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In [[science]], [[engineering]], and other quantitative disciplines, '''order of approximation''' refers to formal or informal expressions for how accurate an [[approximation]] is.
==Usage in science and engineering==
In formal expressions, the [[English_numerals#Ordinal_numbers|ordinal number]] used before the word [[Order_(mathematics)#Analysis|order]] refers to the highest [[Power function|power]] in the [[series expansion]] used in the [[Approximation#Etymology_and_usage|approximation]]. The expressions: a '''''zeroth-order''' approximation'', a '''''first-order''' approximation'', a '''''second-order''' approximation'', and so forth are used as [[fixed phrase]]s. The expression a ''zero-order approximation'' is also common. [[Cardinal numeral]]s are occasionally used in expressions like an ''order-zero approximation'', an ''order-one approximation'', etc.
The omission of the word ''order'' leads to [[phrase]]s that have less formal meaning. Phrases like '''first approximation''' or '''to a first approximation''' may refer to ''a roughly approximate value of a quantity''.<ref>''first approximation'' in Webster's Third New International Dictionary, Könemann, {{ISBN|3-8290-5292-8}}.</ref><ref>[http://www.webster-dictionary.org/definition/to%20a%20first%20approximation ''to a first approximation''] in Online Dictionary and Translations Webster-dictionary.org.</ref> The phrase '''to a zeroth approximation''' indicates ''a wild guess''.<ref name=":0">[http://www.webster-dictionary.org/definition/to%20a%20zeroth%20approximation ''to a zeroth approximation''] in Online Dictionary and Translations Webster-dictionary.org.</ref> The expression ''order of approximation'' is sometimes informally used to mean the number of [[significant figure]]s, in increasing order of accuracy, or to the [[order of magnitude]]. However, this may be confusing, as these formal expressions do not directly refer to the order of derivatives.
The choice of series expansion depends on the [[scientific method]] used to investigate a [[Phenomenon#Scientific|phenomenon]]. The expression '''order of approximation''' is expected to indicate progressively more refined approximations of a [[Function_(mathematics)|function]] in a specified [[Interval_(mathematics)|interval]]. The choice of order of approximation depends on the [[Research|research purpose]]. One may wish to simplify a known [[Closed-form_expression#Analytic_expression|analytic expression]] to devise a new application or, on the contrary, try to [[Curve_fitting|fit a curve to data points]]. Higher order of approximation is not always more useful than the lower one. For example, if a quantity is constant within the whole interval, approximating it with a second-order [[Taylor series]] will not increase the accuracy.
In the case of a [[smooth function]], the ''n''th-order approximation is a [[polynomial]] of [[degree of a polynomial|degree]] ''n'', which is obtained by truncating the Taylor series to this degree. The formal usage of ''order of approximation'' corresponds to the omission of some terms of the [[Series_(mathematics)|series]] used in the [[Series_expansion|expansion]] (usually the higher terms). This affects [[Accuracy_and_precision|accuracy]]. The error usually varies within the interval. Thus the numbers ''zeroth'', ''first'', ''second'' etc. used formally in the above meaning do not directly give information about [[percent error]] or [[significant figures]].
=== Zeroth-order ===
''Zeroth-order approximation'' is the term [[scientist]]s use for a first rough answer. Many [[Approximation#Science|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. The zero of "zeroth-order" represents the fact that even the only number given, "a few", is itself loosely defined.
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>
could be – if data point accuracy were reported – an approximate fit to the data, obtained by simply averaging the ''x'' values and the ''y'' values. However, data points represent [[Unit_of_observation#Data_point|results of measurements]] and they do differ from [[Point_(geometry)#Points_in_Euclidean_geometry|points in Euclidean geometry]]. Thus quoting an average value containing three significant digits in the output with just one significant digit in the input data could be recognized as an example of [[false precision]]. With the implied accuracy of the data points of ±0.5, the zeroth order approximation could at best yield the result for ''y'' of ~3.7 ± 2.0 in the interval of ''x'' from −0.5 to 2.5, considering the [[standard deviation]].
If the data points are reported as
: <math>x = [0.00, 1.00, 2.00],</math>
: <math>y = [3.00, 3.00, 5.00],</math>
the zeroth-order approximation results in
: <math>y \sim f(x) = 3.67.</math>
The accuracy of the result justifies an attempt to derive a multiplicative function for that average, for example,
: <math>y \sim x + 2.67.</math>
One should be careful though, because the multiplicative function will be defined for the whole interval. If only three data points are available, one has no knowledge about the rest of the [[Interval (mathematics)|interval]], which may be a large part of it. This means that ''y'' could have another component which equals 0 at the ends and in the middle of the interval. A number of functions having this property are known, for example ''y'' = sin π''x''. [[Taylor series]] is useful and helps predict an [[Closed-form expression|analytic solution]], but the approximation alone does not provide conclusive evidence.
===Higher-order==
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