롬베르크 적분 문서 원본 보기

{{위키데이터 속성 추적}}
'''롬베르크 적분'''({{lang|de|Romberg-Integration}})은 [[수치해석학]]에서 [[정적분]]

:<math> \int_a^b f(x) \, dx </math>

을 추산하기 위한 방법이다.{{Harv|Romberg|1955}} [[리처드슨 외삽법]]{{Harv|Richardson|1911}}을 [[사다리꼴 공식]] 또는 [[직사각형 공식]]에 반복적으로 적용하는 것이 골자이다. 추산하면 삼각배열이 생성된다. 롬베르크 적분은 [[뉴턴-코츠 공식]]의 일종이다. 이 방법을 1955년 논문으로 발표한 [[베르너 롬베르크]]의 이름을 따서 명명되었다.

== 방법 ==
방법은 다음과 같이 귀납적으로 정의된다.

:<math>R(0,0) = \frac{1}{2} (b-a) (f(a) + f(b))</math>

:<math>R(n,0) = \frac{1}{2} R(n-1,0) + h_n \sum_{k=1}^{2^{n-1}} f(a + (2k-1)h_n)</math>

:<math>R(n,m) = R(n,m-1) + \frac{1}{4^{m}-1} (R(n,m-1) - R(n-1,m-1))</math>

또는

:<math>R(n,m) = \frac{1}{4^{m}-1} ( 4^{m} R(n,m-1) - R(n-1,m-1))</math>

이때

:<math> n \ge m \, </math>

:<math> m \ge 1 \, </math>

:<math> h_n = \frac{b-a}{2^n}. </math>

[[점근 표기법|대문자 O 표기법]]으로 <math>R(n, m)</math>의 오차는{{Harv|Mysovskikh|2002}}

:<math> O\left(h_n^{2m+2}\right). \, </math>

== 기하학적 설명 ==
{{빈 문단}}

== 예시 ==
{{빈 문단}}

== 실행 ==
{{빈 문단}}

== 참고 자료 ==
{{참고 자료 시작}}
* {{인용|last1=Richardson|first1=L. F.|title=The Approximate Arithmetical Solution by Finite Differences of Physical Problems Involving Differential Equations, with an Application to the Stresses in a Masonry Dam|journal= Philosophical Transactions of the Royal Society A<!--, Containing Papers of a Mathematical or Physical Character--> |volume=210|issue=459-470|year=1911|pages=307–357|doi=10.1098/rsta.1911.0009|jstor=90994}}
* {{인용|last1=Romberg|first1=W.|title=Vereinfachte numerische Integration|journal=Det Kongelige Norske Videnskabers Selskab Forhandlinger|volume=28|year=1955|location=Trondheim|pages=30–36|issue=7}}
* {{인용|last=Thacher, Jr.|first=Henry C.|title=Remark on Algorithm 60: Romberg integration|journal=Communications of the ACM|volume=7|pages =420–421|date=July 1964|url=http://portal.acm.org/citation.cfm?id=364520.364542|doi=10.1145/364520.364542|issue=7}}
* {{인용|last1=Bauer|first1=F.L.|last2=Rutishauser|last3=Stiefel|first3=E.|title=New aspects in numerical quadrature|editor-last=Metropolis|editor-first=N. C., et al.|journal=Experimental Arithmetic, high-speed computing and mathematics, Proceedings of Symposia in Applied Mathematics|publisher=[[American Mathematical Society|AMS]]|year=1963|pages=199–218|first2=H.|issue=15}}
* {{인용|last1=Bulirsch|first1=Roland|last2=Stoer|first2=Josef|title= Handbook Series Numerical Integration. Numerical quadrature by extrapolation|journal=Numerische Mathematik|volume=9|year=1967|pages=271–278 |url=http://www-gdz.sub.uni-goettingen.de/cgi-bin/digbib.cgi?PPN362160546_0009|doi=10.1007/bf02162420}}
* {{인용|last=Mysovskikh|first=I.P.|contribution=Romberg method|editor-last=Hazewinkel|editor-first=Michiel|title=Encyclopaedia of Mathematics|publisher=[[Springer-Verlag]]|year=2002|isbn=1-4020-0609-8|url=http://eom.springer.de/r/r082570.htm}}
* {{인용|last1=Press|first1=WH|last2=Teukolsky|first2=SA|last3=Vetterling|first3=WT|last4=Flannery|first4=BP|year=2007|title=Numerical Recipes: The Art of Scientific Computing|edition=3rd|publisher=Cambridge University Press|publication-place=New York|isbn=978-0-521-88068-8|chapter=Section 4.3. Romberg Integration|chapter-url=http://apps.nrbook.com/empanel/index.html?pg=166|access-date=2015-04-01|archive-date=2012-03-19|archive-url=https://web.archive.org/web/20120319193839/http://apps.nrbook.com/empanel/index.html?pg=166|url-status=}}
{{참고 자료 끝}}

== 외부 링크 ==
* [http://www.mathworks.com/matlabcentral/fileexchange/loadFile.do?objectId=34&objectType=file ROMBINT] {{웹아카이브|url=https://web.archive.org/web/20080609193225/http://www.mathworks.com/matlabcentral/fileexchange/loadFile.do?objectId=34&objectType=file}} – code for [[MATLAB]] (author: Martin Kacenak)
* [https://web.archive.org/web/20070517005126/http://math.fullerton.edu/mathews/n2003/RombergMod.html Module for Romberg integration]
* [http://www.hvks.com/Numerical/webintegration.html Free online integration tool using Romberg, Fox–Romberg, Gauss–Legendre and other numerical methods]

{{전거 통제}}
{{토막글|수학}}

[[분류:수치해석학]]
[[분류:수치적분]]