SIAM Review, Volume 64, Issue 3, Page 640-649, August 2022.
The solution of systems of linear(ized) equations lies at the heart of many problems in scientific computing. In particular, for large systems, iterative methods are a primary approach. For many symmetric (or self-adjoint) systems, there are effective solution methods based on the conjugate gradient method (for definite problems) or MINRES (for indefinite problems) in combination with an appropriate preconditioner, which is required in almost all cases. For nonsymmetric systems there are two principal lines of attack: the use of a nonsymmetric iterative method such as GMRES or transformation into a symmetric problem via the normal equations and application of LSQR. In either case, an appropriate preconditioner is generally required. We consider the possibilities here, particularly the idea of preconditioning the normal equations via approximations to the original nonsymmetric matrix. We highlight dangers that readily arise in this approach. Our comments also apply in the context of linear least squares problems.

中文翻译：

---

关于正态方程和最小二乘预处理的一些评论

SIAM 评论，第 64 卷，第 3 期，第 640-649 页，2022 年 8 月。
线性（化）方程组的解是科学计算中许多问题的核心。特别是对于大型系统，迭代方法是一种主要方法。对于许多对称（或自伴）系统，有基于共轭梯度法（用于确定问题）或 MINRES（用于不定问题）结合适当的预条件子的有效求解方法，几乎​​在所有情况下都需要。对于非对称系统，有两条主要攻击路线：使用非对称迭代方法（如 GMRES）或通过正规方程和应用 LSQR 转换为对称问题。在任何一种情况下，通常都需要适当的预处理器。我们考虑这里的可能性，特别是通过对原始非对称矩阵的近似来预处理正规方程的想法。我们强调了这种方法中容易出现的危险。我们的评论也适用于线性最小二乘问题。