Asymptotic analysis - Citizendia

This article is about the mathematical method of asymptotic analysis. For information about asymptotic geometry, see asymptotic curve. In the Differential geometry of surfaces, an asymptotic curve is a Curve always Tangent to an asymptotic direction of the surface (where they exist

In pure and applied mathematics, particularly the analysis of algorithms, real analysis, and engineering, asymptotic analysis is a method of describing limiting behaviour. Broadly speaking pure mathematics is Mathematics motivated entirely for reasons other than application Applied mathematics is a branch of Mathematics that concerns itself with the mathematical techniques typically used in the application of mathematical knowledge to other domains To analyze an Algorithm is to determine the amount of resources (such as time and storage necessary to execute it In Mathematics, the concept of a " limit " is used to describe the Behavior of a function as its argument either "gets close" Similar limiting behaviour is sometimes expressed in the language of equivalence relations. In Mathematics, an equivalence relation is a Binary relation between two elements of a set which groups them together as being "equivalent" Moreover, asymptotic analysis refers to solving problems approximately up to such equivalences. For example, given complex-valued functions f and g of a natural number variable n, one writes

$f \sim g \quad (\mbox{as } n\to\infty)$

to express the fact that

$\lim_{n\to\infty} \frac{f(n)}{g(n)} = 1$

and $f$ and $g$ are called asymptotically equivalent as n → ∞. This defines an equivalence relation (on the set of functions being nonzero for all n large enough - most mathematicians prefer the definition $f\sim g\iff f-g=o(g)$ in terms of Landau notation, which avoids this restriction). In mathematics big O notation (so called because it uses the symbol O) describes the limiting behavior of a function for very small or very large arguments The equivalence class of f consists of all functions g which "behave like" f, in the limit.

Asymptotic notation has been developed to provide a convenient language for the handling of statements about order of growth. In mathematics big O notation (so called because it uses the symbol O) describes the limiting behavior of a function for very small or very large arguments It is also called Landau notation, since it became popular first in research in analytic number theory, from about 1900 onwards, introduced by Edmund Landau (originated though by Paul Bachmann). In mathematics big O notation (so called because it uses the symbol O) describes the limiting behavior of a function for very small or very large arguments In Mathematics, analytic number theory is a branch of Number theory that uses methods from Mathematical analysis to solve number-theoretical problems Edmund Georg Hermann (Yehezkel Landau ( February 14, 1877 – February 19, 1938) was a German Jewish Mathematician Paul Gustav Heinrich Bachmann ( June 22 1837 &ndash March 31 1920) was a German Mathematician. See also Big O notation, for a treatment more from the point of view of analysis of algorithms. In mathematics big O notation (so called because it uses the symbol O) describes the limiting behavior of a function for very small or very large arguments In Mathematics, Computing, Linguistics and related subjects an algorithm is a sequence of finite instructions often used for Calculation The asymptotic point of view is basic in computer science, where the question is typically how to describe the resource implication of scaling-up the size of a computational problem, beyond the 'toy' level. Computer science (or computing science) is the study and the Science of the theoretical foundations of Information and Computation and their

An asymptotic expansion of a function f(x) 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 f. In Mathematics an asymptotic expansion, asymptotic series or Poincaré expansion (after Henri Poincaré) is a formal series of functions which In Mathematics, a series is often represented as the sum of a Sequence of terms That is a series is represented as a list of numbers with In Mathematics, a series is often represented as the sum of a Sequence of terms That is a series is represented as a list of numbers with The idea is that successive terms provide a more and more accurate description of the order of growth of f. An example is Stirling's approximation. In Mathematics, Stirling's approximation (or Stirling's formula) is an approximation for large Factorials It is named in honour of James Stirling

In symbols, it means we have

$f \sim g_1$

but also

$f \sim g_1 + g_2$

and

$f \sim g_1 + \cdots + g_k$

for each fixed k, while some limit is taken, usually with the requirement that g_k+1 = o(g_k), which means the (g_k) form an asymptotic scale. In Mathematics an asymptotic expansion, asymptotic series or Poincaré expansion (after Henri Poincaré) is a formal series of functions which The requirement that the successive sums improve the approximation may then be expressed as $f - (g_1 + \cdots + g_k) = o(g_k)$ .

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. However, this optimal partial sum will usually have more terms as the argument approaches the limit value.

Asymptotic expansions typically arise in the approximation of certain integrals (saddle-point method, method of steepest descent) or in the approximation of probability distributions (Edgeworth series). For the optimization method called "steepest descent" see Gradient descent. For the optimization method called "steepest descent" see Gradient descent. The Gram-Charlier A series and the Edgeworth series, named in honor of Francis Ysidro Edgeworth, are series that approximate a Probability distribution The famous Feynman graphs in quantum field theory are another example of asymptotic expansions which often do not converge. A Feynman graph is a graph suitable to be a Feynman diagram in a particular application of Quantum field theory. In quantum field theory (QFT the forces between particles are mediated by other particles

References

Erdélyi, A. In Computational complexity theory, asymptotic computational complexity is the usage of the Asymptotic analysis for the estimation of computational complexity of Asymptotic Expansions. New York: Dover, 1987.
^ J. P. Boyd, The Devil's Invention: Asymptotic, Superasymptotic and Hyperasymptotic Series, Acta Applicandae Mathematicae: An International Survey Journal on Applying Mathematics and Mathematical Applications 56, 1-98 (1999) PDF of preprint

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