{"title":"Fast computation of function composition derivatives for flatness-based control of diffusion problems","authors":"Stephan Scholz, Lothar Berger","doi":"10.1186/s13362-024-00143-y","DOIUrl":null,"url":null,"abstract":"The chain rule is a standard tool in differential calculus to find derivatives of composite functions. Faà di Bruno’s formula is a generalization of the chain rule and states a method to find high-order derivatives. In this contribution, we propose an algorithm based on Faà di Bruno’s formula and Bell polynomials (Bell in Ann Math 29:38–46, 1927; Parks and Krantz in A primer of real analytic functions, 2012) to compute the structure of derivatives of function compositions. The application of our method is showcased using trajectory planning for the heat equation (Laroche et al. in Int J Robust Nonlinear Control 10(8):629–643, 2000).","PeriodicalId":1,"journal":{"name":"Accounts of Chemical Research","volume":null,"pages":null},"PeriodicalIF":16.4000,"publicationDate":"2024-08-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Accounts of Chemical Research","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.1186/s13362-024-00143-y","RegionNum":1,"RegionCategory":"化学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"CHEMISTRY, MULTIDISCIPLINARY","Score":null,"Total":0}
引用次数: 0
Abstract
The chain rule is a standard tool in differential calculus to find derivatives of composite functions. Faà di Bruno’s formula is a generalization of the chain rule and states a method to find high-order derivatives. In this contribution, we propose an algorithm based on Faà di Bruno’s formula and Bell polynomials (Bell in Ann Math 29:38–46, 1927; Parks and Krantz in A primer of real analytic functions, 2012) to compute the structure of derivatives of function compositions. The application of our method is showcased using trajectory planning for the heat equation (Laroche et al. in Int J Robust Nonlinear Control 10(8):629–643, 2000).
链式法则是微分学中求复合函数导数的标准工具。Faà di Bruno 公式是对链式法则的概括，提出了一种求高阶导数的方法。在这篇论文中，我们提出了一种基于 Faà di Bruno 公式和贝尔多项式（Bell，发表于 Ann Math 29:38-46, 1927 年；Parks 和 Krantz，发表于 A primer of real analytic functions, 2012 年）的算法，用于计算函数合成导数的结构。我们的方法在热方程的轨迹规划中得到了应用（Laroche 等人，载于 Int J Robust Nonlinear Control 10(8):629-643, 2000）。
期刊介绍：
Accounts of Chemical Research presents short, concise and critical articles offering easy-to-read overviews of basic research and applications in all areas of chemistry and biochemistry. These short reviews focus on research from the author’s own laboratory and are designed to teach the reader about a research project. In addition, Accounts of Chemical Research publishes commentaries that give an informed opinion on a current research problem. Special Issues online are devoted to a single topic of unusual activity and significance.
Accounts of Chemical Research replaces the traditional article abstract with an article "Conspectus." These entries synopsize the research affording the reader a closer look at the content and significance of an article. Through this provision of a more detailed description of the article contents, the Conspectus enhances the article's discoverability by search engines and the exposure for the research.
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