Order of approximation

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.

Usage in science and engineering

In formal expressions, the ordinal number used before the word order refers to the highest power in the series expansion used in the approximation. The expressions: a zeroth-order approximation, a first-order approximation, a second-order approximation, and so forth are used as fixed phrases. The expression a zero-order approximation is also common. Cardinal numerals are occasionally used in expressions like an order-zero approximation, an order-one approximation, etc.

In the case of a smooth function, the nth-order approximation is a polynomial of 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 used in the expansion. This affects accuracy. The error usually varies within the interval. Thus the terms (zeroth, first, second, etc.) used above meaning do not directly give information about percent error or significant figures. For example, in the Taylor series expansion of the exponential function, $${\displaystyle e^{x}=\underbrace {1} _{0^{\text{th}}}+\underbrace {x} _{1^{\text{st}}}+\underbrace {\frac {x^{2}}{2!}} _{2^{\text{nd}}}+\underbrace {\frac {x^{3}}{3!}} _{3^{\text{rd}}}+\underbrace {\frac {x^{4}}{4!}} _{4^{\text{th}}}+\ldots \;,}$$ 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 ${\displaystyle |x|<1,}$ each higher order term is smaller than the previous. If ${\displaystyle |x|<<1,\,}$ 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.

Zeroth-order

A zeroth-order approximation of a function (that is, mathematically determining a formula to fit multiple data points) will be constant, or a flat line with no slope: 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 an approximate fit to the data, obtained by simply averaging the x values and the y values. However, data points represent results of measurements and they do differ from 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.

First-order

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:

${\displaystyle x=[0.0,1.0,2.0],}$

${\displaystyle y=[3.0,3.0,5.0],}$

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

is an approximate fit to the data.

Second-order

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:

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

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

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

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. Such a fit would still need to be treated with caution, given the small number of data points.

Higher-order

Continuing the above, a third-order approximation would be required to perfectly fit four data points, and so on. See polynomial interpolation. Higher-order approximations are less commonly used than low-order ones.

Colloquial usage

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. 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.

See also

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

References