A dynamical method of DAEs for the smallest eigenvalue problem

This article gives a new method based on the dynamical system of differential-algebraic equations for the smallest eigenvalue problem of a symmetric matrix. First, the smallest eigenvalue problem is converted into an equivalent constrained optimization problem. Second, from the Karush–Kuhn–Tucker conditions for this special equality-constrained problem, a special continuous dynamical system of differential-algebraic equations is obtained. Lastly, based on the implicit Euler method and an analogous trust-region technique, we obtain a prediction-correction method to compute a steady-state solution of this special system of differential-algebraic equations, and consequently obtain the smallest eigenvalue of the original problem. We also analyze the local superlinear property for this new method, and present the promising numerical results, in comparison with other methods.