SIAM Review, Volume 68, Issue 1, Page 211-212, February 2026. This valuable and unique book delivers a comprehensive lecture on a wide range of control theory issues in relation to matrix computing. Individual problems are illustrated with examples of sufficient dimensionality to ensure they can be manually recalculated, while still illustrating all the intricacies of the relevant calculations and algorithms. The book also contains numerous drawings and diagrams that clarify the various issues.
{"title":"Book Review:; Time-Variant and Quasi-Separable Systems","authors":"Jerzy S. Respondek","doi":"10.1137/25m1758283","DOIUrl":"https://doi.org/10.1137/25m1758283","url":null,"abstract":"SIAM Review, Volume 68, Issue 1, Page 211-212, February 2026. <br/> This valuable and unique book delivers a comprehensive lecture on a wide range of control theory issues in relation to matrix computing. Individual problems are illustrated with examples of sufficient dimensionality to ensure they can be manually recalculated, while still illustrating all the intricacies of the relevant calculations and algorithms. The book also contains numerous drawings and diagrams that clarify the various issues.","PeriodicalId":49525,"journal":{"name":"SIAM Review","volume":"121 1","pages":""},"PeriodicalIF":10.2,"publicationDate":"2026-02-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146138548","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
SIAM Review, Volume 68, Issue 1, Page 3-90, February 2026. Abstract. “One of the ways to help make computer science respectable is to show that it is deeply rooted in history [math]” (Donald E. Knuth, Comm. ACM, 15 (1972), p. 671). A great many of the “respectable” modern numerical methods proceed iteratively, and we give an overview of them in the final section . Teaching and learning science from a historical perspective also leads to a “respectable” deeper understanding. The first problems requiring iterative processes were square-root calculations in Babylon, Greece, and India. More complicated problems such as sine tables in the Arabic, Indian, and medieval calculations, including Kepler’s Problem, were performed with fixed point iterations. With Newton, Raphson, and Simpson we enter the “respectable” realm of methods based on derivatives. Mourraille and Cayley contribute geometric insights in both [math] and [math], while Fourier, Cauchy, and Kantorovich provide rigorous error estimations. Surprisingly, even linear problems became interesting for very large dimensions, beginning with the work of Gauss, Seidel, Young, Richardson, and Krylov to domain decomposition and multigrid methods. We explain all of these methods and illustrate them using the “Montreal test problem.”
SIAM评论，第68卷，第1期，第3-90页，2026年2月。摘要。“使计算机科学受人尊敬的方法之一是表明它深深植根于历史[数学]”（Donald E. Knuth, Comm. ACM, 15(1972)，第671页）。许多“值得尊敬的”现代数值方法都是迭代进行的，我们将在最后一节对它们进行概述。从历史的角度来教授和学习科学也会带来“可敬的”更深层次的理解。第一个需要迭代过程的问题是巴比伦、希腊和印度的平方根计算。更复杂的问题，如阿拉伯、印度和中世纪计算中的正弦表，包括开普勒问题，都是用定点迭代来完成的。随着牛顿、拉夫森和辛普森的出现，我们进入了基于衍生方法的“体面”领域。Mourraille和Cayley在[数学]和[数学]两方面都贡献了几何见解，而Fourier、Cauchy和Kantorovich则提供了严格的误差估计。令人惊讶的是，从Gauss、Seidel、Young、Richardson和Krylov的领域分解和多重网格方法开始，即使是线性问题在非常大的维度上也变得有趣起来。我们将解释所有这些方法，并使用“蒙特利尔测试问题”来说明它们。
{"title":"Landmarks in the History of Iterative Methods","authors":"Martin J. Gander, Philippe Henry, Gerhard Wanner","doi":"10.1137/24m1680428","DOIUrl":"https://doi.org/10.1137/24m1680428","url":null,"abstract":"SIAM Review, Volume 68, Issue 1, Page 3-90, February 2026. <br/> Abstract. “One of the ways to help make computer science respectable is to show that it is deeply rooted in history [math]” (Donald E. Knuth, Comm. ACM, 15 (1972), p. 671). A great many of the “respectable” modern numerical methods proceed iteratively, and we give an overview of them in the final section . Teaching and learning science from a historical perspective also leads to a “respectable” deeper understanding. The first problems requiring iterative processes were square-root calculations in Babylon, Greece, and India. More complicated problems such as sine tables in the Arabic, Indian, and medieval calculations, including Kepler’s Problem, were performed with fixed point iterations. With Newton, Raphson, and Simpson we enter the “respectable” realm of methods based on derivatives. Mourraille and Cayley contribute geometric insights in both [math] and [math], while Fourier, Cauchy, and Kantorovich provide rigorous error estimations. Surprisingly, even linear problems became interesting for very large dimensions, beginning with the work of Gauss, Seidel, Young, Richardson, and Krylov to domain decomposition and multigrid methods. We explain all of these methods and illustrate them using the “Montreal test problem.”","PeriodicalId":49525,"journal":{"name":"SIAM Review","volume":"182 1","pages":""},"PeriodicalIF":10.2,"publicationDate":"2026-02-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146138553","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
SIAM Review, Volume 68, Issue 1, Page 127-149, February 2026. Abstract. We present a variational framework for studying functions learned by deep neural networks with rectified linear unit nonlinearities. We introduce a function space built from compositions of functions of second-order Radon-domain bounded variation. The compositional form of these functions captures the structure of deep neural networks. We prove a representer theorem that shows that deep neural networks with finite width solve regularized data-fitting problems over this space. The critical width is controlled by the square of the number of training data. This perspective explains the effect of weight-decay regularization in neural network training, the importance of skip connections, and the role of sparsity in neural networks. By considering the function-space perspective, we provide sharp links between deep learning and variational methods.
{"title":"Compositional Function Spaces for Deep Learning","authors":"Rahul Parhi, Robert D. Nowak","doi":"10.1137/25m1802948","DOIUrl":"https://doi.org/10.1137/25m1802948","url":null,"abstract":"SIAM Review, Volume 68, Issue 1, Page 127-149, February 2026. <br/> Abstract. We present a variational framework for studying functions learned by deep neural networks with rectified linear unit nonlinearities. We introduce a function space built from compositions of functions of second-order Radon-domain bounded variation. The compositional form of these functions captures the structure of deep neural networks. We prove a representer theorem that shows that deep neural networks with finite width solve regularized data-fitting problems over this space. The critical width is controlled by the square of the number of training data. This perspective explains the effect of weight-decay regularization in neural network training, the importance of skip connections, and the role of sparsity in neural networks. By considering the function-space perspective, we provide sharp links between deep learning and variational methods.","PeriodicalId":49525,"journal":{"name":"SIAM Review","volume":"11 1","pages":""},"PeriodicalIF":10.2,"publicationDate":"2026-02-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146138822","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
SIAM Review, Volume 67, Issue 4, Page 917-918, December 2025. In the current academic landscape, nearly every mathematician will at some point be called upon to contribute—be it through teaching or research—to the burgeoning fields of data science and machine learning. Acquiring the necessary fundamentals in these areas ought to be straightforward. However, for many mathematicians, a significant language barrier arises when encountering the more computer science oriented literature. Bohn, Garcke, and Griebel tackle this challenge from a thoroughly mathematical perspective. Their notation is impeccable, consistently clarifying whether the subject at hand is a scalar, vector, matrix, or function. Concepts are introduced with unwavering rigor, distinguishing between well-posed and ill-posed problems, as well as between algorithms backed by convergence results and those that remain heuristic in nature.
{"title":"Book Review:; Algorithmic Mathematics in Machine Learning","authors":"Volker H. Schulz","doi":"10.1137/25m1741121","DOIUrl":"https://doi.org/10.1137/25m1741121","url":null,"abstract":"SIAM Review, Volume 67, Issue 4, Page 917-918, December 2025. <br/> In the current academic landscape, nearly every mathematician will at some point be called upon to contribute—be it through teaching or research—to the burgeoning fields of data science and machine learning. Acquiring the necessary fundamentals in these areas ought to be straightforward. However, for many mathematicians, a significant language barrier arises when encountering the more computer science oriented literature. Bohn, Garcke, and Griebel tackle this challenge from a thoroughly mathematical perspective. Their notation is impeccable, consistently clarifying whether the subject at hand is a scalar, vector, matrix, or function. Concepts are introduced with unwavering rigor, distinguishing between well-posed and ill-posed problems, as well as between algorithms backed by convergence results and those that remain heuristic in nature.","PeriodicalId":49525,"journal":{"name":"SIAM Review","volume":"1 1","pages":""},"PeriodicalIF":10.2,"publicationDate":"2025-11-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145448276","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
SIAM Review, Volume 67, Issue 4, Page 914-915, December 2025. This textbook on classical numerical analysis is a true gem for students, educators, and practitioners in applied mathematics. With its broad scope and meticulous organization, it serves as a cornerstone reference for a wide range of topics from numerical linear algebra to numerical differential equations, optimization, and approximation theory. Whether you are teaching or attending an entry-level graduate course, this textbook offers all the essential tools to build a solid foundation in numerical analysis.
{"title":"Book Review:; Classical Numerical Analysis: A Comprehensive Course","authors":"Guosheng Fu","doi":"10.1137/24m1700983","DOIUrl":"https://doi.org/10.1137/24m1700983","url":null,"abstract":"SIAM Review, Volume 67, Issue 4, Page 914-915, December 2025. <br/> This textbook on classical numerical analysis is a true gem for students, educators, and practitioners in applied mathematics. With its broad scope and meticulous organization, it serves as a cornerstone reference for a wide range of topics from numerical linear algebra to numerical differential equations, optimization, and approximation theory. Whether you are teaching or attending an entry-level graduate course, this textbook offers all the essential tools to build a solid foundation in numerical analysis.","PeriodicalId":49525,"journal":{"name":"SIAM Review","volume":"78 1","pages":""},"PeriodicalIF":10.2,"publicationDate":"2025-11-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145448278","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
SIAM Review, Volume 67, Issue 4, Page 865-872, December 2025. Abstract.For many of the classic problems of linear algebra, effective and efficient numerical algorithms exist, particularly for situations where dimensions are not too large. The linear least squares problem is one such example: excellent algorithms exist when [math] factorization is feasible. However, for large-dimensional (often sparse) linear least squares problems there currently exist good solution algorithms only for well-conditioned problems or for problems where there are lots of data but only a few variables in the solution. Such approaches ubiquitously employ normal equations and so have to contend with conditioning issues. We explore some alternative approaches that we characterize as not-normal equations where conditioning may not be such an issue.
{"title":"Least Squares and the Not-Normal Equations","authors":"Andrew J. Wathen","doi":"10.1137/23m161851x","DOIUrl":"https://doi.org/10.1137/23m161851x","url":null,"abstract":"SIAM Review, Volume 67, Issue 4, Page 865-872, December 2025. <br/> Abstract.For many of the classic problems of linear algebra, effective and efficient numerical algorithms exist, particularly for situations where dimensions are not too large. The linear least squares problem is one such example: excellent algorithms exist when [math] factorization is feasible. However, for large-dimensional (often sparse) linear least squares problems there currently exist good solution algorithms only for well-conditioned problems or for problems where there are lots of data but only a few variables in the solution. Such approaches ubiquitously employ normal equations and so have to contend with conditioning issues. We explore some alternative approaches that we characterize as not-normal equations where conditioning may not be such an issue.","PeriodicalId":49525,"journal":{"name":"SIAM Review","volume":"42 1","pages":""},"PeriodicalIF":10.2,"publicationDate":"2025-11-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145448282","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Athanasios C. Antoulas, Ion Victor Gosea, Charles Poussot-Vassal
SIAM Review, Volume 67, Issue 4, Page 737-770, December 2025. Abstract.The Loewner framework is an interpolatory approach for the approximation of linear and nonlinear systems. The purpose here is to extend this framework to linear parametric systems with an arbitrary number [math] of parameters. To achieve this, a new generalized multivariate rational function realization is proposed. We then introduce the [math]-dimensional multivariate Loewner matrices and show that they can be computed by solving a set of coupled Sylvester equations. The null space of these Loewner matrices allows the construction of multivariate rational functions in barycentric form. The principal result of this work is to show how the null space of [math]-dimensional Loewner matrices can be computed using a sequence of one-dimensional Loewner matrices. Thus, a decoupling of the variables is achieved, which leads to a drastic reduction of the computational burden. Equally importantly, this burden is alleviated by avoiding the explicit construction of large-scale [math]-dimensional Loewner matrices of size [math]. The proposed methodology achieves the decoupling of variables, leading (i) to a reduction in complexity from [math] to below [math] when [math] and (ii) to memory storage bounded by the largest variable dimension rather than their product, thus taming the curse of dimensionality and making the solution scalable to very large data sets. This decoupling of the variables leads to a result similar to the Kolmogorov superposition theorem for rational functions. Thus, making use of barycentric representations, every multivariate rational function can be computed using the composition and superposition of single-variable functions. Finally, we suggest two algorithms (one direct and one iterative) to construct, directly from data, multivariate (or parametric) realizations ensuring (approximate) interpolation. Numerical examples highlight the effectiveness and scalability of the method.
{"title":"On the Loewner Framework, the Kolmogorov Superposition Theorem, and the Curse of Dimensionality","authors":"Athanasios C. Antoulas, Ion Victor Gosea, Charles Poussot-Vassal","doi":"10.1137/24m1656657","DOIUrl":"https://doi.org/10.1137/24m1656657","url":null,"abstract":"SIAM Review, Volume 67, Issue 4, Page 737-770, December 2025. <br/> Abstract.The Loewner framework is an interpolatory approach for the approximation of linear and nonlinear systems. The purpose here is to extend this framework to linear parametric systems with an arbitrary number [math] of parameters. To achieve this, a new generalized multivariate rational function realization is proposed. We then introduce the [math]-dimensional multivariate Loewner matrices and show that they can be computed by solving a set of coupled Sylvester equations. The null space of these Loewner matrices allows the construction of multivariate rational functions in barycentric form. The principal result of this work is to show how the null space of [math]-dimensional Loewner matrices can be computed using a sequence of one-dimensional Loewner matrices. Thus, a decoupling of the variables is achieved, which leads to a drastic reduction of the computational burden. Equally importantly, this burden is alleviated by avoiding the explicit construction of large-scale [math]-dimensional Loewner matrices of size [math]. The proposed methodology achieves the decoupling of variables, leading (i) to a reduction in complexity from [math] to below [math] when [math] and (ii) to memory storage bounded by the largest variable dimension rather than their product, thus taming the curse of dimensionality and making the solution scalable to very large data sets. This decoupling of the variables leads to a result similar to the Kolmogorov superposition theorem for rational functions. Thus, making use of barycentric representations, every multivariate rational function can be computed using the composition and superposition of single-variable functions. Finally, we suggest two algorithms (one direct and one iterative) to construct, directly from data, multivariate (or parametric) realizations ensuring (approximate) interpolation. Numerical examples highlight the effectiveness and scalability of the method.","PeriodicalId":49525,"journal":{"name":"SIAM Review","volume":"135 1","pages":""},"PeriodicalIF":10.2,"publicationDate":"2025-11-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145448288","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Rikha Rahim, Ahmad F. Sihombing, Ika W. Palupi, Nona T. Sapulette
SIAM Review, Volume 67, Issue 4, Page 915-917, December 2025. This book offers a fresh and innovative approach to competitive system modeling by introducing strategic aggression as a central factor in population dynamics. Through rigorous mathematical analysis, the authors provide valuable insights for researchers and academics in applied mathematics, economics, and social sciences. Moreover, the model’s relevance to real-world phenomena such as the increasing frequency and duration of civil conflicts over recent decades further enhances the book’s significance, making it a valuable resource for those seeking to understand conflict dynamics through a mathematical lens. We confirm that we have no affiliations with the book’s authors or editors. However, we recognize that this book aligns well with one of the courses in our research group, the Industrial and Financial Mathematics Research Group, specifically in the study of dynamic systems, where we also explore extensions of the Lotka–Volterra model by incorporating aggressive strategy considerations.
{"title":"Book Review:; A New Lotka–Volterra Model of Competition With Strategic Aggression: Civil Wars When Strategy Comes into Play","authors":"Rikha Rahim, Ahmad F. Sihombing, Ika W. Palupi, Nona T. Sapulette","doi":"10.1137/25m1740838","DOIUrl":"https://doi.org/10.1137/25m1740838","url":null,"abstract":"SIAM Review, Volume 67, Issue 4, Page 915-917, December 2025. <br/> This book offers a fresh and innovative approach to competitive system modeling by introducing strategic aggression as a central factor in population dynamics. Through rigorous mathematical analysis, the authors provide valuable insights for researchers and academics in applied mathematics, economics, and social sciences. Moreover, the model’s relevance to real-world phenomena such as the increasing frequency and duration of civil conflicts over recent decades further enhances the book’s significance, making it a valuable resource for those seeking to understand conflict dynamics through a mathematical lens. We confirm that we have no affiliations with the book’s authors or editors. However, we recognize that this book aligns well with one of the courses in our research group, the Industrial and Financial Mathematics Research Group, specifically in the study of dynamic systems, where we also explore extensions of the Lotka–Volterra model by incorporating aggressive strategy considerations.","PeriodicalId":49525,"journal":{"name":"SIAM Review","volume":"35 1","pages":""},"PeriodicalIF":10.2,"publicationDate":"2025-11-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145448277","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
SIAM Review, Volume 67, Issue 4, Page 913-913, December 2025. This is the second book I have reviewed in the Oxford University Press A Very Short Introduction series. The first one was Eric Lauga’s Fluid Mechanics: A Very Short Introduction, reviewed in this journal a year ago. These A Very Short Introduction books are pocket-sized and written by expert authors, and (judging by the book list published by the Oxford University Press) they present all kinds of interesting and challenging topics in a readable way. Earl’s book is no exception—its author has succeeded in making a few highly technical topics accessible.
{"title":"Book Review:; Mathematical Analysis: A Very Short Introduction","authors":"Anita T. Layton","doi":"10.1137/24m1676211","DOIUrl":"https://doi.org/10.1137/24m1676211","url":null,"abstract":"SIAM Review, Volume 67, Issue 4, Page 913-913, December 2025. <br/> This is the second book I have reviewed in the Oxford University Press A Very Short Introduction series. The first one was Eric Lauga’s Fluid Mechanics: A Very Short Introduction, reviewed in this journal a year ago. These A Very Short Introduction books are pocket-sized and written by expert authors, and (judging by the book list published by the Oxford University Press) they present all kinds of interesting and challenging topics in a readable way. Earl’s book is no exception—its author has succeeded in making a few highly technical topics accessible.","PeriodicalId":49525,"journal":{"name":"SIAM Review","volume":"135 1","pages":""},"PeriodicalIF":10.2,"publicationDate":"2025-11-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145448279","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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