Editing Richardson extrapolation (section)

{{Short description|Sequence acceleration method in numerical analysis}}
[[File:Richardson extra 2d.gif|thumb|An example of Richardson extrapolation method in two dimensions.]]
In [[numerical analysis]], '''Richardson extrapolation''' is a [[Series acceleration|sequence acceleration]] method used to improve the [[rate of convergence]] of a [[sequence]] of estimates of some value <math>A^\ast = \lim_{h\to 0} A(h)</math>. In essence, given the value of <math>A(h)</math> for several values of <math>h</math>, we can estimate <math>A^\ast</math> by extrapolating the estimates to <math>h=0</math>. It is named after [[Lewis Fry Richardson]], who introduced the technique in the early 20th century,<ref>{{cite journal
 | last=Richardson | first=L. F. | author-link=Lewis Fry Richardson
 | title=The approximate arithmetical solution by finite differences of physical problems including differential equations, with an application to the stresses in a masonry dam
 | journal=Philosophical Transactions of the Royal Society A
 | year=1911 | volume=210
 | issue=459–470 | pages=307–357
 | doi=10.1098/rsta.1911.0009
| doi-access=free}}</ref><ref>{{cite journal
 | last1=Richardson | first1=L. F. | author-link=Lewis Fry Richardson
 | title=The deferred approach to the limit
 | journal=Philosophical Transactions of the Royal Society A
 | year=1927 | volume=226
 | issue=636–646 | pages=299–349
 | doi=10.1098/rsta.1927.0008
 | last2=Gaunt
 | first2=J. A.
| doi-access=free
 }}</ref> though the idea was already known to [[Christiaan Huygens]] in [[Christiaan_Huygens#De_Circuli_Magnitudine_Inventa|his calculation]] of <math>\pi</math>.<ref>{{Citation|last=Brezinski|first=Claude|title=Some pioneers of extrapolation methods|date=2009-11-01|url=https://www.worldscientific.com/doi/10.1142/9789812836267_0001|work=The Birth of Numerical Analysis|pages=1–22|publisher=WORLD SCIENTIFIC|doi=10.1142/9789812836267_0001|isbn=978-981-283-625-0}}</ref> In the words of [[Garrett Birkhoff|Birkhoff]] and [[Gian-Carlo Rota|Rota]], "its usefulness for practical computations can hardly be overestimated."<ref>Page 126 of {{cite book | last=Birkhoff | first=Garrett | author-link=Garrett Birkhoff |author2=Gian-Carlo Rota |author2-link=Gian-Carlo Rota | title=Ordinary differential equations | publisher=John Wiley and sons | year=1978 | edition=3rd | isbn=0-471-07411-X | oclc= 4379402}}</ref> 

Practical applications of Richardson extrapolation include [[Romberg integration]], which applies Richardson extrapolation to the [[trapezoid rule]], and the [[Bulirsch–Stoer algorithm]] for solving ordinary differential equations.