  This paper is concerned with the least squares inverse eigenvalue problem of
reconstructing a linear parameterized real symmetric matrix from the prescribed
partial eigenvalues in the sense of least squares, which was originally
proposed by Chen and Chu [SIAM J. Numer. Anal., 33 (1996), pp. 2417--2430]. We
provide a Riemannian inexact Gausss-Newton method for solving the least squares
inverse eigenvalue problem. The global and local convergence analysis of the
proposed method is discussed. Also, a preconditioned conjugate gradient method
with an efficient preconditioner is proposed for solving the Riemannian
Gauss-Newton equation. Finally, some numerical tests, including an application
in the inverse Sturm-Liouville problem, are reported to illustrate the
efficiency of the proposed method.
