1 Numerical Differentiation 1.1. Methods and Implementation Write a subroutine that takes in a grid spacing h and a polynomial degree n and returns the n + 1 numbers that correspond to the symmetric finite difference stencil/filter for the operator . For n = 2,4,6,8 demonstrate that the filter you have constructed gives the 'exact result (to within numerical precision) for polynomials of degree n independent of h. Try this for two different polynomials in each case. Try a polynomial of degree n + 1 for each filter. What do you notice? Can you explain this result? 1.2 Analysis For functions that are not exactly representable by sufficiently low order functions the convergence will depend upon h. Investigate the convergence of the filters you have constructed for n = 2,4,6,8 for two such functions. Plot the convergence of with respect to 1/h for each filter. Show that the convergence is what you would expect