Since the development of the conjugate gradient (CG) method in 1952 by Hestenes and Stiefel, CG, has become an indispensable tool in computational mathematics for solving positive definite linear systems. On the other hand, the conjugate residual (CR) method, closely related CG and introduced by Stiefel in 1955 for the same settings, remains relatively less known outside the numerical linear algebra community. Since their inception, these methods -- henceforth collectively referred to as conjugate direction methods -- have been extended beyond positive definite to indefinite, albeit consistent, settings. Going one step further, in this paper, we investigate theoretical and empirical properties of these methods under inconsistent systems. Among other things, we show that small modifications to the original algorithms allow for the pseudo-inverse solution. Furthermore, we show that CR is essentially equivalent to the minimum residual method, proposed by Paige and Saunders in 1975, in such contexts. Lastly, we conduct a series of numerical experiments to shed lights on their numerical stability (or lack thereof) and their performance for inconsistent systems. Surprisingly, we will demonstrate that, unlike CR and contrary to popular belief, CG can exhibit significant numerical instability, bordering on catastrophe in some instances.

翻译：自1952年Hestenes与Stiefel提出共轭梯度（CG）法以来，该方法已成为计算数学领域求解正定线性系统的核心工具。而与之密切相关的共轭残差（CR）法虽由Stiefel于1955年针对同一问题场景提出，但在数值线性代数领域之外仍鲜为人知。自诞生之初，这两类方法（以下统称为共轭方向方法）便被拓展至不定系统（虽需满足相容性条件）。本文更进一步，探讨了这些方法在不相容系统中的理论与实证特性。我们特别阐明：对原始算法进行适度修正即可获得伪逆解；在此类背景下CR法与Paige和Saunders于1975年提出的最小残差法本质等价；最后通过系列数值实验揭示其数值稳定性（或不稳定性）及对不相容系统的处理效能。令人意外的是，我们将证明与普遍认知相反：不同于CR法，CG法在某些情形下可能表现出严重的数值不稳定性，甚至逼近灾难性崩溃。