SIAM Review, Volume 63, Issue 2, Page 393-393, January 2021.
This issue of SIAM Review presents two papers in the Education section. The first paper, “Rotations in Three Dimensions,” is written by Milton F. Maritz. Current textbooks in calculus and linear algebra typically discuss rotation on the plane and do not present rotation matrices in higher dimensions. The author provides two ways to derive the $3 \times 3$ matrix for a 3D rotation which are based on geometrical arguments. The first approach requires only basic calculus, while the second one makes use of matrix differential equations. Additional questions, such as identifying the rotation angle when the rotation matrix is given, are discussed as well. Several exercises and recommendations to the instructor are provided. This paper is written in an intuitive and accessible style, so that it could be useful in an undergraduate linear algebra course or for an undergraduate seminar. The second paper is “Vandermonde with Arnoldi.” It is presented by Pablo D. Brubeck, Yuji Nakatsukasa, and Lloyd N. Trefethen. When a function is approximated by a polynomial of high degree, one could form the Vandermonde matrix for a set of arguments and formulate a linear model which is then fit to the respective values of the function. The Vandermonde matrix has a large condition number because its columns are powers of the arguments. Therefore, standard numerical techniques for solving the resulting system of normal equations are frequently unstable and inefficient. Numerical stability can be achieved by using the Arnoldi orthogonalization procedure. The authors discuss this idea beyond polynomial approximation. One example constitutes the approximation of a function by Fourier series, and another one deals with the solution of the 2D Laplace equation which is approximated by the real part of a complex polynomial. In their concluding remarks, the authors survey relevant literature and point to further applications of that stabilization technique.

中文翻译：

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教育

SIAM 评论，第 63 卷，第 2 期，第 393-393 页，2021 年 1 月。
本期 SIAM 评论在教育部分介绍了两篇论文。第一篇论文“三维旋转”由米尔顿·F·马里茨 (Milton F. Maritz) 撰写。当前的微积分和线性代数教科书通常讨论平面上的旋转，而不提供更高维度的旋转矩阵。作者提供了两种方法来导出基于几何参数的 3D 旋转的 $3 \times 3$ 矩阵。第一种方法只需要基本的微积分，而第二种方法使用矩阵微分方程。还讨论了其他问题，例如在给定旋转矩阵时识别旋转角度。提供了一些练习和对教师的建议。本文以直观易懂的风格撰写，以便它可以用于本科线性代数课程或本科研讨会。第二篇论文是“范德蒙德与阿诺迪”。它由 Pablo D. Brubeck、Yuji Nakatsukasa 和 Lloyd N. Trefethen 提出。当一个函数被一个高阶多项式逼近时，可以为一组参数形成 Vandermonde 矩阵，并制定一个线性模型，然后拟合函数的各个值。Vandermonde 矩阵有一个很大的条件数，因为它的列是参数的幂。因此，用于求解正规方程组的标准数值技术常常不稳定且效率低下。数值稳定性可以通过使用 Arnoldi 正交化过程来实现。作者在多项式近似之外讨论了这个想法。一个例子是通过傅立叶级数逼近一个函数，另一个例子是处理由复多项式的实部逼近的二维拉普拉斯方程的解。在他们的结论性评论中，作者调查了相关文献并指出了该稳定技术的进一步应用。