SIAM Review, Volume 62, Issue 1, Page 229-230, January 2020.
The Education section of SIAM Review presents three papers in this issue. In the first paper Robert M. Corless and Leili Rafiee Sevyeri discuss “The Runge Example for Interpolation and Wilkinson's Examples for Rootfinding.” The authors use several classical examples in numerical analysis to discuss propagation of error and the role of sensitivity and the conditioning of the functions subject to numerical procedures. The first example was brought to the fore by Carl Runge in 1901 when exploring the approximation error occurring in polynomial interpolation. The effect illustrated by this example is also referred to as Runge's phenomenon. It consists of oscillation at the edges of the domain of the function, which is observed when the function is approximated by interpolation using a high-degree polynomial over a uniform grid. This example plays an important role and is used in many numerical analysis textbooks because it illustrates pitfalls and limitations of high-degree polynomial interpolation. It reveals that using higher degree polynomials does not always improve the accuracy of the approximation. The second example is a classical polynomial, introduced by J. H. Wilkinson in his paper “The Evaluation of the Zeros of Ill-Conditioned Polynomials. Part I" [Numer. Math., 1 (1959), pp. 150--166]. It illustrates a difficulty when finding the root of a polynomial, which stems from the sensitivity of the polynomial at the location of the roots, with respect to perturbations in its coefficients. Wilkinson's polynomial is also frequently used in the context of eigenvalue calculation. In that case, computing eigenvalues of a matrix by first calculating the coefficients of the matrix's characteristic polynomial and then calculating its roots may result in a very ill-conditioned problem even if the original problem is well conditioned. The characteristic polynomial can be very sensitive to perturbations in its coefficients near its roots. The authors bring the theory of backward error analysis and conditioning to the reader's attention. This approach explains the phenomena of the presented examples and is in harmony with the numerical experience thus far. The second paper is titled “Mathematics + Cancer: An Undergraduate `Bridge' Course in Applied Mathematics.” It is presented by Tracy L. Stepien, Eric J. Kostelich, and Yang Kuang. The authors share their insightful experience with a one-semester modeling course based on mathematical models for cancer occurrence and growth, as well as models describing the effect of certain cancer treatments. The targeted audience is undergraduate students who have completed standard courses on calculus, ordinary differential equations, linear algebra, and introductory probability and statistics. The offer of such a course supports various goals. It helps develop modeling skills for students in applied mathematics while at the same time exposing students to many mathematical ideas and enabling them to make informed choices about future specialization in the various branches of applied mathematics and statistics. Additionally, the students come in contact with compelling scientific as well as social problems and see the impact of applied mathematics on real life. Finally, a research component in the course provides invaluable experience to the enrolled students. The paper discusses the structure and organization of the course. The main portion of it provides an overview of various modules whose suggested length is one or two weeks. The topics encompass statistical models regarding the occurrence of cancer depending on age, the growth of the population of cancer cells, the evolution of resistance to treatment, tumor dynamics at the cellular and tissue levels, the measurement and assessment of treatment efficacy, short-term forecasts of tumor progression, the use of statistics in designing clinical trials, and many others. Finally, the public health perspective is brought into the picture: how much cancer might be preventable by a healthy life style, how much is due to environmental exposure, and how much might be due to hereditary reasons or random mutations. The reader will find a sample syllabus, homework problem sets, and computer lab descriptions in the supplementary materials. The authors themselves aim to give students experience in reading research papers and do not use textbooks. However, they mention suitable textbooks for those teachers who would prefer to follow one. Extensive references to relevant literature are included. Finally, the authors point out that their course might be used as a template for other courses of similar type based on topics drawn from other areas of the mathematical sciences. The third contribution is the paper by Antônio Neto on “Matrix Analysis and Omega Calculus.” The first ideas giving rise to Omega calculus were presented by MacMahon in his book Combinatory Analysis published by Cambridge University Press in 1915--1916. The main subject there is the solution of linear Diophantine systems composed of equalities and inequalities; in this context, the author involves results on partitions of natural numbers. In that framework, the so-called Omega operator is applied systematically to each equation and inequality. The final outcome of the procedure is a generating function describing all the solutions of the Diophantine system. The methodology is referred to as MacMahon's partition analysis (MPA) or Omega calculus. Frequently, the latter is identified with the Omega package, which contains numerical methods based on MPA. In the current paper, Antônio Neto presents an extension of Omega operator calculus to matrix-valued functions. The focus is placed on the matrix exponential functions, which have many applications. The extended Omega operator is defined based on the matrix exponential series and the Neumann series with the use of the Frobenius norm for square matrices. Application of the Omega operator to the theory of differential equations, to graph theory, and to other areas is discussed. The paper contains a compact and easy-to-apply method to compute multiple integrals involving matrix exponentials based on MPA. The author stipulates that the presented definition and method generalize and unify previous work. The material is accessible to advanced undergraduate students in applied mathematics who have knowledge in calculus including Taylor series of scalar functions, various norms, matrix operations and inversion, and matrix exponentials.

中文翻译：

---

教育

SIAM评论，第62卷，第1期，第229-230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，Robert M. Corless和Leili Rafiee Sevyeri讨论了“内插的Runge示例和Rootfinding的Wilkinson示例”。作者在数值分析中使用了几个经典示例来讨论误差的传播，灵敏度的作用以及受数值程序影响的函数的条件。第一个例子由卡尔·朗格（Carl Runge）在1901年提出，他研究了多项式插值法中出现的近似误差。此示例说明的效果也称为龙格现象。它由函数域边缘的振荡组成，当使用均匀网格上的高阶多项式通过插值法对函数进行逼近时，可以观察到振荡。该示例起着重要作用，并在许多数值分析教科书中使用，因为它说明了高次多项式插值的陷阱和局限性。结果表明，使用高阶多项式并不总是会提高逼近精度。第二个示例是经典多项式，由JH Wilkinson在他的论文“病态多项式的零点的评估”中介绍。第一部分” [Numer。Math。，1（1959），第150--166页。”它说明了在找到多项式根时的困难，这是由于多项式在根位置处的敏感性所致。关于其系数的摄动。在特征值计算的背景下，威尔金森多项式也经常使用。通过先计算矩阵特征多项式的系数，然后计算其根来计算矩阵的特征值，即使原始问题条件良好，也可能会导致问题非常严重。特征多项式对其根附近的系数扰动可能非常敏感。作者将后向错误分析和条件调整的理论提请读者注意。该方法解释了所提供示例的现象，并且与到目前为止的数值经验相一致。第二篇论文的标题是“数学与癌症：应用数学本科课程”。它由Tracy L. Stepien，Eric J. Kostelich和Yang Kuang提出。作者在一个学期的建模课程中分享了他们的深刻见识，该课程基于癌症发生和发展的数学模型以及描述某些癌症治疗效果的模型，为期一个学期。目标受众是已完成有关微积分，常微分方程，线性代数以及入门概率和统计学的标准课程的本科学生。提供这样的课程支持各种目标。它有助于为应用数学专业的学生发展建模技能，同时使学生了解许多数学思想，并使他们能够对应用数学和统计学的各个分支的未来专业作出明智的选择。另外，学生将接触到引人注目的科学以及社会问题，并了解应用数学对现实生活的影响。最后，课程中的研究部分为注册学生提供了宝贵的经验。本文讨论了该课程的结构和组织。它的主要部分概述了建议长度为一到两周的各种模块。主题包括关于癌症发生的统计模型，这些模型取决于年龄，癌细胞群体的增长，对治疗的抵抗力的演变，细胞和组织水平的肿瘤动态，治疗效果的测量和评估，短期肿瘤进展的预测，在设计临床试验中使用统计数据等。最后，从公共卫生的角度来看：健康的生活方式可以预防多少癌症，环境暴露可以预防多少癌症，遗传性原因或随机突变可以预防多少癌症。读者将在补充材料中找到示例教学大纲，作业问题集和计算机实验室描述。作者自己的目的是为学生提供阅读研究论文的经验，而不使用教科书。但是，他们为那些愿意听一本的老师提到了合适的教科书。包括对相关文献的大量引用。最后，作者指出，基于从数学科学其他领域得出的主题，他们的课程可以用作其他类似类型课程的模板。第三个贡献是AntônioNeto关于“矩阵分析和Omega微积分”的论文。麦克马洪（MacMahon）在1915年至1916年由剑桥大学出版社出版的《组合分析》一书中提出了引起欧米茄微积分的最初想法。这里的主要课题是由相等和不等式组成的线性丢番图系统的解。在这种情况下，作者涉及自然数分区的结果。在那个框架中，所谓的Omega算子被系统地应用于每个方程和不等式。该过程的最终结果是一个生成函数，描述了丢番图系统的所有解。该方法称为MacMahon的分区分析（MPA）或Omega演算。通常，后者使用Omega软件包标识，其中包含基于MPA的数值方法。在当前论文中，AntônioNeto提出了将Omega算术微积分扩展到矩阵值函数的方法。重点放在具有许多应用的矩阵指数函数上。扩展的Omega算子是基于矩阵指数级数和Neumann级数定义的，并使用Frobenius范数用于平方矩阵。讨论了Omega算子在微分方程理论，图论以及其他领域的应用。本文包含一种紧凑且易于应用的方法，可以基于MPA计算涉及矩阵指数的多个积分。作者规定，提出的定义和方法可以概括和统一以前的工作。