Skip to content

Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6610 • Analysis of Numerical Methods I

Fall 2008 • Question 3

Question

Math 6610 • Analysis of Numerical Methods I Fall 2008 • Question 3

Interpolation error Let \(I=[a,b]\) and \(f\in C^\infty [a,b]\). Let \(x_i\), \(i=0,\ldots ,n\) be \(n+1\) distinct points in \(I\). Let \(p(x)\) be the unique polynomial of degree at most \(n\) satisfying \(p(x_i)=f(x_i)\), \(i=0,\ldots ,n\). Given \(x\) in \([a,b]\), show that there exists some point \(\xi \in [a,b]\) such that \[ f(x)-p(x) = \frac {\prod _{i=0}^n(x-x_i)}{(n+1)!}\,f^{(n+1)}(\xi ). \tag {3} \]

Raw LaTeX

{Interpolation error} Let \(I=[a,b]\) and \(f\in C^\infty[a,b]\). Let
\(x_i\), \(i=0,\ldots,n\) be \(n+1\) distinct points in \(I\). Let \(p(x)\) be the unique
polynomial of degree at most \(n\) satisfying \(p(x_i)=f(x_i)\), \(i=0,\ldots,n\).
Given \(x\) in \([a,b]\), show that there exists some point \(\xi\in[a,b]\) such that
\[
  f(x)-p(x)
  =
  \frac{\prod_{i=0}^n(x-x_i)}{(n+1)!}\,f^{(n+1)}(\xi).
  \tag{3}
\]

Topic Tags

Linear Programming and Optimization Polynomial Interpolation and Error Formulas