A quadratically convergent algorithm for inverse eigenvalue problems with multiple eigenvalues

In 2017, for inverse symmetric eigenvalue problems, a new quadratically convergent algorithm has been derived from simple matrix equations. Although this algorithm has some nice features compared with the other quadratically convergent methods, it is not applied to multiple eigenvalues. In this paper, we improve this algorithm with the aid of an optimization problem for the eigenvectors associated with multiple eigenvalues. The proposed algorithm is adapted to an arbitrary set of given eigenvalues. The main contribution is our convergence theorem formulated in a different manner from previous work for the existing quadratically convergent methods. Our theorem ensures the quadratic convergence in a neighborhood of the solutions that satisfy a mild condition.