Subject: Calculus

Orders Of Approximation

When we talk of quantity, we normally use approximations in dealing with them. When our teacher asks for our age, we say like 12 years old. When someone asks for our height, we say like 4 and a half feet or something. Such quantities often bears approximations because you don't want to say "I am 12 years, 4 months, 3 days, 8 minutes and 4 seconds old at this moment!!". Or else "I am 4.54323 feet tall". That's why approximations are important and just makes our life easier.

However, approximation begets accuracy. The more the approximation, the more the inaccuracy. In Science and Mathematics, we dealt with a lot of approximations but we keep a good account of it by making approximations of some quantities acceptable to some degree. Such degrees of approximations are referred to as orders of approximation. It is an informal term for how precise an approximation is.

Orders of approximation can be best explained by considering the sets of points below.

The points (4, 3), (2, 1), (-2, -1) above can be approximated to a function by mathematically determining a formula to fit the data points. Degrees of approximations are as follows:

Zeroth Order Approximation

In approximating the data sets above into a function, zero degree approximation is done by just taking the average of the y values. Thus, y \approx f(x) \approx 1. Notice that the approximation is just a constant. In fact, all zeroth order approximations are either a constant, a flat line (see figure below) with zero slope which are polynomials with zero order, hence the name zero order approximation.

Zero order approximations are often called as an "order of magnitude" approximation and "first educated guess". These kinds of approximation gives rough estimates of quantity that's why errors can never be disassociated with it as can be seen from the graph above.

First Order Approximation

First Order approximation is better than the previous one. Here, we use linear regression or linear fitting formula to take the fit of the data sets. As you notice in the graph below, the fitting line is more closer to the data points compared above.

Approximating data points using first order approximation is done by fitting the points through a line with a slope, which is just a polynomial of degree 1.

Second Order Approximation

"Decent quality answer", this is the term used by scientist referring to the second order approximation. A second-order approximation is a parabola: a polynomial of degree 2, hence the name. This is more accurate than the first order approximation as seen below. The curve fit is much closer to the data points.

Higher Order Approximation

While accuracy is more apparent as the order of approximation is getting higher, the degree of relevance is not anymore important. Imagine writing the value of pi to the 50^{th} decimal places!. That's why higher order approximations are not taken into practice most of the time because some of its terms are often neglected.

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