Editing Automatic differentiation (section)

== Applications ==
Currently, for its efficiency and accuracy in computing first and higher order [[derivative]]s, auto-differentiation is a celebrated technique with diverse applications in [[scientific computing]] and [[mathematics]]. It should therefore come as no surprise that there are numerous computational implementations of auto-differentiation. Among these, one mentions [[INTLAB]], [[Sollya]], and [[InCLosure]].<ref name="Rump.1999">[[Siegfried M. Rump]] (1999). INTLAB–INTerval LABoratory. In T. Csendes, editor, Developments in Reliable Computing, pages 77–104. Kluwer Academic Publishers, Dordrecht.</ref><ref name="Chevillard.Joldes.Lauter.2010">S. Chevillard, M. Joldes, and C. Lauter. Sollya (2010). An Environment for the Development of Numerical Codes. In K. Fukuda, J. van der Hoeven, M. Joswig, and N. Takayama, editors, Mathematical Software - ICMS 2010, volume 6327 of Lecture Notes in Computer Science, pages 28–31, Heidelberg, Germany. Springer.</ref><ref name="Dawood.Inc4.2022">[[Hend Dawood]] (2022). [[InCLosure]] (Interval enCLosure)–A Language and Environment for Reliable Scientific Computing. Computer Software, Version 4.0. Department of Mathematics. Faculty of Science, Cairo University, Giza, Egypt, September 2022. url: https://doi.org/10.5281/zenodo.2702404.</ref> In practice, there are two types (modes) of algorithmic differentiation: a forward-type and a reversed-type.<ref name="Dawood.Megahed.2023"/><ref name="Dawood.Megahed.2019"/> Presently, the two types are highly correlated and complementary and both have a wide variety of applications in, e.g., non-linear [[optimization]], [[sensitivity analysis]], [[robotics]], [[machine learning]], [[computer graphics]], and [[computer vision]].<ref name="Dawood-attribution"/><ref name="Fries.2019">Christian P. Fries (2019). Stochastic Automatic Differentiation: Automatic Differentiation for Monte-Carlo Simulations. Quantitative Finance, 19(6):1043–1059. doi: 10.1080/14697688.2018.1556398. url: https://doi.org/10.1080/14697688.2018.1556398.</ref><ref name="Dawood.Megahed.2023"/><ref name="Dawood.Megahed.2019"/><ref name="Dawood.Dawood.2020">[[Hend Dawood]] and [[Yasser Dawood]] (2020). Universal Intervals: Towards a Dependency-Aware Interval Algebra. In S. Chakraverty, editor, Mathematical Methods in Interdisciplinary Sciences. chapter 10, pages 167–214. John Wiley & Sons, Hoboken, New Jersey. ISBN 978-1-119-58550-3. doi: 10.1002/9781119585640.ch10. url: https://doi.org/10.1002/9781119585640.ch10.</ref><ref name="Dawood.2014">[[Hend Dawood]] (2014). Interval Mathematics as a Potential Weapon against Uncertainty. In S. Chakraverty, editor, Mathematics of Uncertainty Modeling in the Analysis of Engineering and Science Problems. chapter 1, pages 1–38. IGI Global, Hershey, PA. ISBN 978-1-4666-4991-0.</ref> Automatic differentiation is particularly important in the field of [[machine learning]]. For example, it allows one to implement [[backpropagation]] in a [[neural network (machine learning)|neural network]] without a manually-computed derivative.