Individual Project 1

This project is based on writing a Matlab function to approximate . This function will be computed using its Taylor series about the point:

You may choose whichever version of the A-grade work you prefer.

 C Grade:  Verify that the formula for the Taylor series is correct (use Matlab or Maple). Implement the formula as a Matlab function, using my function getexp.m as a template. Continue adding terms until either (a) adding a new term has no effect, (b) the term involving has been used. Test your function for x = 0, 1, 5, 10, 50, 100, 1000. Present your results in a table that includes the value of x, your answer, Matlab's answer, the error in your result, and the number of terms used. Comment on your results, and (if appropriate) explain any failures in your calculations.

 B Grade: For what values of x should the series converge to the correct answer? For what values of x does the series converge to the correct answer? What is the scaling factor for ? Can you use the alternating series theorem to predict the accuracy of your result? For , how accurately should you be able to calculate ? Is loss of significance an issue in this calculation?

 A Grade (Version 1): Modify your function so that it uses a Taylor series about a point , where k is some integer chosen so that is as close as possible to x. Perform tests similar to (but more extensive than) those in the C-grade section to assess the performance of your function. (In Matlab, you can use pi to get the value of .) Use Matlab's flops function to compare the efficiency of your function to the built-in function.

 A Grade (Version 2): Investigate (in the library or over the internet) other techniques for estimating . Report on their advantages and disadvantages.

 A Grade (Version 3): Modify your function so that it transforms x into , where y is an integer multiple of , and then calculates by applying the Taylor series to . Perform tests similar to (but more extensive than) those in the C-grade section to assess the performance of your function. In addition to reporting your estimate of , report also the expected accuracy of the result.

 Preliminary deadline: Tuesday, February 3

 Final deadline: Friday, February 13, 9:30am

 Links:

OR481/Math 446

Stephen Nash

(snash@gmu.edu)