Module for Numerical Differentiation, Part II

Check out the new Numerical Analysis Projects page.

Background. Numerical differentiation formulas can be derived by first constructing the Lagrange interpolating polynomial   through three points, differentiating the Lagrange polynomial, and finally evaluating    at the desired point.  The truncation error is be investigated, but round off error from computer arithmetic using computer numbers will be studied in another lab.

Three point rule for .  The centered formula for  the first derivative, based on three points is  

    ,  

and the error bound is    where  .  

Five point rule for .  The centered formula for  the first derivative, based on five points is  

    .  
    

Big "O" error term for .  

    .  

Richardson's Extrapolation.  Richardson's extrapolation relates the five point rule and the three point rule,  ,  that was studied previously.  

    .  

Enter the three point formula for numerical differentiation.

Enter the function, use  .

Use the results of Example 2 in the previous module to construct the error bound .  

Animations (Numerical Differentiation  Numerical Differentiation).  Internet hyperlinks to animations.

Project III.  Investigate the numerical differentiation formulae    and error bound    where  .   
The truncation error is be investigated, but round off error from computer arithmetic using computer numbers will be studied in another lab.

Enter the five point formula for numerical differentiation.

Enter the function, use  .

Example 1.  Find the formula for the  .  
Use  .

Solution 1.

Example 2.  Graph  .  Find the bound  .   Look at a graph and estimate the value  , be sure to take the absolute value if necessary.

Solution 2.

Example 3 (a).  Compute numerical approximations for the derivatives , using step sizes .  
3 (b).  Plot the numerical approximation over the interval .  Compare it with the graph of over the interval .  

Solution 3.

Example 4.  Plot the absolute error    over the interval  , and estimate the maximum absolute error over the interval.
4 (a).  Compute the error bound    and observe that    over  .  

Solution 4.

Example 5.  Investigate the behavior of  .  If the step size is reduced by a factor of    then the error bound is reduced by  .  This is the    behavior.

Solution 5.

Example 6.  Compare the error bounds for the three point and five point formulas.
6 (a).  Which is smaller ?  Explain your answer.  
6 (b).  Which is smaller ?   Explain your answer.  

Solution 6.

Project IV.  Investigate Richardson's extrapolation for numerical differentiation.

Example 7.  In general, show that  .  

Solution 7.

Enter the function, use  .

Example 8.  Find the approximations  ,   and then use the extrapolation formula  .  

Compute    directly.  Finally, compare these numerical approximations for the derivative with  

Solution 8.

Old Lab Project (Numerical Differentiation  Numerical Differentiation).  

Internet hyperlinks to an old lab project.  

Research Experience for Undergraduates

Numerical Differentiation  Numerical Differentiation  

Internet hyperlinks to web sites and a bibliography of articles.  

Richardson Extrapolation  Richardson Extrapolation  

Internet hyperlinks to web sites and a bibliography of articles.  
  

Downloads (Richardson's Extrapolation Richardson's Extrapolation).  

Download this Mathematica notebook.  

(c) John H. Mathews 2003