Mechanical Engineering Community Mechanics Corner

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Mechanics Corner
A Journal of Applied Mechanics & Mathematics by DrD, #33

The use of polynomials to fit engineering data is a common engineering practice. In school, we learn that "A data set consisting of n data points ((x_{i},y_{i}), i=1,2,3,…n) can be exactly fitted with a polynomial of degree n-1. Thus three data points can be fitted exactly with a quadratic expression, four data points can be fitted exactly with a cubic expression, and so on. If this approach is pursued much further, something ugly appears: while a polynomial of degree n-1 will pass exactly through n data points, for large values of n, it will oscillate wildly in between the data points. Since one of the most common reason for using a polynomial fit in the first place is for interpolation -- to be able to estimate a function value at locations between the known data points -- this wild oscillation is devastating. It is at this point that least squares fitting is usually introduced to give an approximate fit using a much lower order polynomial. A different approach is employed here.

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