SIAM Review, Volume 62, Issue 2, Page 437-438, January 2020.
The Education section of SIAM Review presents three papers in this issue. In the first paper Lloyd N. Trefethen reviews “Eight Perspectives on the Exponentially Ill-Conditioned Equation $\varepsilon y'' - x y' + y = 0$.” This paper illustrates how an ensemble of mathematical techniques can come together to provide interesting insights into a mathematical structure, looking at it through various lenses. Equations of this type were formulated long ago and can be considered as versions of equations related to Hermite polynomials. When analyzing the equation, we start by addressing the existence and uniqueness of its solution. The next step takes us to its numerical solution. Applying a numerical method faces the challenge of ill-conditioning, and, at that point, backward error analysis becomes an important tool and provides crucial help. To gain another perspective, we may conduct asymptotic analysis. The problem discussed in the paper has an analytic solution, but this is not always the case. The method of boundary layer analysis allows us to treat more general problems with small parameters. That method provides a means to obtain approximate solutions that are accurate for small values of $\varepsilon$. The solutions become exact in the limit when $\varepsilon \downarrow 0$. Further perspective is brought by the theory of dynamical systems. While continuous-time dynamical systems are described by differential equations and therefore deal with the same subject, the theory of dynamical systems emphasizes the geometrical analyses of the problems. In that context, we observe slow-fast, critical, attracting, and repelling manifolds looking at our differential equation. Further discussion includes the framework of Sturm--Liouville operators, spectral theory, sensitivity analysis via adjoints and singular value decomposition, PDE theory, and some perspectives from physics. The author concludes that many other mathematical tools, some subject to increased interest recently, may also be relevant and could bring interesting new observations to this classical problem. The second paper, “First-Order Perturbation Theory for Eigenvalues and Eigenvectors,” by Anne Greenbaum, Ren-Cang Li, and Michael L. Overton, discusses the behavior of the eigenvalues and the normalized eigenvectors of a complex square matrix, which is subjected to perturbations. This question---the focus of many scientific studies---has been addressed in two different ways, both reflected upon in this paper. The first approach is based on the analytic perturbation theory. We consider a matrix-valued analytic function depending on a complex parameter in a particular area; the matrix of interest is the image of a given parameter value. The widely known classical results pertain to the existence of convergent power expansions of eigenvalues and eigenvectors for Hermitian matrices or self-adjoint linear operators subjected to real analytic perturbations. The recent advances in numerical linear algebra inspire the second approach to perturbation theory for matrices. This line of research leads to perturbation bounds rather than expansions. The goals are to identify methods for bounding the change in the eigenvalues and the associated eigenvectors when a given matrix is subjected to a perturbation with a given norm and structure. In this paper, the authors consider general square matrices, which are not necessarily Hermitian or normal ones. The authors provide a theorem addressing the perturbations of a simple eigenvalue and the corresponding right and left eigenvectors. The statement of the perturbation theorem is first discussed informally and then supplied with two different proofs. In this way, the reader is provided with a formal as well as an intuitive understanding of the essence and the flavor of both methodologies. Additionally, the paper contains a discussion on the numerical verification of the formulae for the derivatives of the eigenvalues and the eigenvectors. The authors provide two examples illustrating that the results do not hold when the eigenvalue is not simple. The paper includes a fairly extensive reference list, helpful to those who seek to gain more in-depth insight. The third contribution, written by Ling Guo, Akil Narayan, and Tao Zhou, is “Constructing Least-Squares Polynomial Approximations.” This paper deals with a popular and widely used method for constructing a mathematical model using observed data. The goal is to identify a mathematical model of a function $f$ by choosing an element from a prescribed finite-dimensional vector space such as the space of polynomials of a fixed degree. It is assumed that $f$ is a $w$-weighted square-integrable real-valued function over a given domain. The function $f$ is deterministic, but it is observed at random points. The approximation error is evaluated with respect to the $L^2_w$-norm. The function samples are gathered from observational data, which is assumed to be exact, not noisy. The authors argue that the lack of noise makes this setup different from the setup of statistical regression. In the large-sample limit, it is expected that the approximation converges to the projection of $f$ onto the finite-dimensional model space, rather than to $f$ itself. After formulating the problem, the analysis in the one-dimensional case is presented: functions of a single variable are approximated by using polynomials. This example helps to explain the notation and the general problem setting, as well as to understand the challenges which occur in high dimensions. The approximation error is calculated approximately based on discretizing the definition domain. The main portion of the paper deals with the behavior of the approximate solution and of the error when the sample size increases. Various strategies for sampling are discussed, which bring the authors to the following conclusions. Preference should be given to the technique using a random choice of sample points over points from a grid formed by tessellation. Furthermore, it makes sense to generate biased random samples depending on the weighting density $w$. In the last section of the paper, the authors mention alternatives and further extensions for computing approximations in high dimensions known in the extant literature.

中文翻译：

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教育

SIAM评论，第62卷，第2期，第437-438页，2020年1月。
《 SIAM评论》的“教育”部分介绍了本期的三篇论文。在第一篇论文中，劳埃德·N·特雷菲森（Lloyd N. Trefethen）评述了“指数病态方程$ \ varepsilon y的八个透视图”-xy'+ y = 0 $。” 本文说明了如何结合各种数学技术，以各种角度对数学结构提供有趣的见解。这种类型的方程式是很久以前制定的，可以视为与Hermite多项式有关的方程式的版本。在分析方程式时，我们首先要解决其解的存在性和唯一性。下一步将我们带到其数值解。应用数值方法面临病态挑战，因此，向后误差分析已成为重要的工具并提供了关键的帮助。为了获得另一个观点，我们可以进行渐近分析。本文讨论的问题具有解析解，但并非总是如此。边界层分析方法使我们可以用较小的参数来处理更一般的问题。该方法提供了一种获得近似解的方法，该近似解对于$ \ varepsilon $的较小值而言是准确的。当$ \ varepsilon \ downarrow 0 $时，解在极限中变得精确。动力学系统理论带来了进一步的视角。连续时间动力系统由微分方程描述，因此涉及同一主题，而动力系统理论则强调对问题的几何分析。在这种情况下，我们观察微分方程时观察到了慢速，临界，吸引和排斥流形。进一步的讨论包括Sturm-Liouville算子的框架，光谱理论，通过伴随和奇异值分解的灵敏度分析，PDE理论以及物理学的一些观点。作者得出的结论是，许多其他数学工具（可能最近也受到了越来越多的关注）可能也具有相关性，并且可以为这个经典问题带来有趣的新发现。第二篇论文，“ Anne Greenbaum”，“ Ren-Cang Li”和“ Michael L. Overton”的“特征值和特征向量的一阶摄动理论”讨论了复杂平方矩阵的特征值和归一化特征向量的行为。扰动。这个问题-许多科学研究的重点-已通过两种不同的方式得到解决，这两种方式都在本文中得到了反映。第一种方法基于解析扰动理论。我们考虑一个矩阵值的解析函数，该函数取决于特定区域中的复杂参数。感兴趣的矩阵是给定参数值的图像。广为人知的经典结果与存在实解析扰动的Hermitian矩阵或自伴线性算子的特征值和特征向量的收敛幂展开有关。数值线性代数的最新进展启发了矩阵摄动理论的第二种方法。这一研究领域导致了摄动界而不是扩张。目的是确定当给定矩阵受到给定范数和结构的扰动时，用于约束特征值和相关特征向量变化的方法。在本文中，作者考虑了一般的平方矩阵，这些矩阵不一定是Hermitian矩阵或正常矩阵。作者提供了一个定理，用于解决简单特征值的扰动以及相应的左右特征向量。摄动定理的陈述首先被非正式地讨论，然后提供两个不同的证明。以这种方式，为读者提供了两种方法的本质和风格的形式化的以及直观的理解。此外，本文还讨论了特征值和特征向量的导数公式的数值验证。作者提供了两个示例，说明在特征值不简单时结果不成立。该文件包括相当广泛的参考清单，对寻求获得更深入的见解的人很有帮助。郭岭，阿基尔·纳拉扬和陶周撰写的第三篇论文是“构造最小二乘多项式逼近”。本文讨论了一种使用观测数据构建数学模型的流行且广泛使用的方法。目的是通过从规定的有限维向量空间（例如固定度的多项式空间）中选择一个元素来确定函数$ f $的数学模型。假定$ f $是给定域上$ w $加权的平方可积实值函数。函数$ f $是确定性的，但可以在随机点观察到。针对$ L ^ 2_w $-范数评估近似误差。功能样本是从观测数据中收集的，这些观测数据被认为是准确的，而不是嘈杂的。作者认为，噪声的缺乏使该设置不同于统计回归的设置。在大样本范围内，预计近似值收敛到$ f $在有限维模型空间上的投影，而不是$ f $本身。提出问题后，进行一维情况分析：使用多项式逼近单个变量的函数。该示例有助于说明概念和一般问题的设置，并有助于理解高维度中出现的挑战。近似误差是基于离散化定义域来近似计算的。本文的主要部分讨论了近似解的行为以及样本量增加时的误差。讨论了各种抽样策略，这使作者得出以下结论。应该优先使用该技术，该方法是使用随机选择的采样点，而不是通过细分形成的网格中的点。此外，根据加权密度$ w $生成有偏差的随机样本是有意义的。在本文的最后一部分，作者提到了现有文献中已知的用于计算高维近似值的替代方法和进一步的扩展。