Homework 15, Due Friday March 14, 2008.
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In this exercise, you are to code a gradient 
search with the added wrinkle, discussed in class, 
that the distance you go in the gradient direction -- 
as an iterative step in maxmimizing a function --
is itself found by a univariate maximization.

Your task is to write an R function with the 
inputs:

ffcn  :  a scalar function of a vector parameter
fprim : the analytical gradient of ffcn with 
        respect to its vector argument (ie with 
        p-dimensional vector value when the 
        argument of ffcn is p dimensional .
xzer  : a starting value for x.
step  : the maximum step length
grdtol: the (small) real number to be used as a 
        stop criterion in your iteration.
iter  : the maximum number of allowed iterations

Your R function should iterate for up to iter steps, 
stopping earlier if  fprim(x_k)  ever has norm as 
small as  grdtol . The outputs should be: your 
optimized (or final)  x_k value and its fprim value 
and (optionally) a numerical approximation to the 
Hessian matrix of ffcn at x_k; also include in
the output list your starting value and the number 
of iteration-steps actually used.

Show that your code works to maximize a nontrivial 
function with 2-dimensional argument, and on another 
with 5-dimensional argument. Both times, the 
function could be (but need not) be defined as 
log-likelihood from a statistical regression problem, 
if you like, but then take the error density to be 
something like logistic or t or the log of a Weibull 
r.v., not normal.