Computer Science 6676, Spring 1999                                         March 3, 1999

Assignment 4

Due Wednesday, March 17

  1. Perform the modified Cholesky factorization algorithm given in class (and in the new CHOLDECOMP) on the following matrix, and show the intermediate and final results.
  2.    4    2 -2
       2   3    8
    -2    8   11
  1. Chapter 6, problems 6, 8, 9.


Class Project, Phase I

Due Wednesday, March 17

  1. Code the following routines: MACHINEPS; MODELHESS (new) and CHOLDECOMP (new); CHOLSOLVE, LSOLVE, LTSOLVE; FDGRAD; FDHESSF; test problems 1-5 plus the Lennard-Jones problem given below.
  1. First debug each of the above routines individually. In the case of CHOLDECOMP, check that it runs correctly on the three examples given in class. Please hand in results that verify this. Then write a driver to run Newton's method each of your test problems. I.e. it should start from close starting points and simply calculate the (modified) Newton step using MODELHESS, CHOLDECOMP, and CHOLSOLVE, and set the new iterate to the one given by the full Newton step. It should stop when ||x _+ - x _c|| _inf   <=  (macheps)^(1/3) or after 20 iterations. Experiment with a few starting points for each of the 6 test problems to get a feel for where this does and doesn't converge from. Turn in successful runs from close starting points for each of the 6 problems, showing your iterates and the function and gradient value at each. You can choose small values of n for the problems where n  is variable.
Lennard-Jones Problem:
    x  in  R ^(3m)  =  ( v_ 1 , v_2 , ... , v_ m )   where each   v_i  in  R^3
   d _(i,j) = || v_i - v_j || _2 ,   1 <= i < j <= m
   f(x) = sum from i=1 to m  { sum from j=i+1 to m  { ( 1/(d_(i,j)^12) )- (2/(d_(i,j )^6))) } }