en>BethNaught: clean up, typo(s) fixed: a interval → an interval using AWB

2015-01-11T15:44:00Z

clean up, typo(s) fixed: a interval → an interval using AWB

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en>Spinningspark: Reverted edits by 188.154.99.222 (talk) to last version by Foobarnix

2014-02-04T16:02:33Z

Reverted edits by 188.154.99.222 (talk) to last version by Foobarnix

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2014-01-19T10:09:21Z

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← Older revision		Revision as of 12:09, 19 January 2014
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			In [[science]], [[engineering]], and other quantitative disciplines, '''orders of approximation''' refer to formal or informal terms for how precise an [[approximation]] is, and to indicate progressively more refined approximations: in increasing order of precision, a '''zeroth-order''' approximation, a '''first-order''' approximation, a '''second-order''' approximation, and so forth.

			Formally, an ''n''th-order approximation is one where the [[order of magnitude]] of the error is at most <math>x^{n+1}</math>, or in terms of [[big O notation]], the error is <math>O(x^{n+1}).</math>
			In suitable circumstances, approximating a function by a [[Taylor polynomial]] of degree ''n'' yields an ''n''th-order approximation, by [[Taylor's theorem]]: a first-order approximation is a [[linear approximation]], and so forth.

			The term is also used more loosely, as detailed below.

			==Usage in science and engineering==

			===Zeroth-order ===
			''Zeroth-order approximation'' (also 0th order) is the term [[scientist]]s use for a first [[Approximation\|educated guess]] at an answer. Many simplifying assumptions are made, and when a number is needed, an order-of-magnitude answer (or zero [[significant figure]]s) is often given. For example, you might say "the town has '''a few thousand''' residents", when it has 3,914 people in actuality. This is also sometimes referred to as an [[order of magnitude\|order-of-magnitude]] approximation.

			A zeroth-order approximation of a [[function (mathematics)\|function]] (that is, [[mathematics\|mathematically]] determining a [[formula]] to fit multiple [[data point]]s) will be [[Constant (mathematics)\|constant]], or a flat [[line (mathematics)\|line]] with no [[slope]]: a polynomial of degree 0. For example,

			:<math>x=[0,1,2]\,</math>
			:<math>y=[3,3,5]\,</math>
			:<math>y\sim f(x)=3.67\,</math>

			is an approximate fit to the data, obtained by simply averaging the y-values. Other methods for selecting a constant approximation can be used.

			===First-order===
			''First-order approximation'' (also 1st order) is the term scientists use for a further educated guess at an answer. Some simplifying assumptions are made, and when a number is needed, an answer with only one significant figure is often given ("the town has 4×10<sup>3</sup> or '''four thousand''' residents").

			A first-order approximation of a function (that is, mathematically determining a formula to fit multiple data points) will be a [[linear approximation]], straight line with a slope: a polynomial of degree 1. For example,

			:<math>x=[0,1,2]\,</math>
			:<math>y=[3,3,5]\,</math>
			:<math>y\sim f(x)=x+2.67\,</math>

			is an approximate fit to the data.

			===Second-order===
			''Second-order approximation'' (also 2nd order) is the term scientists use for a decent-quality answer. Few simplifying assumptions are made, and when a number is needed, an answer with two or more significant figures ("the town has 3.9×10<sup>3</sup> or '''thirty nine hundred''' residents") is generally given. In [[mathematical finance]], second-order approximations are known as [[convexity correction]]s.

			A second-order approximation of a function (that is, mathematically determining a formula to fit multiple data points) will be a [[quadratic polynomial]], geometrically, a [[parabola]]: a polynomial of degree 2. For example,

			:<math>x=[0,1,2]\,</math>
			:<math>y=[3,3,5]\,</math>
			:<math>y\sim f(x)=x^2-x+3\,</math>

			is an approximate fit to the data. In this case, with only three data points, a parabola is an exact fit.

			===Higher-order===
			While higher-order approximations exist and are crucial to a better understanding and description of reality, they are not typically referred to by number.

			Continuing the above, a third-order approximation would be required to perfectly fit four data points, and so on. See [[polynomial interpolation]].

			These terms are also used colloquially by scientists and engineers to describe phenomena that can be neglected as not significant (e.g. "Of course the rotation of the Earth affects our experiment, but it's such a high-order effect that we wouldn't be able to measure it" or "At these velocities, relativity is a fourth-order effect that we only worry about at the annual calibration.") In this usage, the ordinality of the approximation is not exact, but is used to emphasize its insignificance; the higher the number used, the less important the effect.

			== See also ==
			* [[Linearization]]
			* [[Perturbation theory]]
			* [[Taylor series]]

			[[Category:Perturbation theory]]
			[[Category:Numerical analysis]]

en>Pragmaknowledge at 16:15, 9 August 2012

2012-08-09T16:15:05Z

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Improper integral - Revision history

en>BethNaught: clean up, typo(s) fixed: a interval → an interval using AWB

en>Spinningspark: Reverted edits by 188.154.99.222 (talk) to last version by Foobarnix

188.154.99.222: /* External links */

en>Pragmaknowledge at 16:15, 9 August 2012