  We give a novel analytic analysis of the worst-case complexity of the
gradient method with exact line search and the Polyak stepsize, respectively,
which previously could only be established by computer-assisted proof. Our
analysis is based on studying the linear convergence of a family of gradient
methods, whose stepsizes include the one determined by exact line search and
the Polyak stepsize as special instances. The asymptotic behavior of the
considered family is also investigated which shows that the gradient method
with the Polyak stepsize will zigzag in a two-dimensional subspace spanned by
the two eigenvectors corresponding to the largest and smallest eigenvalues of
the Hessian.
