Laplace's method
In mathematics, Laplace's method, named after Pierre-Simon Laplace, is a technique used to approximate integrals of the form where is a twice-differentiable function, M is a large number, and the endp
Laplace principle (large deviations theory)
In mathematics, Laplace's principle is a basic theorem in large deviations theory which is similar to Varadhan's lemma. It gives an asymptotic expression for the Lebesgue integral of exp(−θφ(x)) over
Saddlepoint approximation method
The saddlepoint approximation method, initially proposed by Daniels (1954) is a specific example of the mathematical saddlepoint technique applied to statistics. It provides a highly accurate approxim
Spillover (experiment)
In experiments, a spillover is an indirect effect on a subject not directly treated by the experiment. These effects are useful for policy analysis but complicate the statistical analysis of experimen
Asymptotology
Asymptotology has been defined as “the art of dealing with applied mathematical systems in limiting cases” as well as “the science about the synthesis of simplicity and exactness by means of localizat
Gregory coefficients
Gregory coefficients Gn, also known as reciprocal logarithmic numbers, Bernoulli numbers of the second kind, or Cauchy numbers of the first kind, are the rational numbers that occur in the Maclaurin s
Galactic algorithm
A galactic algorithm is one that outperforms any other algorithm for problems that are sufficiently large, but where "sufficiently large" is so big that the algorithm is never used in practice. Galact
Perturbation problem beyond all orders
In mathematics, perturbation theory works typically by expanding unknown quantity in a power series in a small parameter. However, in a perturbation problem beyond all orders, all coefficients of the
Series acceleration
In mathematics, series acceleration is one of a collection of sequence transformations for improving the rate of convergence of a series. Techniques for series acceleration are often applied in numeri
Distinguished limit
In mathematics, a distinguished limit is an appropriately chosen scale factor used in the method of matched asymptotic expansions.
Tilted large deviation principle
In mathematics — specifically, in large deviations theory — the tilted large deviation principle is a result that allows one to generate a new large deviation principle from an old one by "tilting", i
Watson's lemma
In mathematics, Watson's lemma, proved by G. N. Watson (1918, p. 133), has significant application within the theory on the asymptotic behavior of integrals.
Hajek projection
In statistics, Hájek projection of a random variable on a set of independent random vectors is a particular measurable function of that, loosely speaking, captures the variation of in an optimal way.
Dawson–Gärtner theorem
In mathematics, the Dawson–Gärtner theorem is a result in large deviations theory. Heuristically speaking, the Dawson–Gärtner theorem allows one to transport a large deviation principle on a “smaller”
Big O notation
Big O notation is a mathematical notation that describes the limiting behavior of a function when the argument tends towards a particular value or infinity. Big O is a member of a invented by Paul Bac
Schilder's theorem
In mathematics, Schilder's theorem is a generalization of the Laplace method from integrals on to functional Wiener integration. The theorem is used in the large deviations theory of stochastic proces
Shanks transformation
In numerical analysis, the Shanks transformation is a non-linear series acceleration method to increase the rate of convergence of a sequence. This method is named after Daniel Shanks, who rediscovere
Contraction principle (large deviations theory)
In mathematics — specifically, in large deviations theory — the contraction principle is a theorem that states how a large deviation principle on one space "pushes forward" (via the pushforward of a p
Activation energy asymptotics
Activation energy asymptotics (AEA), also known as large activation energy asymptotics, is an asymptotic analysis used in the combustion field utilizing the fact that the reaction rate is extremely se
Stirling's approximation
In mathematics, Stirling's approximation (or Stirling's formula) is an approximation for factorials. It is a good approximation, leading to accurate results even for small values of . It is named afte
Varadhan's lemma
In mathematics, Varadhan's lemma is a result from large deviations theory named after S. R. Srinivasa Varadhan. The result gives information on the asymptotic distribution of a statistic φ(Zε) of a fa
Rate function
In mathematics — specifically, in large deviations theory — a rate function is a function used to quantify the probabilities of rare events. It is required to have several properties which assist in t
Asymptotic expansion
In mathematics, an asymptotic expansion, asymptotic series or Poincaré expansion (after Henri Poincaré) is a formal series of functions which has the property that truncating the series after a finite
Asymptotic analysis
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 functio
Leading-order term
The leading-order terms (or corrections) within a mathematical equation, expression or model are the terms with the largest order of magnitude. The sizes of the different terms in the equation(s) will
Akra–Bazzi method
In computer science, the Akra–Bazzi method, or Akra–Bazzi theorem, is used to analyze the asymptotic behavior of the mathematical recurrences that appear in the analysis of divide and conquer algorith
Borel's lemma
In mathematics, Borel's lemma, named after Émile Borel, is an important result used in the theory of asymptotic expansions and partial differential equations.
Euler–Maclaurin formula
In mathematics, the Euler–Maclaurin formula is a formula for the difference between an integral and a closely related sum. It can be used to approximate integrals by finite sums, or conversely to eval
Iterated logarithm
In computer science, the iterated logarithm of , written (usually read "log star"), is the number of times the logarithm function must be iteratively applied before the result is less than or equal to
Richardson extrapolation
In numerical analysis, Richardson extrapolation is a sequence acceleration method used to improve the rate of convergence of a sequence of estimates of some value . In essence, given the value of for
Sequence transformation
In mathematics, a sequence transformation is an operator acting on a given space of sequences (a sequence space). Sequence transformations include linear mappings such as convolution with another sequ
Stokes phenomenon
In complex analysis the Stokes phenomenon, discovered by G. G. Stokes , is that the asymptotic behavior of functions can differ in different regions of the complex plane, and that these differences ca
Limit (mathematics)
In mathematics, a limit is the value that a function (or sequence) approaches as the input (or index) approaches some value. Limits are essential to calculus and mathematical analysis, and are used to
Variational perturbation theory
In mathematics, variational perturbation theory (VPT) is a mathematical method to convert divergent power series in a small expansion parameter, say , into a convergent series in powers , where is a c
Method of matched asymptotic expansions
In mathematics, the method of matched asymptotic expansions is a common approach to finding an accurate approximation to the solution to an equation, or system of equations. It is particularly used wh
Freidlin–Wentzell theorem
In mathematics, the Freidlin–Wentzell theorem (due to Mark Freidlin and Alexander D. Wentzell) is a result in the large deviations theory of stochastic processes. Roughly speaking, the Freidlin–Wentze
Method of steepest descent
In mathematics, the method of steepest descent or saddle-point method is an extension of Laplace's method for approximating an integral, where one deforms a contour integral in the complex plane to pa
Transseries
In mathematics, the field of logarithmic-exponential transseries is a non-Archimedean ordered differential field which extends comparability of asymptotic growth rates of elementary nontrigonometric f
Homotopy analysis method
The homotopy analysis method (HAM) is a semi-analytical technique to solve nonlinear ordinary/partial differential equations. The homotopy analysis method employs the concept of the homotopy from topo
Asymptotic homogenization
In mathematics and physics, homogenization is a method of studying partial differential equations with rapidly oscillating coefficients, such as where is a very small parameter and is a 1-periodic coe
Slowly varying envelope approximation
In physics, slowly varying envelope approximation (SVEA, sometimes also called slowly varying asymmetric approximation or SVAA) is the assumption that the envelope of a forward-travelling wave pulse v
Large deviations theory
In probability theory, the theory of large deviations concerns the asymptotic behaviour of remote tails of sequences of probability distributions. While some basic ideas of the theory can be traced to
Extrapolation
In mathematics, extrapolation is a type of estimation, beyond the original observation range, of the value of a variable on the basis of its relationship with another variable. It is similar to interp
Perturbation theory
In mathematics and applied mathematics, perturbation theory comprises methods for finding an approximate solution to a problem, by starting from the exact solution of a related, simpler problem. A cri
Quadratic growth
In mathematics, a function or sequence is said to exhibit quadratic growth when its values are proportional to the square of the function argument or sequence position. "Quadratic growth" often means
L-notation
L-notation is an asymptotic notation analogous to big-O notation, denoted as for a bound variable tending to infinity. Like big-O notation, it is usually used to roughly convey the rate of growth of a
Divergent series
In mathematics, a divergent series is an infinite series that is not convergent, meaning that the infinite sequence of the partial sums of the series does not have a finite limit. If a series converge
Hardy field
In mathematics, a Hardy field is a field consisting of germs of real-valued functions at infinity that are closed under differentiation. They are named after the English mathematician G. H. Hardy.
Multiple-scale analysis
In mathematics and physics, multiple-scale analysis (also called the method of multiple scales) comprises techniques used to construct uniformly valid approximations to the solutions of perturbation p
WKB approximation
In mathematical physics, the WKB approximation or WKB method is a method for finding approximate solutions to linear differential equations with spatially varying coefficients. It is typically used fo
Method of dominant balance
In mathematics, the method of dominant balance is used to determine the asymptotic behavior of solutions to an ordinary differential equation without fully solving the equation. The process is iterati
Riemann–Lebesgue lemma
In mathematics, the Riemann–Lebesgue lemma, named after Bernhard Riemann and Henri Lebesgue, states that the Fourier transform or Laplace transform of an L1 function vanishes at infinity. It is of imp
Exponentially equivalent measures
In mathematics, exponential equivalence of measures is how two sequences or families of probability measures are "the same" from the point of view of large deviations theory.
Master theorem (analysis of algorithms)
In the analysis of algorithms, the master theorem for divide-and-conquer recurrences provides an asymptotic analysis (using Big O notation) for recurrence relations of types that occur in the analysis
Linear predictive analysis
Linear predictive analysis is a simple form of first-order extrapolation: if it has been changing at this rate then it will probably continue to change at approximately the same rate, at least in the