SIAM Review, Volume 64, Issue 1, Page 179-179, February 2022.
The Education section in this issue presents two contributions. The first paper, “$LU$ and $CR$ Elimination,” is written by Gilbert Strang and Cleve Moler. This is an expository paper which is helpful to instructors and students interested in linear algebra and matrix manipulations. The focus of the paper is the column-row factorization for any matrix $A$ of rank $r$. The matrix is represented as $A=CR$, where the matrix $C$ contains the first $r$ independent columns of $A$, and the matrix $R$ contains the nonzero rows of the reduced row echelon form of $A$. The authors explain how to obtain the factorization and contrast it with $LU$, $QR$ factorizations, the single-value decomposition, and another factorization, $A= CW^{-1}B$, which they call magic. In section 6, the authors point to an important timely application pertaining to randomized algorithms for a low-rank approximation of large sparse matrices. The paper is clearly and nicely written. It includes a discussion about relevant MATLAB functions, examples, and several references to randomized linear algebra literature. The second paper is “Newton's Method in Mixed Precision,” presented by C. T. Kelley. The paper discusses the impact of precision in calculations on the rate of convergence of Newton's method for solving a system of nonlinear equations $F(x)=0.$ The vector-function $F$ is differentiable and its Jacobian $F' (x)$ is nonsingular. It is assumed that the step $s$ in each iteration is determined by solving the linear equation $F' (x)s= - F(x)$ via Gaussian elimination with column pivoting. The paper starts with the classical result on the local convergence of the method under standard assumptions and then passes onto analyzing the effect of errors arising in calculation of the function and in approximation of the Jacobian. The author's derivations predict no significant difference in the convergence of the nonlinear iteration between a double-precision analytic Jacobian with double precision in the linear solver and a forward-difference approximate Jacobian with a single precision in the linear solver. An interesting part of the paper deals with the question of estimating the effect of the backward error in the solver. The author observes that the worst case estimates are too pessimistic and rarely seen in practice. Invoking recent techniques in probabilistic rounding analysis, he derives more realistic results. The theoretical statements are supplemented by a numerical illustration in section 3. An additional example demonstrates how the theory breaks down if the Jacobian is singular at the solution.

中文翻译：

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教育

SIAM 评论，第 64 卷，第 1 期，第 179-179 页，2022 年 2 月。
本期的教育部分提出了两个贡献。第一篇论文“$LU$ and $CR$ Elimination”由 Gilbert Strang 和 Cleve Moler 撰写。这是一篇说明性论文，对对线性代数和矩阵运算感兴趣的教师和学生很有帮助。本文的重点是任何秩为 $r$ 的矩阵 $A$ 的列行分解。矩阵表示为$A=CR$，其中矩阵$C$ 包含$A$ 的前$r$ 个独立列，矩阵$R$ 包含$A$ 的缩减行梯形形式的非零行. 作者解释了如何获得分解并将其与 $LU$、$QR$ 分解、单值分解和另一个分解 $A= CW^{-1}B$（他们称之为魔法）进行对比。在第 6 节中，作者指出了一个重要的及时应用，该应用与用于大型稀疏矩阵的低秩近似的随机算法有关。这篇论文写得很清楚，很好。它包括对相关 MATLAB 函数、示例以及对随机线性代数文献的若干参考的讨论。第二篇论文是“混合精度中的牛顿法”，由 CT Kelley 发表。本文讨论了计算精度对求解非线性方程组 $F(x)=0 的牛顿法收敛速度的影响。 )$ 是非奇异的。假设每次迭代中的步长 $s$ 是通过使用列旋转的高斯消元求解线性方程 $F' (x)s= - F(x)$ 来确定的。本文从标准假设下该方法局部收敛的经典结果开始，然后分析了在函数计算和雅可比逼近中出现的误差的影响。作者的推导预测线性求解器中双精度双精度解析雅可比行列式与线性求解器中单精度前向差分近似雅可比行列式之间的非线性迭代收敛没有显着差异。本文有趣的部分涉及估计求解器中后向误差的影响的问题。作者观察到，最坏情况的估计过于悲观，在实践中很少见。他引用了概率舍入分析中的最新技术，得出了更现实的结果。