I'm studying for a test, and this is one of the practice problems:
I have the answer, but can't make much sense of it:What straight line is closest to x^3 over the interval −1 ≤ x ≤ 1?
I figure this has something to do with least squares, since you're trying to fit a line to a curve, but other than that I'm lost.Solution: The answer is:
((x^3, 1)/(1, 1)) · 1 + ((x^3, x)/(x, x)) · x = ((1/2)/(2/3))x .
Here, (f,g) = Integral from -1 to 1 of (f·g dx).
Could someone please provide a detailed explanation of the above solution? It would highly benefit my understanding of the material. Thank you!
If 'closest' is in the 'least square sense' , then the coefficients of 'best polynomial' of order n approximating y(x) in [-1,1] minimize the quantity...
(1)
... where is the Legendre Polynomial of order k. The minimizing (1) are found with standard procedure to be...
(2)
In Your case is , , and , so that...
(3)
Kind regards