Editing Polynomial interpolation (section)

===Equally spaced intervals===
In the case of equally spaced interpolation nodes where <math>x_i = a + ih</math>, for <math>i=0,1,\ldots,n,</math> and where <math>h = (b-a)/n,</math> the product term in the interpolation error formula can be bound as<ref>{{cite web| url=https://www-users.cselabs.umn.edu/Spring-2021/csci5302/Notes/Classnotes/interp.pdf| title=Notes on Polynomial Interpolation}}</ref>
<math display="block">\left|\prod_{i=0}^n (x-x_i)\right| = \prod_{i=0}^n \left|x-x_i\right| \leq \frac{n!}{4} h^{n+1}.</math>

Thus the error bound can be given as
<math display="block"> \left|R_n(x)\right| \leq \frac{h^{n+1}}{4(n+1)} \max_{\xi\in[a,b]} \left|f^{(n+1)}(\xi)\right| </math>

However, this assumes that <math>f^{(n+1)}(\xi)</math> is dominated by <math>h^{n+1}</math>, i.e. <math>f^{(n+1)}(\xi) h^{n+1} \ll 1</math>. In several cases, this is not true and the error actually increases as {{math|''n'' → ∞}} (see [[Runge's phenomenon]]). That question is treated in the section [[#Convergence properties|''Convergence properties'']].