1

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.

eChung00
  • 2,963
  • 8
  • 29
  • 42

1 Answers1

2

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.

:)

Novice
  • 719
  • 1
    So in my question, the degree $n=4$ since there are 4 points, and the error term would be $f(x)-p_3(x) = \frac{f^{(4)}(x)}{4!}\Pi_{i=0}^3(x-x_i)$?? – eChung00 Oct 20 '13 at 18:20
  • Yes. If you apply the formula directly with n=3. Note that it is $f^{4}(a)$, where $a$ is in the interior of the set of x values that you interpolate. You can differentiate the given $f$ four times, apply Extreme Value Theorem if necessary (for your $f$, just choose the largest x value in the interval that you are given), then compute the upperbound.:) – Novice Oct 20 '13 at 18:25
  • Normally, you cannot find the $a$ explicitly, but its existence is given by Rolle's theorem. – Novice Oct 20 '13 at 18:25
  • I have one more question. In my problem, is degree 4 or 3?? I thought since there are 4 points, the degree would be 4 and we can find a unique interpolant with degree 3. I am confused with the first sentence in your comment where it says "...directly with n=3." thanks – eChung00 Oct 20 '13 at 18:31
  • You have four points. The max polynomial degree that you can come out with is 3 (since there are four unknown coefficients in a degree 3 polynomial). The website says "a polynomial of degree at most n that interpolates f at n + 1 distinct points". Therefore, for your case, n=3.:) – Novice Oct 20 '13 at 18:36
  • thanks a lot!! It was great help! – eChung00 Oct 21 '13 at 23:46