Symmetric Tridiagonal Inverse Quadratic Eigenvalue Problems with Partial Eigendata

In this paper we concern the inverse problem of constructing the n-by-n real symmetric tridiagonal matrices C and K so that the monic quadratic pencil Q(λ) := λI + λC + K (where I is the identity matrix) possesses the given partial eigendata. We first provide the sufficient and necessary conditions for the existence of an exact solution to the inverse problem from the self-conjugate set of prescribed four eigenpairs. To find a physical solution for the inverse problem where the matrices C and K are weakly diagonally dominant and have positive diagonal elements and negative off-diagonal elements, we consider the inverse problem from the partial measured noisy eigendata. We propose a regularized smoothing Newton method for solving the inverse problem. The global and quadratic convergence of our approach is established under some mild assumptions. Some numerical examples and a practical engineering application in vibrations show the efficiency of our method.