Richardson Extrapolation

  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.

Theorem  (Three point rule for ).  The central difference formula for  the first derivative, based on three points is  

    ,  

and the remainder term is

    .

Together they make the equation  ,  and the truncation error bound is  

      

where  .  This gives rise to the Big "O" notation for the error term for  :

    .   

Theorem  (Five point rule for ).  The central difference formula for  the first derivative, based on five points is  

    ,  

and the remainder term is

    .

Together they make the equation  ,  and the truncation error bound is  

      

where  .  This gives rise to the Big "O" notation for the error term for  :

    .  

Theorem  (Richardson's Extrapolation for ).  The central difference formula for  the first derivative, based on five points is a linear combination of   and .

    ,  

where    and  .

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.

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.

Example  Consider the function  .   Find the formula for the third derivative , it will be used in our explorations for the remainder term and the truncation error bound.  Graph  .  Find the bound  .  Look at it's graph and estimate the value  , be sure to take the absolute value if necessary.
Solution

Project IV.  

    Investigate Richardson's extrapolation for numerical differentiation.

Example   In general, show that  .  
Solution