Nonnegative inverse eigenvalue problems with partial eigendata

In this paper we consider the inverse problem of constructing an
$n$-by-$n$ real nonnegative matrix $A$ from the prescribed partial eigendata. We first give the solvability conditions for the inverse problem without the nonnegative constraint
and then discuss the associated best approximation problem.
To find a nonnegative solution, we reformulate the inverse problem as a monotone complementarity problem and propose a nonsmooth Newton-type method for solving its equivalent nonsmooth equation. Under some mild assumptions, the global and quadratic convergence of our method is established. We also apply our method to the symmetric nonnegative inverse problem and to the cases of prescribed lower bounds and of prescribed entries. Numerical tests demonstrate the efficiency of the proposed method and support our theoretical findings.