Order of approximation: Difference between revisions

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{{Short description|Expressions for approximation accuracy}}

{{original research|date=November 2024}}

{{Order-of-approx}}

{{unclear|date=March 2016}}

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==

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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 numbersterms (''zeroth'', ''first'', ''second,'' etc.) used formally in the above meaning do not directly give information about [[percent error]] or [[significant figures]]. For example, in the [[w:Taylor's theorem|Taylor series]] expansion of the [[Exponential_function#Formal_definition|exponential function]],

<math display="block">e^x=\underbrace{1}_{0^\text{th}}+\underbrace{x}_{1^\text{st}}+\underbrace{\frac{x^2}{2!}}_{2^\text{nd}}+\underbrace{\frac{x^3}{3!}}_{3^\text{rd}} + \underbrace{\frac{x^4}{4!}}_{4^\text{th}} + \ldots\;, </math>

the zeroth-order term is <math>1;</math> the first-order term is <math>x,</math> second-order is <math>x^2/2,</math> and so forth. If <math>|x|<1,</math> each higher order term is smaller than the previous. If <math>|x|<<1,\,</math> then the first order approximation,

<math display="block">e^x\approx 1+x, </math>

is often sufficient. But at <math>x=1,</math> the first-order term, <math>x,</math> is not smaller than the zeroth-order term, <math>1.</math> And at <math>x=2,</math> even the second-order term, <math>2^3/3!=4/3,\,</math> is greater than the zeroth-order term.

=== 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,

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: <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]] isare useful and helpshelp predict an [[Closed-form expression|analytic solutionsolutions]], but the approximationapproximations alone doesdo not provide conclusive evidence.

===First-order===

''First-order approximation'' is the term scientists use for a slightly better answer.<ref name=":0" /> Some simplifying assumptions are made, and when a number is needed, an answer with only one significant figure is often given ("the town has {{val|4|e=3}}, or ''four thousand'', residents"). In the case of a first-order approximation, at least one number given is exact. In the zeroth-order example above, the quantity "a few" was given, but in the first-order example, the number "4" is given.

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&nbsp;1. For example:

: <math>x = [0.00, 1.00, 2.00],</math>

: <math>y = [3.00, 3.00, 5.00],</math>

: <math>y \sim f(x) = x + 2.67</math>

is an approximate fit to the data.

In this example there is a zeroth-order approximation that is the same as the first-order, but the method of getting there is different; i.e. a wild stab in the dark at a relationship happened to be as good as an "educated guess".

===Second-order===

''Second-order approximation'' 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 {{val|3.9|e=3}}, or ''thirty-nine hundred'', residents") is generally given. In [[mathematical finance]], second-order approximations are known as [[Convexity (finance)|convexity corrections]]. As in the examples above, the term "2nd order" refers to the number of exact numerals given for the imprecise quantity. In this case, "3" and "9" are given as the two successive levels of precision, instead of simply the "4" from the first order, or "a few" from the zeroth order found in the examples above.

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&nbsp;2. For example:

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* [[Chapman–Enskog_theory#Mathematical_Formulation | Chapman–Enskog method]]

* [[Big O notation]]

* [[Order of accuracy]]

==References==