I have a question on one interpolation problem. The problem is below.
For the given points, $x_0 = -1, x_1 = 0, x_2 = 3$ and $x_3 = 4,$ find the error term $e_3(\bar{x}) = f(\bar{x}) – p_3(\bar{x})$ for cubic interpolation of $f(x)=x^5 -5x^4.$ Give an upper bound on the absolute value of interpolating error $|e_3(\bar{x})|$ at $\bar{x} = 2.$
I found the interpolant $p_3(x) = -6+6(x+1)-15x(x+1)+x(x+1)(x-3).$
For the error term, do I have to subtract the interpolant from the original function?? How do I find the upper bound??
Thanks.
Best Answer
The error term is related to the article in wiki:
http://en.wikipedia.org/wiki/Polynomial_interpolation#Interpolation_error
at the section "Interpolation error"
Since you are given the original function and the closed interval containing the points that you interpolate the function with, then you can apply the formula in wiki directly to calculate an upperbound for the error at x=2.
:)