M371 - Alexiades
                  Lab 3:  Linear Systems
          [ For each lab, create a new dir (Lab#) and work in it ]
At the doctor's advice, you are supposed to consume 100% of the
Recommended Daily Allowance (RDA) of protein, carbohydrate and 
calcium, 80% of fat, and 75% of salt. You select 5 food items
for the day and you need to figure out how many servings of each
you need to eat to stay with your diet.  The nutritional content
of the foods you selected, per serving, is as follows:

			% of RDA per serving

	Food    protein  carbohydrate  fat  salt  calcium
	----    -------  ------------  ---  ----  -------
	 A	   0         15        10    10     15
	 B	  30         20        20    10     20
	 C	  20         25        10    20     15
	 D	  30         15        30    10     20
	 E	  10         15         0    20      5

1. Set up the problem as a linear system  Ax=b.
   Try to arrange the matrix A so that no pivoting will be 
   necessary, if you can.

2. Use Matlab to find the LU factorization of A:  [L,U]=lu(A) .
   (>> diary on  to record your work in file "diary",
    specify format short for readability).
 
   Is L really lower triangular ?  If not, use 
		[L,U,P] = lu(A)
   to find the necessary permutation matrix P, then
   find the LU factorization of the matrix PA.
   Is L lower triangular now ?

3. Use Matlab to solve your system Ax=b.
   Round the values to integers. Is this a feasible diet ?  Why ?
   What can you do to get a proper diet for the day ?

4. Solve the system with forward and backward substitutions,
   in Matlab, using L and U appropriately.
   Does the solution agree with the one found in 3. ?
 

Submit, all in a single text file "lab3.txt" (can use Matlab notation for matrices): 
	a. Your matrix A and vector b from 1.
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	b. The L and U from 2, and if for A or for PA.
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	c. Answers to 3, unrounded and rounded.
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	d. What systems you solved, in symbols (like: Ly = b, ....), 
	   and the solution, from 4.
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