The performance of gradient methods has been considerably improved by the introduction of delayed parameters. Recently, the revealing of second-order information has given rise to the Cauchy-based methods with alignment, which are generally considered as the state of the art of gradient methods. This paper investigates 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 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.

通过引入延迟参数，梯度方法的性能已得到显着改善。近来，二阶信息的揭示引起了基于柯西（Cauchy）方法的对准，这些方法通常被认为是梯度方法的最新技术。本文研究了最小梯度和渐近最佳步长的光谱特性，然后提出了三种不使用柯西步长的快速对准方法。提供了收敛结果，数值实验表明，新方法为基于经典柯西的方法提供了替代方法。特别是，在某些情况下，对齐梯度方法具有优于Krylov子空间方法的优点，这使其在实践中具有吸引力。