Numerical Integration Assignment

Due Friday, April 20, 2012 by 5pm

The file Integrate.java contains a program to do some numerical integration. Currently the program implements the composite Trapezoid rule and Simpson's rule and has built in the following 7 functions:

1. f(x) = 4 / (1 + x²)
2. f(x) = Math.sqrt(x) * Math.log(x)
3. f(x) = cos(x)
4. f(x) = 1 / (1 + 100x²)
5. f(x) = Math.sqrt( |x| )
6. f(x) = (100/x²)sin(10/x)
7. f(x) = x¹⁶cos(x¹⁶)

1. Implement the composite Midpoint Rule. The stub for the method is already in the program. Replace the "return 0" statement with the code.

2. Use the program to estimate PI using the first function (the integral of that function from 0 to 1 is PI) using the three quadrature rules (Midpoint, Trapezoid, and Simpson) as follows. All will be computed when you choose option I in the program. You must first select the function. Use 0 and 1 as the endpoints of the intervals. For each use a variety of values of n (and hence h).
   1. Examine the error for each approximation and see how well it matches the theoretical error.
   2. For each, find a value of n beyond which there is no further improvement in the accuracy of the answer.

   Get output from the program to support your answers (don't waste paper -- decide what output shows what you want then print just that). (Be sure to actually write some answers -- I don't want just output with no analysis!)

3. In class, we derived a way of using the midpoint rule and the trapezoid rule to estimate the error in the midpoint rule (and hence also in trapezoid since the two are related). Add code to the integrate method to compute and print this error estimate. (Print it after the table showing the results of the three composite methods.)

4. Use the program to evaluate the integrals of functions 2 (using limits of integration of 0 and 1) and 3 - 5 (using limits of integration of -1 and 1) using several values of n (to get different sizes of interval subdivision). Answer the following questions about error and accuracy:
   1. Examine the error for each approximation. Compare it to both the theoretical and, in the case of the midpoint rule, to the approximation.
   2. Explain any unexpected results (anything that doesn't appear to be quite right even if you expected it!)
   3. For each find a value of n beyond which there is no further improvement.

5. Use the Adaptive Simpson method on function 5 (the square root of the absolute value) from -1 to 1. Use a tolerance of 0.0001.
   1. On the graph of the function (Mathematica printout) mark the subdivisions used by the method on the x-axis.
   2. Explain why the subdivisions appear as they do.
   3. Verify that using composite Simpson you would need at least n = 1000 to get results as good.

6. Use the Monte Carlo method to evaluate the integral of function #1 (and thus estimate PI). Try different numbers n of sample points. Record your results - number of sample points and the approximate integral. Try to determine the relationship between the number of points and the error. Also compare the amount of work for this method to the others.

7. Use all methods (including Romberg) on function #7 from 0.8 to 1.3 (the part of the function in the graph from Mathematica). The correct answer is -0.0510397. Compare the methods both in terms of accuracy and number of calculations. (Note: the "correct" answer built into the program is for different a and b - you can change it to -0.0510397 so the error is calculated for you for the composite methods). For Romberg, look both down and across in the table generated to see when/where the solution gets best.