Let the function $f(x) = \ln(x)$ be approximated by an interpoation polynomial of degree of 9 with 10 nodes uniformly distributed in the interval $[1,2]$. What bound can be placed on the error?
I've been trying to think over the problem for the past day, but i'm still not sure sure how I am supposed to do this problem.
On http://en.wikipedia.org/wiki/Polynomial_interpolation#Interpolation_error I find the error expression of $$\frac{f^{(n+1)}(\xi)}{(n+1)!}\prod_{i=0}^n{(x-x_i)}$$ In this case $f(x)=\ln x$ so $$f^{(n+1)}(x)=n! (-1)^{n}x^{-(n+1)}$$ We have $$|f^{(n+1)}(x)|<n!$$ and $$\prod_{i=0}^n{(x-x_i)}<1$$ So in this case an error bound is just $$\frac{n!}{(n+1)!}={1\over 10}$$