Numerical Methods using Mathematica

Module for Numerical Differentiation, Part I

Check out the new Numerical Analysis Projects page.

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.  The truncation error will 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  .

Proof of the Three point rule for f '

Big "O" error term for .

.

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

and the error bound is where .

Proof of the Three point rule for f ''

Big "O" error term for .

.

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

Project I. Investigate the numerical differentiation formula and 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.

Enter the function, use .

Example 1. Find the formula for the third derivative .
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 derivative  ,  using step sizes  .
3 (b).  Compute numerical approximations for the derivatives ,  using step sizes .
3 (c).  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  .
4 (b).  Since we the function f[x] and its derivative is well known, and we have the graph for ,  we can observe that the maximum error on the given interval occurs at x=0.  Thus we can do better that "theory", we see 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.

Project II. Investigate the numerical differentiation formulae and 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.

Enter the function, use .

Example 6. Find the formula for the fourth derivative of f(x).
Use .

Solution 6.

Example 7. Graph . Find the bound . Look at a graph and estimate the value , be sure to take the absolute value if necessary.

Solution 7.

Example 8 (a). Compute numerical approximations for the derivatives , using step sizes .
8 (b). Plot the numerical approximation over the interval . Compare it with the graph of over the interval .

Solution 8.

Example 9.  Plot the absolute error    over the interval  ,  and estimate the maximum absolute error over the interval.
9 (a).  Compute the error bound    and observe that    over  .
9 (b).  Since we the function f[x] and its derivative is well known, and we have the graph for ,  we can observe that the maximum error on the given interval occurs at x=0.  Thus we can do better that "theory", we see that   over  .