Finger exercises, to be discussed Wed Oct 2, 2002
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Variations on "golden.c", discussed in class.

1. We do not have to maintain the array y[i] of values of f(x), if we
are willing to run the occasional risk of re-evaluating f() at the
same x values.

Modify "golden.c" by wiping out all of the stuff related to y[i] and
eval[i], and simply decide how to shorten the interval from aprime to
bprime, based on looking at the values

    f(aprime + c*(bprime-aprime))  vs  f(aprime + (1-c)*(bprime-aprime))

in each iteration in the while loop in main().  

(a) You must pick a value of c between 0 and 1 to make the program
    work.  What do you think the optimal value of c would be?  Does
    anything prevent you from using, or coming close to, this value?

(b) You may (or may not) want to change the stopping condition for the
    while loop as well (e.g. use a relative stopping criterion rather
    than an absolute one; or stop when the y-values stabilize rather
    than when the x-values stabilize, etc.).  What do you think makes
    sense here?

(c) Compare the performance of this "almost-golden.c" program with
    golden.c itself.  (e.g. how many function evaluations does
    "almost-golden.c" require, for the same decimal accuracy as
    "golden.c"?)

2. Write a program "bisection.c" that finds the maximum of f(x) (unimodal
in [a,b]) by finding the roots of df(x) = f'(x), based on your experience
with "golden.c" and "almost-golden.c".  Compare the performance of the
three programs.

3. If you have time, please think about writing programs for some of the
other univariate methods we are discussing:
  function iteration
  newton's method, 
  regula falsi, 
  the illinois method