Abstract. The problems of computing least squares approximations for various types of real and symmetric matrices subject to spectral constraints share a common structure. This paper describes a general procedure in using the projected gradient method. It is shown that the projected gradient of the objective function on the manifold of constraints usually can be formulated explicitly. This gives rise to the construction of a descent flow that can be followed numerically. The explicit form also facilitates the computation of the second-order optimality conditions. Examples of applications are discussed. With slight modifications, the procedure can be extended to solve least squares problems for general matrices subject to singular-value constraints.
We propose solving the inverse eigenvalue problem for symmetric nonnegative matrices by means of a di erential equation. If the given spectrum is feasible, then a symmetric nonnegative matrix can be constructed simply by f o l l o wing the solution curve of the di erential system. The choice of the vector eld is based on the idea of minimizing the distance between the cone of symmetric nonnegative matrices and the isospectral surface determined by the given spectrum. We explicitly describe the projected gradient of the objective function. Using center manifold theory, w e also show that the !-limit set of any solution curve is a single point. Some numerical examples are presented.
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