Assignment 3 -- Solution and Common Mistakes

Here is a script file and a printout of Maple session, where I checked the the results.

Comments to the tests:

Test 1: regular values, no command line arguments

Tests 2-3: invalid data (Test 2: invalid x, Test 3: invalid y)

Tests 4-9: testing command line argument (CLA) switch

Test 4: CLA= 1 (arithmetic only)

Test 5: CLA= 2 (geometric only)

Test 6: CLA= 3 (harmonic only)

Test 7: CLA= 4 (AGM only)

Tests 8-9: invalid CLA (Test 8: CLA=-1, Test 9: CLA=5)

Tests 10-13: numerical stress-tests (trying to fail the program)

Test 10: x=2, y=2E+7. The program never ends (failed).

Test 11: x=2, y=2E+6: Passed. Results compare well with Maple's.

Test 12: (Trying to approach the program's bound closer): x=2, y=1E+7: Passed. Results compare well with Maple's.

Test 13: (switching the arguments): x=1E+7, y=2: Passed.

Common Mistakes

• In AGM loop: the sequence
         x=a(x,y);
         y=g(x,y);
  results in the use of already updated value of x in calculation of the geometric mean.
  Mathematically, this piece of code corresponds to the recurrence
         x_n+1= a(x_n, y_n),        y_n+1= g(x_n+1, y_n),
  rather than to
         x_n+1= a(x_n, y_n),        y_n+1= g(x_n, y_n).


• The Harmonic Mean h(x,y) is defined by the equation
         1/h= arithmetic(1/x,1/y).
  Some students computed the arithmetic mean of the reciprocals but forgot to invert the result.


• Some programs aren't mathematically correct. Mistakes could be found or at least noted had the authors use Maple or other tools (a handbook, for instance, or the Internet) to check numerical answers.