Approximation


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approximation

[ə¦präk·sə¦mā·shən]
(mathematics)
A result that is not exact but is near enough to the correct result for some specified purpose.
A procedure for obtaining such a result.

Approximation

 

replacement of certain mathematical objects by others which are in one sense or another close to the initial objects. Approximation makes it possible to study the numerical characteristics and qualitative properties of the object, reducing the problem to a study of simpler or more convenient objects—for example, objects whose characteristics are easily computed or whose properties are already known. The theory of numbers studies Diophantine approximations—in particular, approximations of irrational numbers by rational numbers. Approximations of curves, surfaces, spaces, and mappings are investigated in geometry and topology. Some branches of mathematics are wholly devoted to approximations, as, for example, the approximation and interpolation of functions and numerical methods of analysis. The role of approximation in mathematics is continually growing. Presently, approximation can be viewed as one of the basic concepts of mathematics.

S. B. STECHKIN

References in periodicals archive ?
Cardinality violation: Li [12] gave a novel linear program called rectangle LP and presented an improved approximation algorithm (exp(0(1/[[epsilon].sup.2]))) using at most (1 + [epsilon])k facilities for hard uniform CkM problem.
which provides an approximation solution of the SwiftHohenberg equation projected onto the low-frequency parts.
The general procedure of the simple approximation is to substitute (3) into (1), develop in powers of [epsilon], and put all coefficients of the powers of [epsilon] equal to zero.
Before the implementation of Milstein scheme we need to mention some facts about the two-level approximation.
In order to obtain the weighted approximation Theorem 6, let p = {[p.sub.n]}, q = {[q.sub.n]} be sequences satisfying the conditions (A), (B), or (C) of Lemma 4 and [lim.sub.n[right arrow][infinity]] [p.sup.n.sub.n] = a, a [member of] (0,1].
In this paper we introduce in an abstract setting the Rogosinski- and Blackman-type operators and find the order of approximation via a modulus of continuity (smoothness), which is defined by a general cosine operator functions.
A sampling approach to saddlepoint approximation has been presented in [12], while an improved third-moment saddlepoint approximation appears in [13].
1, in which M1 is the method of single NR combined with eight-piecewise linear approximation in highly non-linear range.
Figure 6 shows the relative error in the first-order approximation over the entire range, while Fig.
The exact formula can be difficult to use in practice, so we will use the following approximation.
where [??] is the mapping of kernel approximation. The original feature x can be mapped into the approximated Hilbert space by [??].