Fast gradient methods with alignment for symmetric linear systems without using Cauchy step

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