MAT360 Section Summary: 4.1b

More Numerical Differentiation

1. Summary

   Last time we looked at two- and three- point formulas. This time we want to go beyond those, to three and five point formulas.

2. Definitions
   • Other three-point formulas: Assume that h>0.
     • forward:

       

     • backward:

       

     Recall that the centered-difference formula is a three-point formula, with the coefficient of the term equal to zero, and whose error term is of opposite sign and about twice as good (i.e., half as much):

     

     This suggests that we might mix and match to create one of the following

   • five-point formulas
     • If we combine the backward and forward three-point formulas with four times the centered difference formula,

       

       then we might hope that these errors will essentially cancel, and we end up with

       

       Notice the exciting development: we went from an method to an method, dependent on the fifth derivative of f.

       Notice also that, although this is called a five-point method, only four points actually figure into the derivative calculations.

     • forward:

       

       This can be obtained using Taylor series and by carefully selecting the coefficients of the , so as to get cancellation up to the fifth derivative terms. Then again, assuming continuity of the fifth derivative we can use the Intermediate Value Theorem to arrive at the error term.

       

       We need a linear combination of these things that gives us

       

     • Obviously there's a corresponding backward formula where we merely replace the formula above by the one obtained by setting h to -h.
     These formulas are useful at the endpoints of data sets, where we don't have the neighboring points that we would need for a centered derivative approximation.

     Each is an exercise in linear algebra, actually, and not so terribly complicated. Here it is in lisp:

   • Higher order formulas: Higher order terms can be arrived at via the Taylor series expansions, too: for example, the approximation

     

     comes right of the Taylor series for

     

     and

     

     Again, here it is in lisp: the same procedure, with a few options:

3. Properties/Tricks/Hints/Etc.

   One interesting observation is that if an error term is dependent on the derivative term , then the approximation will be exact for polynomial functions of degree n-1. So, if you knew that a certain phenomenon would theoretically be modelled by a cubic function, then we can get the derivatives exactly right using position data and the appropriate form of the approximation to the derivatives (e.g. a five-point scheme).