  The performance of gradient methods has been considerably improved by the
introduction of delayed parameters. After two and a half decades, the revealing
of second-order information has recently given rise to the Cauchy-based methods
with alignment, which reduce asymptotically the search spaces in smaller and
smaller dimensions. They are generally considered as the state of the art of
gradient methods. This paper reveals the spectral properties of minimal
gradient and asymptotically optimal steps, and then suggests three fast methods
with alignment without using the Cauchy step. The convergence results are
provided, and numerical experiments show that the new methods provide
competitive and more stable alternatives to the classical Cauchy-based methods.
In particular, alignment gradient methods present advantages over the Krylov
subspace methods in some situations, which makes them attractive in practice.
