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Fit approximation
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Orders of approximation Scale analysis Big O notation Curve fitting False precision Significant figures
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Approximation Generalization error Taylor polynomial Scientific modelling
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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 in terms of the number of parameters used to construct the approximation. This article focuses on the approximation of smooth real-valued functions of one variable - the notion extends to functions between other spaces, such as between Euclidean spaces of varying dimension.

In formal expressions, theordinal number used before the wordorder refers to the highestpower in theseries expansion used in theapproximation. The expressions: azeroth-order approximation, afirst-order approximation, asecond-order approximation, and so forth are used asfixed phrases. The expression azero-order approximation is also common.Cardinal numerals are occasionally used in expressions like anorder-zero approximation, anorder-one approximation, etc.

In the case of asmooth function, thenth-order approximation is apolynomial ofdegreen, which is obtained by truncating the Taylor series to this degree. The formal usage oforder of approximation corresponds to the omission of some terms of theseries used in theexpansion. This affectsaccuracy. The error usually varies within the interval. Thus the terms (zeroth,first,second, etc.) used above meaning do not directly give information aboutpercent error orsignificant figures. For example, in theTaylor series expansion of theexponential function,$${\displaystyle e^x=\underset{0^{\text{th}}}{\underbrace{1}}+\underset{1^{\text{st}}}{\underbrace{x}}+\underset{2^{\text{nd}}}{\underbrace{\frac{x^2}{2!}}}+\underset{3^{\text{rd}}}{\underbrace{\frac{x^3}{3!}}}+\underset{4^{\text{th}}}{\underbrace{\frac{x^4}{4!}}}+\dots ,}$$the zeroth-order term is${\displaystyle 1;}$ the first-order term is${\displaystyle x,}$ second-order is${\displaystyle x^2/2,}$ and so forth. If each higher order term is smaller than the previous. If then the first order approximation,$${\displaystyle e^x\approx 1+x,}$$is often sufficient. But at${\displaystyle x=1,}$ the first-order term,${\displaystyle x,}$ is not smaller than the zeroth-order term,${\displaystyle 1.}$ And at${\displaystyle x=2,}$ even the second-order term,${\displaystyle 2^3/3!=4/3,}$ is greater than the zeroth-order term.

A zeroth-order approximation of afunction (that is,mathematically determining aformula to fit multipledata points) will beconstant, or a flatline with noslope: a polynomial of degree 0. For example,

${\displaystyle x=[0,1,2],}$

${\displaystyle y=[3,3,5],}$

${\displaystyle y\sim f(x)=3.67}$

could be – if data point accuracy were reported – an approximate fit to the data, obtained by simply averaging thex values and they values. However, data points representresults of measurements and they do differ frompoints 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 offalse precision. With the implied accuracy of the data points of ±0.5, the zeroth order approximation could at best yield the result fory of ~3.7 ± 2.0 in the interval ofx from −0.5 to 2.5, considering thestandard deviation.

If the data points are reported as

${\displaystyle x=[0.00,1.00,2.00],}$

${\displaystyle y=[3.00,3.00,5.00],}$

the zeroth-order approximation results in

${\displaystyle y\sim f(x)=\mathrm{3.67.}}$

The accuracy of the result justifies an attempt to derive a multiplicative function for that average, for example,

${\displaystyle y\sim x+\mathrm{2.67.}}$

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 theinterval, which may be a large part of it. This means thaty 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 exampley = sin πx.Taylor series are useful and help predictanalytic solutions, but the approximations alone do not provide conclusive evidence.

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:

${\displaystyle x=[0.00,1.00,2.00],}$

${\displaystyle y=[3.00,3.00,5.00],}$

${\displaystyle y\sim f(x)=x+2.67}$

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".

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

${\displaystyle x=[0.00,1.00,2.00],}$

${\displaystyle y=[3.00,3.00,5.00],}$

${\displaystyle y\sim f(x)=x^2-x+3}$

is an approximate fit to the data. In this case, with only three data points, a parabola is an exact fit based on the data provided. However, the data points for most of the interval are not available, which advises caution (see "zeroth 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. Seepolynomial interpolation.

These terms are also usedcolloquially 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. The terminology, in this context, represents a high level of precision required to account for an effect which is inferred to be very small when compared to the overall subject matter. The higher the order, the more precision is required to measure the effect, and therefore the smallness of the effect in comparison to the overall measurement.

• Linearization
• Perturbation theory
• Taylor series
• Chapman–Enskog method
• Big O notation
• Order of accuracy

• Perturbation theory
• Numerical analysis

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