优化中对称线性系统对称线性导导法框架框架 (A Framework of Conjugate Direction Methods for Symmetric Linear Systems in Optimization)

In this paper we introduce a parameter dependent class of Krylov-based methods, namely CD, for the solution of symmetric linear systems. We give evidence that in our proposal we generate sequences of conjugate directions, extending some properties of the standard Conjugate Gradient (CG) method, in order to preserve the conjugacy. For specific values of the parameters in our framework we obtain schemes equivalent to both the CG and the scaled-CG. We also prove the finite convergence of the algorithms in CD, and we provide some error analysis. Finally, preconditioning is introduced for CD, and we show that standard error bounds for the preconditioned CG also hold for the preconditioned CD.

翻译：在本文中,我们引入了一种基于Krylov法的参数依附类方法,即CD,用于解决对称线性系统。我们提供了证据,证明在我们的提案中,我们生成了共融方向序列,扩展了标准的共化梯度(CG)法的某些特性,以维护共化法的某些特性。对于我们框架中参数的具体值,我们获得了与CG和缩放CG相等的计划。我们还证明了CD中算法的有限趋同,我们提供了一些错误分析。最后,为CD引入了先决条件,我们表明先决条件的CG标准误差界限也维持了先决条件的CD。