Английская Википедия:Asymptotic analysis

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In mathematical analysis, asymptotic analysis, also known as asymptotics, is a method of describing limiting behavior.

As an illustration, suppose that we are interested in the properties of a function Шаблон:Math as Шаблон:Mvar becomes very large. If Шаблон:Math, then as Шаблон:Mvar becomes very large, the term Шаблон:Math becomes insignificant compared to Шаблон:Math. The function Шаблон:Math is said to be "asymptotically equivalent to Шаблон:Math, as Шаблон:Math". This is often written symbolically as Шаблон:Math, which is read as "Шаблон:Math is asymptotic to Шаблон:Math".

An example of an important asymptotic result is the prime number theorem. Let Шаблон:Math denote the prime-counting function (which is not directly related to the constant pi), i.e. Шаблон:Math is the number of prime numbers that are less than or equal to Шаблон:Mvar. Then the theorem states that <math display="block">\pi(x)\sim\frac{x}{\ln x}.</math>

Asymptotic analysis is commonly used in computer science as part of the analysis of algorithms and is often expressed there in terms of big O notation.

Definition

Formally, given functions Шаблон:Math and Шаблон:Math, we define a binary relation <math display="block">f(x) \sim g(x) \quad (\text{as } x\to\infty)</math> if and only if Шаблон:Harv <math display="block">\lim_{x \to \infty} \frac{f(x)}{g(x)} = 1.</math>

The symbol Шаблон:Math is the tilde. The relation is an equivalence relation on the set of functions of Шаблон:Mvar; the functions Шаблон:Mvar and Шаблон:Mvar are said to be asymptotically equivalent. The domain of Шаблон:Mvar and Шаблон:Mvar can be any set for which the limit is defined: e.g. real numbers, complex numbers, positive integers.

The same notation is also used for other ways of passing to a limit: e.g. Шаблон:Math, Шаблон:Math, Шаблон:Math. The way of passing to the limit is often not stated explicitly, if it is clear from the context.

Although the above definition is common in the literature, it is problematic if Шаблон:Math is zero infinitely often as Шаблон:Mvar goes to the limiting value. For that reason, some authors use an alternative definition. The alternative definition, in little-o notation, is that Шаблон:Math if and only if <math display="block">f(x)=g(x)(1+o(1)).</math>

This definition is equivalent to the prior definition if Шаблон:Math is not zero in some neighbourhood of the limiting value.[1][2]

Properties

If <math>f \sim g</math> and <math>a \sim b</math>, then, under some mild conditions,Шаблон:Explain the following hold:

  • <math>f^r \sim g^r</math>, for every real Шаблон:Mvar
  • <math>\log(f) \sim \log(g)</math> if <math>\lim g \neq 1 </math>
  • <math>f\times a \sim g\times b</math>
  • <math>f / a \sim g / b</math>

Such properties allow asymptotically-equivalent functions to be freely exchanged in many algebraic expressions.

Examples of asymptotic formulas

  • Factorial <math display="block">n! \sim \sqrt{2\pi n}\left(\frac{n}{e}\right)^n</math> —this is Stirling's approximation
  • Partition function Шаблон:Pb For a positive integer n, the partition function, p(n), gives the number of ways of writing the integer n as a sum of positive integers, where the order of addends is not considered. <math display="block">p(n)\sim \frac{1}{4n\sqrt{3}} e^{\pi\sqrt{\frac{2n}{3}}}</math>
  • Airy function Шаблон:Pb The Airy function, Ai(x), is a solution of the differential equation Шаблон:Math; it has many applications in physics. <math display="block">\operatorname{Ai}(x) \sim \frac{e^{-\frac{2}{3} x^\frac{3}{2}}}{2\sqrt{\pi} x^{1/4}}</math>
  • Hankel functions <math display="block">\begin{align}
H_\alpha^{(1)}(z) &\sim \sqrt{\frac{2}{\pi z}} e^{ i\left(z - \frac{2\pi\alpha - \pi}{4}\right)} \\
H_\alpha^{(2)}(z) &\sim \sqrt{\frac{2}{\pi z}} e^{-i\left(z - \frac{2\pi\alpha - \pi}{4}\right)}

\end{align}</math>

Asymptotic expansion

Шаблон:Main An asymptotic expansion of a function Шаблон:Math is in practice an expression of that function in terms of a series, the partial sums of which do not necessarily converge, but such that taking any initial partial sum provides an asymptotic formula for Шаблон:Mvar. The idea is that successive terms provide an increasingly accurate description of the order of growth of Шаблон:Mvar.

In symbols, it means we have <math>f \sim g_1,</math> but also <math>f - g_1 \sim g_2</math> and <math>f - g_1 - \cdots - g_{k-1} \sim g_{k}</math> for each fixed k. In view of the definition of the <math>\sim</math> symbol, the last equation means <math>f - (g_1 + \cdots + g_k) = o(g_k)</math> in the little o notation, i.e., <math>f - (g_1 + \cdots + g_k)</math> is much smaller than <math>g_k.</math>

The relation <math>f - g_1 - \cdots - g_{k-1} \sim g_{k}</math> takes its full meaning if <math>g_{k+1} = o(g_k)</math> for all k, which means the <math>g_k</math> form an asymptotic scale. In that case, some authors may abusively write <math>f \sim g_1 + \cdots + g_k</math> to denote the statement <math>f - (g_1 + \cdots + g_k) = o(g_k).</math> One should however be careful that this is not a standard use of the <math>\sim</math> symbol, and that it does not correspond to the definition given in Шаблон:Section link.

In the present situation, this relation <math>g_{k} = o(g_{k-1})</math> actually follows from combining steps k and k−1; by subtracting <math>f - g_1 - \cdots - g_{k-2} = g_{k-1} + o(g_{k-1})</math> from <math>f - g_1 - \cdots - g_{k-2} - g_{k-1} = g_{k} + o(g_{k}),</math> one gets <math>g_{k} + o(g_{k})=o(g_{k-1}),</math> i.e. <math>g_{k} = o(g_{k-1}).</math>

In case the asymptotic expansion does not converge, for any particular value of the argument there will be a particular partial sum which provides the best approximation and adding additional terms will decrease the accuracy. This optimal partial sum will usually have more terms as the argument approaches the limit value.

Examples of asymptotic expansions

  • Gamma function <math display="block">\frac{e^x}{x^x \sqrt{2\pi x}} \Gamma(x+1) \sim 1+\frac{1}{12x}+\frac{1}{288x^2}-\frac{139}{51840x^3}-\cdots
\ (x \to \infty)</math>
  • Exponential integral <math display="block">xe^xE_1(x) \sim \sum_{n=0}^\infty \frac{(-1)^nn!}{x^n} \ (x \to \infty) </math>
  • Error function <math display="block"> \sqrt{\pi}x e^{x^2}\operatorname{erfc}(x) \sim 1+\sum_{n=1}^\infty (-1)^n \frac{(2n-1)!!}{n!(2x^2)^n} \ (x \to \infty)</math> where Шаблон:Math is the double factorial.

Worked example

Asymptotic expansions often occur when an ordinary series is used in a formal expression that forces the taking of values outside of its domain of convergence. For example, we might start with the ordinary series <math display="block">\frac{1}{1-w}=\sum_{n=0}^\infty w^n</math>

The expression on the left is valid on the entire complex plane <math>w \ne 1</math>, while the right hand side converges only for <math>|w|< 1</math>. Multiplying by <math>e^{-w/t}</math> and integrating both sides yields <math display="block"> \int_0^\infty \frac{e^{-\frac{w}{t}}}{1 - w} \, dw = \sum_{n=0}^\infty t^{n+1} \int_0^\infty e^{-u} u^n \, du</math>

The integral on the left hand side can be expressed in terms of the exponential integral. The integral on the right hand side, after the substitution <math>u=w/t</math>, may be recognized as the gamma function. Evaluating both, one obtains the asymptotic expansion <math display="block">e^{-\frac{1}{t}} \operatorname{Ei}\left(\frac{1}{t}\right) = \sum _{n=0}^\infty n! \; t^{n+1} </math>

Here, the right hand side is clearly not convergent for any non-zero value of t. However, by keeping t small, and truncating the series on the right to a finite number of terms, one may obtain a fairly good approximation to the value of <math>\operatorname{Ei}(1/t)</math>. Substituting <math>x = -1/t</math> and noting that <math>\operatorname{Ei}(x) = -E_1(-x)</math> results in the asymptotic expansion given earlier in this article.

Asymptotic distribution

Шаблон:Main

In mathematical statistics, an asymptotic distribution is a hypothetical distribution that is in a sense the "limiting" distribution of a sequence of distributions. A distribution is an ordered set of random variables Шаблон:Math for Шаблон:Math, for some positive integer Шаблон:Math. An asymptotic distribution allows Шаблон:Math to range without bound, that is, Шаблон:Math is infinite.

A special case of an asymptotic distribution is when the late entries go to zero—that is, the Шаблон:Math go to 0 as Шаблон:Math goes to infinity. Some instances of "asymptotic distribution" refer only to this special case.

This is based on the notion of an asymptotic function which cleanly approaches a constant value (the asymptote) as the independent variable goes to infinity; "clean" in this sense meaning that for any desired closeness epsilon there is some value of the independent variable after which the function never differs from the constant by more than epsilon.

An asymptote is a straight line that a curve approaches but never meets or crosses. Informally, one may speak of the curve meeting the asymptote "at infinity" although this is not a precise definition. In the equation <math>y = \frac{1}{x},</math> y becomes arbitrarily small in magnitude as x increases.

Applications

Asymptotic analysis is used in several mathematical sciences. In statistics, asymptotic theory provides limiting approximations of the probability distribution of sample statistics, such as the likelihood ratio statistic and the expected value of the 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.

Asymptotic analysis is a key tool for exploring the ordinary and partial differential equations which arise in the mathematical modelling of real-world phenomena.[3] An illustrative example is the derivation of the 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, Шаблон:Mvar: in the boundary layer case, this is the nondimensional ratio of the boundary layer thickness to a typical length scale of the problem. Indeed, applications of asymptotic analysis in mathematical modelling often[3] 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

Debruijn illustrates the use of asymptotics in the following dialog between Miss N.A., a Numerical Analyst, and Dr. A.A., an Asymptotic Analyst:

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

<math>|f(x)-x^{-1}|<57000x^{-2} \qquad (x>100).</math>

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

<math>|f(x)-x^{-1}|<20x^{-2} \qquad (x>100).</math>

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.[4]

See also

Шаблон:Div col

Шаблон:Div col end

Notes

  1. ↑ Шаблон:SpringerEOM
  2. ↑ Шаблон:Harvtxt
  3. ↑ 3,0 3,1 Howison, S. (2005), Practical Applied Mathematics, Cambridge University Press
  4. ↑ Шаблон:Cite book

References

External links

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