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The theory for finding the nth-order derivatives is the same as that for finding simple derivatives: Given a sequence of equally spaced points (either around or preceding a given point x0), find the interpolating polynomial of the appropriate degree, differentiate the interpolating polynomial n times (instead of once) and evaluate that interpolating polynomial at the point.
For the sake of completeness, we will list the centred and backward divided-difference formulae here. In each case, the format of Richardson extrapolation we found could be used to improve the approximations for the O(h2) formulae on this page.
Centred Divided-Difference Formulae
For each derivative, centred divided-difference formulae are O(h2) and O(h4), respectively.
Second Derivatives (3 and 5 point interpolations)
Third Derivatives (5 and 7 point interpolations)
Fourth Derivatives (5 and 7 point interpolations)
Fifth Derivatives (7 and 9 point interpolations)
Backward Divided-Difference Formulae
For each derivative, backward divided-difference formulae are O(h) and O(h2), respectively. Note that in all cases, the O(h) formula is equivalent to the corresponding O(h2) centred divided-difference formula, only shifted by h.
Second Derivatives (3 and 4 point interpolations)
Third Derivatives (4 and 5 point interpolations)
Fourth Derivatives (5 and 6 point interpolations)
Fifth Derivatives (6 and 7 point interpolations)
Copyright ©2005 by Douglas Wilhelm Harder. All rights reserved.
