Numerical Differentiation

Background.

Numerical differentiation formulas 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.  In this module the truncation error will be investigated, but round off error from computer arithmetic using computer numbers will be studied in another module.

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  (Three point rule for ).  The central difference formula for the second 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 or the error term for  :

     .  

Project I.  

    Investigate the numerical differentiation formula    and truncation error bound    where  .   The truncation error is investigated.  The round off error from computer arithmetic using computer numbers will be studied in another module.

Enter the three point formula for numerical differentiation.

Aside.  From a mathematical standpoint, we expect that the limit of the difference quotient is the derivative. Such is the case, check it out.

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 II.  

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

Enter the formula for numerical differentiation.

Aside.  It looks like the formula is a second divided difference, i.e. the difference quotient of two difference quotients.  Such is the case.

Aside.  From a mathematical standpoint, we expect that the limit of the second divided difference is the second derivative. Such is the case.

Example.  Consider the function  .   Find the formula for the fourth 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