Order of approximation

Fit approximation
Concepts
	Orders of approximation; Scale analysis · Big O notation; Curve fitting · False precision; Significant figures
Other fundamentals
	Approximation · Generalization error; Taylor polynomial; Scientific modelling

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. Informally, it is simply the level of precision used to represent quantities which are not perfectly known.

Formally, an nth-order approximation is one where the order of magnitude of the error is at most on the order of $x^{n+1}$ , or in terms of big O notation, the error is $O(x^{{n+1}}).$ In the case of a smooth function, the nth-order Taylor approximation is a polynomial of degree n, which is obtained by truncating the Taylor series of the function to this degree.

Usage in science and engineering

Zeroth-order

Zeroth-order approximation (also 0th order) is the term scientists use for a first educated guess at an answer. Many simplifying assumptions are made, and when a number is needed, an order-of-magnitude answer (or zero significant figures) 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 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 (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,

x=[0,1,2]\,

y=[3,3,5]\,

y\sim f(x)=3.67\,

is an approximate fit to the data, obtained by simply averaging the x-values and the y-values then deriving a multiplicative function for that average. Other methods for selecting a constant approximation can be used, such as taking the average of x-values and y-values and then working out the average difference between them:

y\sim \ x+2.67

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 4×10³ or four thousand residents"). In the case of a first-order approximation, at least one number given is exact. IE: In the zeroth order above, the quantity "a few" was given but in the first order example, the number "4" was 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 1. For example,

x=[0,1,2]\,

y=[3,3,5]\,

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'.

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 3.9×10³ or thirty nine hundred residents") is generally given. In mathematical finance, second-order approximations are known as 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 2. For example,

x=[0,1,2]\,

y=[3,3,5]\,

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.

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. 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.

Fit approximation

Concepts
Orders of approximation Scale analysis · Big O notation Curve fitting · False precision Significant figures
Other fundamentals
Approximation · Generalization error Taylor polynomial Scientific modelling