Interpolation for Derivatives and Integrals

The idea to use interpolation methods to evaluate definite integrals and to compute derivatives is simple: integrate or differentiate the interpolating polynomial. On a picture we can immediately see that, while integration in this way is numerically sound, the straightforward way to perform numerical differentiation based on interpolation does not work because of the oscillations in the interpolating polynomial.

In this lecture we showed with difference operators that that the divided differences used in Newton interpolation can be interpreted as approximations for the derivatives at the first point.

In particular, if we interpolate at equidistant points x0, x0+h, x0+2*h, .. ,x0+n*h, then we have that 


   f[x0,x0+h,x0+2*h,..,x0+n*h]*n!

                n
           Delta f(x0)
       = ----------------
                 n
                h 

              n
       = +/- D f(x0)
where D is the usual derivative operator and where the operator Delta is defined by 
         Delta f(x0) = f(x0+h) - f(x0).
So when we performed Newton interpolation, we were already computing numerical approximations for the derivatives.