Orders of approximation

Often in science, engineering, or other quantitative disciplines, it is necessary to make approximations with various degrees of precision. These approximations can be classified based on the order of magnitude of the rounding error involved. It is an application of the concepts in big O notation.

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Zeroth order approximation

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.

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. For example,

[itex]x=[0,1,4][itex]
[itex]y=[0,1,2][itex]
[itex]y\sim f(x)=1.67[itex]

is an approximate fit to the data.

First order approximation

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 4,000 residents").

A first order approximation of a function (that is, mathematically determining a formula to fit multiple data points) will be a straight line with a slope. For example,

[itex]x=[0,1,4][itex]
[itex]y=[0,1,2][itex]
[itex]y\sim f(x)=1.5*x[itex]

is an approximate fit to the data.

Second order approximation

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 3,900 residents") is generally given.

A second order approximation of a function (that is, mathematically determining a formula to fit multiple data points) will be a parabola. For example,

[itex]x=[0,1,2][itex]
[itex]y=[3,3,5][itex]
[itex]y\sim f(x)=x^2-x+3[itex]

is an approximate fit to the data. In this case, with only three data points, a parabola is an exact fit.

Higher order approximations

While higher order approximations exist and are crucial to a better understanding and description of reality, they are not typically referred to by number.

A third order approximation would be required to fit four data points, and so on.

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