On the Barzilai and Borwein choice of steplength for the gradient method

In a recent paper, Barzilai and Borwein presented a new choice of steplength for the gradient method. Their choice does not guarantee descent in the objective function and greatly speeds up the convergence of the method. They presented a convergence analysis of their method only in the two-dimensional quadratic case. We establish the convergence of the Barzilai and Borwein gradient method when applied to the minimization of a strictly convex quadratic function of any number of variables.