Editing Carl Friedrich Gauss (section)

==== Numerical analysis ====
Gauss often deduced theorems [[Inductive reasoning|inductively]] from numerical data he had collected empirically.{{sfn|Schlesinger|1933|p=18}} As such, the use of efficient algorithms to facilitate calculations was vital to his research, and he made many contributions to [[numerical analysis]], such as the method of [[Gaussian quadrature]], published in 1816.<ref>{{cite book | last = Gautschi | first = Walter | author-link = Walter Gautschi | edition = 1 | title = E.B. Christoffel. The Influence of his Work on Mathematics and the Physical Science | editor-last1 = Butzer | editor-first1 = Paul B. | editor-last2 = Fehér | editor-first2 = Franziska | editor-link1 = Paul Butzer | chapter = A Survey of Gauss-Christoffel Quadrature Formulae | place = Birkhäuser, Basel | year = 1981 | pages = 72–147 | publisher = Springer | doi = 10.1007/978-3-0348-5452-8_6 | isbn = 978-3-0348-5452-8 | chapter-url = https://link.springer.com/chapter/10.1007/978-3-0348-5452-8_6}}</ref>

In a private letter to [[Christian Ludwig Gerling|Gerling]] from 1823,<ref>{{Cite book|url=https://gdz.sub.uni-goettingen.de/id/PPN335994989?tify=%7B%22pages%22:%5B316%5D,%22view%22:%22info%22%7D|title=Briefwechsel zwischen Carl Friedrich Gauss und Christian Ludwig Gerling|publisher=Elsner}}</ref> he described a solution of a 4x4 system of linear equations with the [[Gauss-Seidel method]] – an "indirect" [[iterative method]] for the solution of linear systems, and recommended it over the usual method of "direct elimination" for systems of more than two equations.<ref>{{cite arXiv | author = Yousef Saad | author-link = Yousef Saad | title = Iterative Methods for Linear Systems of Equations: A Brief Historical Journey | date = 2 August 2019 | class = math.HO | eprint = 1908.01083v1}}</ref>

Gauss invented an algorithm for calculating what is now called [[discrete Fourier transform]]s when calculating the orbits of Pallas and Juno in 1805, 160 years before [[James Cooley|Cooley]] and [[John Tukey|Tukey]] found their similar [[Cooley-Tukey FFT algorithm|Cooley–Tukey algorithm]].<ref>{{cite journal | last1 = Cooley | first1 = James W. | first2 = John W. | last2 = Tukey | title = An algorithm for the machine calculation of complex Fourier series | journal = [[Mathematics of Computation]] | volume = 19 | issue = 90 | pages = 297–301 | year = 1965 | doi = 10.2307/2003354 | jstor = 2003354 | doi-access=free }}</ref> He developed it as a [[trigonometric interpolation]] method, but the paper ''Theoria Interpolationis Methodo Nova Tractata'' was published only posthumously in 1876,<ref>{{cite book | last = Gauss | first = C.F. | date = 1876 | title = Theoria Interpolationis Methodo Nova Tractata | place = Göttingen | language = la | pages = 265–327 | url = https://gdz.sub.uni-goettingen.de/id/PPN235999628?tify=%7B%22pages%22%3A%5B273%5D%2C%22pan%22%3A%7B%22x%22%3A0.524%2C%22y%22%3A0.333%7D%2C%22view%22%3A%22info%22%2C%22zoom%22%3A0.856%7D | publisher = K. Gesellschaft der Wissenschaften zu Göttingen}}</ref> well after [[Joseph Fourier]]'s introduction of the subject in 1807.<ref>{{cite journal | last1 = Heideman | first1 = Michael T. | last2 = Johnson| first2 = Don H. | last3 = Burrus | first3 = C. Sidney | author-link3 = C. Sidney Burrus | title = Gauss and the history of the fast Fourier transform | journal = IEEE ASSP Magazine | year = 1984 | volume = 1 | issue = 4 | pages = 14–21 | doi = 10.1109/MASSP.1984.1162257|s2cid=10032502 | url = http://www.cis.rit.edu/class/simg716/Gauss_History_FFT.pdf | archive-url=https://web.archive.org/web/20130319053449/http://www.cis.rit.edu/class/simg716/Gauss_History_FFT.pdf | archive-date = 19 March 2013 | url-status=live}}</ref>