Order of approximation: Difference between revisions

Revision as of 07:10, 9 April 2023

Revision as of 07:47, 25 June 2023

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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 (mathematics)|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 π''x''. [[Taylor series]] is useful and helps predict an [[Closed-form expression|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.<ref name=":0" /> Some simplifying assumptions are made, and when a number is needed, an answer with only one significant figure is often given ("the town has {{val|4|e=3}}, 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&nbsp;1. For example:

: <math>x = [0.00, 1.00, 2.00],</math>

: <math>y = [3.00, 3.00, 5.00],</math>

: <math>y \sim f(x) = x + 2.67</math>

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===