  In this paper, we give some new thoughts about the classical gradient method
(GM) and recall the proposed fractional order gradient method (FOGM). It is
proven that the proposed FOGM holds a super convergence capacity and a faster
convergence rate around the extreme point than the conventional GM. The
property of asymptotic convergence of conventional GM and FOGM is also
discussed. To achieve both a super convergence capability and an even faster
convergence rate, a novel switching FOGM is proposed. Moreover, we extend the
obtained conclusion to a more general case by introducing the concept of
p-order Lipschitz continuous gradient and p-order strong convex. Numerous
simulation examples are provided to validate the effectiveness of proposed
methods.
