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Danskin's theorem — used in the analysis of minimax problems; Maximum theorem — the maximum and maximizer are continuous as function of parameters, under some conditions; No free lunch in search and optimization; Relaxation (approximation) — approximating a given problem by an easier problem by relaxing some constraints Lagrangian relaxation
Another problem of extrapolation is loosely related to the problem of analytic continuation, where (typically) a power series representation of a function is expanded at one of its points of convergence to produce a power series with a larger radius of convergence. In effect, a set of data from a small region is used to extrapolate a function ...
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 A ( h ) {\displaystyle A(h)} for several values of h {\displaystyle h} , we can estimate A ∗ {\displaystyle A^{\ast }} by extrapolating the ...
For example, an experimental uncertainty analysis of an undergraduate physics lab experiment in which a pendulum can estimate the value of the local gravitational acceleration constant g. The relevant equation [1] for an idealized simple pendulum is, approximately,
If the trend can be assumed to be linear, trend analysis can be undertaken within a formal regression analysis, as described in Trend estimation. If the trends have other shapes than linear, trend testing can be done by non-parametric methods, e.g. Mann-Kendall test, which is a version of Kendall rank correlation coefficient.
In numerical analysis, Aitken's delta-squared process or Aitken extrapolation is a series acceleration method used for accelerating the rate of convergence of a sequence. It is named after Alexander Aitken, who introduced this method in 1926. [1] It is most useful for accelerating the convergence of a sequence that is converging linearly.
In his original work Jan Tauc (/ t aʊ t s /) showed that the optical absorption spectrum of amorphous germanium resembles the spectrum of the indirect transitions in crystalline germanium (plus a tail due to localized states at lower energies), and proposed an extrapolation to find the optical bandgap of these crystalline-like states. [4]
The field of numerical analysis predates the invention of modern computers by many centuries. Linear interpolation was already in use more than 2000 years ago. Many great mathematicians of the past were preoccupied by numerical analysis, [5] as is obvious from the names of important algorithms like Newton's method, Lagrange interpolation polynomial, Gaussian elimination, or Euler's method.