Zeroth-order

Zeroth-order approximation is the term scientists use for a first rough 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,

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

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

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

could be – if data point accuracy were reported – an approximate fit to the data, obtained by simply averaging 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.

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

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

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

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

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 the 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 πx. Taylor series is useful and helps predict an analytic solution, but the approximation alone does not provide conclusive evidence.

First-order

First-order approximation is the term scientists use for a slightly better answer.^[3] 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. 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 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".

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 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:

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

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

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

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").

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.