The central difference formulae use an odd number of points  ,  and an even number of   equations.  Since we want to include the remainder terms, we need to use series expansions of order  .  The remainder term in these formulas all involve even powers of  h  and even and odd derivatives depending on the situation.  Also, the subroutine requires a replacement of the point where the remainder term is evaluated to be instead of  .  

Exploration

(c) John H. Mathews 2004