Numerical methods for inverse singular value problems3

Two numerical methods—one continuous and the other discrete—are proposed for solving inverse singular value problems. The first method consists of solving an ordinary differential equation obtained from an explicit calculation of the projected gradient of a certain objective function. The second method generalizes an iterative process proposed originally by Friedland, Nocedal, and Overton [SIAM J. Numer. Anal., 24 (1987), pp. 634–667] for solving inverse eigenvalue problems. With the geometry understood from the first method, it is shown that the second method (also, the method proposed by Friedland, Nocedal, and Overton for inverse eigenvalue problems) is a variation of the Newton method. While the continuous method is expected to converge globally at a slower rate (in finding a stationary point of the objective function), the discrete method is proved to converge locally at a quadratic rate (if there is a solution). Some numerical examples are presented.