1. Use the Newton code you constructed in Lab2 to find all three roots 
   (one at a time!), in double precision, of the cubic
         	F(x) = x3 − 1.2x2 − 8.55x + 12.15

   You'll probably need to plot the function to figure out good 
   starting points.  Try at least two different starting values 
   and discuss how convergence is affected (count iterations).
   [ For efficiency and accuracy, polynomials should always be 
     evaluated in "nested form" (Horner's Rule), like:
		ax2+bx+c = c + x * (b + a*x) ].

   Check, by hand, that x=−3 is an exact root. 
   Did your solver find  −3  as an exact root ? 
   If yes, how do you know ?
   If not, what could you do to increase the accuracy ?  try it !  did it work ?
   What are the other 2 roots ?   how many iterations did each take ?

    Submit this part as "M475 Lab3.1"

 
2. To practice getting software from the Internet: 
   Try to find the file dpara.f from the "paranoia" collection on netlib
   (most extensive library of free, high quality numerical software, 
   actually residing at UTK by the way!). 
   Read the beginning of dpara.f to see what it does.
   You will need a Fortran compiler:
   Install gfortran on your machine (see course page),
   then compile it:  gfortran dpara.f , and run it:  a.exe

3. To practice computing on a remote computer: 
   • Get a UTK unix account (if you don't already have one).
   • Install SSH on your machine (see course page).
   • Then SSH to unix.utk.edu  and login to your unix account,
   • Either: transfer the file dpara.f to unix.utk.edu using SCP.
     Or you can just copy it from my account on that machine to yours:
	  cp ~vasili/dpara.f . (note: there is a " ." at the end). 
   • Compile it by typing: f77 dpara.f , and run it:  a.out

Report the values of: radix, smallest number, and largest number,
and what machine this is from ( uname -a  on unix machines).

4. It is useful, and often necessary(!), to be able to do the same 
   computation by different tools/methods.

   For a little practice, search the web for any root finding codes.  
   You are likely to find free codes only in Fortran (and some in C), 
   like fzero.f or zeroin.f, there are lots of them out there... 
   In particular, look on netlib.org. 
   For C codes, the GNU Scientific Library is excellent.

   Do the following: 

   a. Use whichever code you downloaded to find the x=-3 root of the 
      cubic function in Part 1. Your UT unix account (unix.utk.edu) has
      both fortran (f77 code.f or f90 code.f) and C (cc code.c). 
      Any Linux OS contains gcc and gfortran. 

   b. Do the same using Matlab or Maple (or Mathematica), they have
      built-in root finders (fzero, fsolve, ... ). 

  Compare and contrast with your Newton solver (for accuracy and efficiency).