Other inner-loop operations

Write subroutines, with prototypes identical to mult(), which perform the following operations:

1. divide(): works just like mult (doing n operations) but, after initializing x to 1,does n divisions of x by 1.000001.
2. multsub(): performs the same operation as mult, but where multsub() makes a subroutine call

   sub(x,1.000001)

   to perform the computation. This measures the time needed to make the subroutine calls. To prevent the compiler from automatically in-lining the subroutine, you should put sub() in a separate file.
3. multif(): within the same loop as for mult(), test whether x<1.00001. If so, multiply by 1.000001; otherwise multiply by 0.99. Note that you must return the value of x: If you forget to return x, many compilers are smart enough to realize that you don't need the loop, and they'll skip it entirely!
4. multal(): works just like mult, but, within the loop, makes a call to allocate/deallocate memory

   p=dvector(0,1);
   free_dvector(p,0,1);

   This gives a feel for the cost of making calls to the operating system.
5. multadd(): works like mult(), but performs multiply-add operations instead,

     x+=x*0.000001;

   Note that each of these counts as two FLOP (one add and one multiply).
6. cmat1(): loops through all n n elements of the matrix M1 and copies them into matrix M3 with a loop which steps through memory sequentially, and repeats this operation n times:

   for (k=0; k<n; k++)
     for (i=0; i<n; i++)
       for (j=0; j<n; j++)
         M3[i][j]=M1[i][j];

   Note that this loop will run a lot slower than mult(), which does a number of multiplies equal to the number of copies here. This gives an idea of the cost of accessing the main memory.
7. cmat2(): The same as cmat1(), but now with the order of the loops changed. This gives an idea of the cost of a cache-miss, equivalently the cache-line length (measured in number of double's). Note that, because some part of your matrices fit in cache, you'll have to run large problems to see the true cost of a cache miss.

Finally, in main, uncomment the timer() calls to your new routines as well as the section of print statements labeled ``Print out time ratios.'' Running your code for various sizes, you should find that most operations proceed at a constant rate, except for the cmat operations. The speed of these depends on the size of your matrices compared to the size of the cache memory!