Editing Gaussian elimination (section)

== History ==
The method of Gaussian elimination appears – albeit without proof – in the Chinese mathematical text [[Rod calculus#System of linear equations|Chapter Eight: ''Rectangular Arrays'']] of ''[[The Nine Chapters on the Mathematical Art]]''. Its use is illustrated in eighteen problems, with two to five equations. The first reference to the book by this title is dated to 179&nbsp;AD, but parts of it were written as early as approximately 150&nbsp;BC.<ref>{{Cite web |title=DOCUMENTA MATHEMATICA, Vol. Extra Volume: Optimization Stories (2012), 9-14 |url=https://www.emis.de/journals/DMJDMV/vol-ismp/10_yuan-ya-xiang.html |access-date=2022-12-02 |website=www.emis.de}}</ref><ref>{{harvnb|Calinger|1999|pp=234–236}}</ref><ref name="princeton">{{cite book | author1=Timothy Gowers | author2=June Barrow-Green | author3=Imre Leader | title=The Princeton Companion to Mathematics |url=https://archive.org/details/princetoncompanio00gowe|date=8 September 2008|publisher=Princeton University Press|isbn=978-0-691-11880-2|page=607}}<!-- |access-date=28 September 2012 --></ref> It was commented on by [[Liu Hui]] in the 3rd century.

According to Grcar<ref>{{harvnb|Grcar|2011a|pp=169–172}}</ref> solution of linear equations by elimination was invented independently in several cultures  in Eurasia starting from antiquity and in Europe definite examples of procedure were published already by late Renaissance (in 1550's). It is quite possible that already then the procedure was considered by mathematicians  elementary and in no need to explanation for professionals, so we may never learn its detailed history except that by then it was practiced in at least several places in Europe.

The method in Europe stems from the notes of [[Isaac Newton]].<ref>{{harvnb|Grcar|2011a|pp=169–172}}</ref><ref>{{harvnb|Grcar|2011b|pp=783–785}}</ref> In 1670, he wrote that all the algebra books known to him lacked a lesson for solving simultaneous equations, which Newton then supplied. Cambridge University eventually published the notes as ''Arithmetica Universalis'' in 1707 long after Newton had left academic life. The notes were widely imitated, which made (what is now called) Gaussian elimination a standard lesson in algebra textbooks by the end of the 18th century. [[Carl Friedrich Gauss]] in 1810 devised a notation for symmetric elimination that was adopted in the 19th century by professional [[Human computer|hand computers]] to solve the normal equations of least-squares problems.<ref>{{harvnb|Lauritzen|p=3}}</ref> The algorithm that is taught in high school was named for Gauss only in the 1950s as a result of confusion over the history of the subject.<ref>{{harvnb|Grcar|2011b|p=789}}</ref>

Some authors use the term ''Gaussian elimination'' to refer only to the procedure until the matrix is in echelon form, and use the term Gauss–Jordan elimination to refer to the procedure which ends in reduced echelon form. The name is used because it is a variation of Gaussian elimination as described by [[Wilhelm Jordan (geodesist)|Wilhelm Jordan]] in 1888. However, the method also appears in an article by Clasen published in the same year. Jordan and Clasen probably discovered Gauss–Jordan elimination independently.<ref>{{Citation | last1=Althoen | first1=Steven C. | last2=McLaughlin | first2=Renate | title=Gauss–Jordan reduction: a brief history | doi=10.2307/2322413 | year=1987 | journal=[[American Mathematical Monthly|The American Mathematical Monthly]] | issn=0002-9890 | volume=94 | issue=2 | pages=130–142 | jstor=2322413 | publisher=Mathematical Association of America}}</ref>