Computer Science 6676, Spring 1999
March 3, 1999
Assignment 4
Due Wednesday, March 17

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.

Chapter 6, problems 6, 8, 9.
Class Project, Phase I
Due Wednesday, March 17

Code the following routines: MACHINEPS; MODELHESS (new) and CHOLDECOMP
(new); CHOLSOLVE, LSOLVE, LTSOLVE; FDGRAD; FDHESSF; test problems 15 plus
the LennardJones problem given below.

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.
LennardJones 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))) } }