Numerical integration techniques

• Getting started
  • Review: Use EXCEL to obtain estimate for the definite integral of exp(-x^2) for x from 0 to 1.
  • Issues: How to choose Delta_x, left or right sums, how good is the result?
  • Add another objective: Want to have estimate to be guaranteed within 0.000 01 of the exact value.
• Data collection
  • Each team calculates estimates with the assigned number of subdivisions. Report results to the instructor's machine (don't truncate decimals, want all decimals ...)
  • You may want to do a sample calculation in EXCEL, but do the real calculation in MATLAB. The main step is to assign names like n=100, b=1, a=0, dx=(b-a)/n, xxx=[a:dx:b] and to do good book-keeping. ALWAYS TEST YOUR IMPLEMENTATION WITH SMALL DATA SETS, e.g. first try out n=5, before trying n=10000.
  • As data emerge on the front screen begin to put your observations into precise statements. Write these down!
  • If time permits start a second example, f(x)=sin(x)/x when x is not zero and f(0)=0, integrated from 0 to pi.
• Data analysis, part 1
  • Observations in teams -- look at many different aspects: upper and lower bounds, increasing and decreasing, how fast do the estimates approach the limit(s). Formulate precise statements, and write these down.
  • Explanations: Why does which technique provide lower/upper estimates; why do the estimates obatined by the respective techniques increase/decrease as the number n of subdivisions grows. Are these mere coincidences, or are there reasons. WRITE down detailed explanations....
  • Whole class: Summarize findings.
  • Using a weighted average to get even better results!....
• Data analysis, part 2
  • Convergence rates. Observations.
  • Convergence rates of left and right sums: In teams: explanation why.... assuming that the function is differentiable, .... draw pictures, identify where to "see" the errors.... WRITE out your explanations in full sentences! Summary in the whole class.
  • If time allows and interest is there: Richardson extrapolation....

Related resources:

• The program "INTEGRAL" of the Arizona Mathematical Software is probably the fastest way to generate data like the ones for the above table (but we wanted to practice using MATLAB, the true professional engineering software). (In the lab, go to \LAB\APPS\AZMATH, alternatively download the small package directly from UA Tucson.)
• In MAPLE, load the student package (with(student);), and use funcitons like leftbox, rightsum etc. These are very useful if you need neat graphics for a written report.