigenvalue problem for matrices and related problems for difference and differential equations

The problem of estimating parameters α1,...,αk of the matrix valued function M(λ,α) given eigenvalue data λ1,...,λp, p ≥ k, is considered. Two algorithms are presented. The first reduces the estimation problem to an unconstrained minimisation and contains as special cases methods suggested by other authors. The second reduces the problem to one of minimisation subject to equality constraints. Examples are given to show that the behaviour of the solutions can be involved so that the application of numerical methods is probably of necessity tentative. The results of some numerical experiments are summarised.