Editing Asymptotic analysis (section)

==Applications==
Asymptotic analysis is used in several [[mathematical sciences]]. In [[statistics]], asymptotic theory provides limiting approximations of the [[probability distribution]] of [[sample statistic]]s, such as the [[Likelihood-ratio test|likelihood ratio]] [[statistic]] and the [[expected value]] of the [[deviance (statistics)|deviance]]. Asymptotic theory does not provide a method of evaluating the finite-sample distributions of sample statistics, however. Non-asymptotic bounds are provided by methods of [[approximation theory]].

Examples of applications are the following.
* In [[applied mathematics]], asymptotic analysis is used to build [[numerical method]]s to approximate [[equation]] solutions.
* In [[mathematical statistics]] and [[probability theory]], asymptotics are used in analysis of long-run or large-sample behaviour of random variables and estimators.
* In [[computer science]] in the [[analysis of algorithms]], considering the performance of algorithms.
* The behavior of [[physical system]]s, an example being [[statistical mechanics]].
* In [[accident analysis]] when identifying the causation of crash through count modeling with large number of crash counts in a given time and space.

Asymptotic analysis is a key tool for exploring the [[ordinary differential equation|ordinary]] and [[partial differential equation|partial]] differential equations which arise in the [[mathematical model]]ling of real-world phenomena.<ref name="Howison">Howison, S. (2005), ''[https://books.google.com/books?id=A2Hy_54Y1MsC&q=%22asymptotic+analysis%22 Practical Applied Mathematics]'', [[Cambridge University Press]]</ref> An illustrative example is the derivation of the [[Boundary layer#Boundary layer equations|boundary layer equations]] from the full [[Navier-Stokes equations]] governing fluid flow. In many cases, the asymptotic expansion is in power of a small parameter, {{mvar|ε}}: in the boundary layer case, this is the [[dimensional analysis|nondimensional]] ratio of the boundary layer thickness to a typical length scale of the problem. Indeed, applications of asymptotic analysis in mathematical modelling often<ref name="Howison" /> center around a nondimensional parameter which has been shown, or assumed, to be small through a consideration of the scales of the problem at hand.

Asymptotic expansions typically arise in the approximation of certain integrals ([[Laplace's method]], [[saddle-point method]], [[method of steepest descent]]) or in the approximation of probability distributions ([[Edgeworth series]]). The [[Feynman graphs]] in [[quantum field theory]] are another example of asymptotic expansions which often do not converge.

=== Asymptotic versus Numerical Analysis ===
De Bruijn illustrates the use of asymptotics in the following dialog between Dr. N.A., a Numerical Analyst, and Dr. A.A., an Asymptotic Analyst:
<blockquote>N.A.: I want to evaluate my function <math>f(x)</math> for large values of <math>x</math>, with a relative error of at most 1%.

A.A.: <math>f(x)=x^{-1}+\mathrm O(x^{-2}) \qquad (x\to\infty)</math>.

N.A.: I am sorry, I don't understand.

A.A.: <math>|f(x)-x^{-1}|<8x^{-2} \qquad (x>10^4).</math>

N.A.: But my value of <math>x</math> is only 100.

A.A.: Why did you not say so? My evaluations give<blockquote><math>|f(x)-x^{-1}|<57000x^{-2} \qquad (x>100).</math></blockquote>

N.A.: This is no news to me. I know already that <math>0<f(100)<1</math>.

A.A.: I can gain a little on some of my estimates. Now I find that<blockquote><math>|f(x)-x^{-1}|<20x^{-2} \qquad (x>100).</math></blockquote>

N.A.: I asked for 1%, not for 20%.

A.A.: It is almost the best thing I possibly can get. Why don't you take larger values of <math>x</math>?

N.A.: !!!  I think it's better to ask my electronic computing machine.

Machine:  f(100) = 0.01137 42259 34008 67153

A.A.: Haven't I told you so? My estimate of 20% was not far off from the 14% of the real error.

N.A.: !!! . . .  !

Some days later, Miss N.A. wants to know the value of f(1000), but her machine would take a month of computation to give the answer. She returns to her Asymptotic Colleague, and gets a fully satisfactory reply.<ref>{{Cite book |last=Bruijn |first=Nicolaas Govert de |title=Asymptotic methods in analysis |date=1981 |publisher=Dover publ |isbn=978-0-486-64221-5 |series=Dover books on advanced mathematics |location=New York |pages=19}}</ref></blockquote>