Recently, I have been reading about the Runge Phenomenon .
The Runge Phenomenon describes a situation in which a polynomial is used to approximate another function using a fixed number of evenly spaced points - however, the polynomial being used to approximate this function can be shown to exhibit oscillatory behavior towards the end points of the range over which the function being approximated on (i.e. higher error):
The above picture demonstrates an example of the Runge Phenomenon - the "red colored" function is approximated using a "$5$th order polynomial" ("blue") and a "$9$th order polynomial" ("green") : as we can see, the quality of the approximation for both the "$5$th order polynomial" and the "$9$th order polynomial" begin to decrease in quality (i.e. higher error) when examined at interpolation points towards the extremities
My Question:. As we know, a "$n$-th order polynomial" is a polynomial function where the highest exponential term can not be greater than "$n$" - but in the picture above, could we determine the mathematical equations of the "blue" and "green" curves?
For instance, suppose we were given :
- The equation of the "red" function: $y = \frac{1}{1+25x^2}$
- "$n$" (e.g. $n = 20$) evenly spaced points between $-1$ and $1$ (e.g. $-1, -0.8, -0.6, \dots 0.8, 1.0$) corresponding to "$x$"
- The value of the "red" function (i.e. $y$ values) at each of these "$n$" points
Given the above information, how would we determine the equation of the "blue" and "green" functions? How exactly are we interpolating? Essentially, (to obtain the equations of the "blue" and "green" functions) are we fitting polynomial regression models to these points?
Can someone please comment on this?
Thanks!
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