A new algorithm for the computation of the smallest eigenvalue of a symmetric matrix and its eigenspace

The problem of nding the smallest eigenvalue and the corresponding eigenspace of a symmetric matrix is stated as a semide nite optimization problem. A straightforward application of nowadays more or less standard routines for the solution of semide nite problems yields a new algorithm for the smallest eigenvalue problem; the approach not only yields the smallest eigenvalue, but also a symmetric positive semide nite (SPSD) matrix whose column space is equal to the eigenspace for the smallest eigenvalue. It is shown that the predictor-corrector method yields a polynomial time algorithm which, with a suitable choice of the step size, asymptotically is quadratically convergent.