Homework Assignment 3
                               CS 350c

                         Unique Number: 52140
                             Spring, 2017

                       Given:  January 31, 2017
                        Due:  February 9, 2017


This homework concerns the speed of the matrix multiplication
algorithm as it is commonly implemented.

Consider the six different matrix multiplication algorithms that
appear on page 645 of "Computer Systems, A Programmer's Perspective"
3rd Edition (For the 2nd Edition, it's on page 626).  You are to
implement all six algorithms, but with a some changes.

  1.  Arrays A, B, and C, should all be declared as:

         #define ROWS          (1000)
         #define COLUMNS       (1000)
         #define ENTRIES       (ROWS*COLUMNS)
         char A[ENTRIES], B[ENTRIES], C[ENTRIES];

  2.  The line(s) with "sum = 0.0;" should be changed to "sum = 0;"

  3.  The summation should be saturating; that is, when the sum
      exceeds 127, the answer should be 127.  When the sum is less
      than -128, the answer should be -128.  Be sure to calculate the
      sums with full precision, then, finally, perform the saturation
      rounding.

Initialize array A with a tri-diagonal set of 1s.  Initialize array B
with a tri-diagonal set of 1s, but then change the main diagonal only
to all -1 entries.  Remember, we are multiplying array A to array B,
and then putting the answer in array C.  Organize your arrays so that
the stored elements are in row-major order.

Record the running time of multiplying array A with array B with the
size of the arrays varying from 100x100, 200x200, ..., 1000x1000.
Thus, you will produce 60 answers.  Then, calculate the time per
element and make a chart like that on page 646, in the 3rd edition of
"Computer Systems, a Programmer's Perspective" (a similar chart
appears on page 628 in the 2nd edition).

Below are the questions that you are asked to answer as a part of this
assignment:

1. Is there a difference (for your matrix multiplication code) produced
   by the compiler for "-O2" and "-O3".  If so, what is this difference?

2. How many memory references does your code make when multiplying two
   arrays of size N?  That is, write an equation (in terms of N) that
   describes the number of array reads and writes required to
   implement one of the matrix multiplication algorithms.

3. Is the number of memory accesses the same for all six algorithms?

4. Would it make sense to modify the algorithms you implemented to use
   blocking?  If so, why?  If not, why?  If unsure, how could this be
   determined?

When you report your results, also carefully document what system you
used.  This work should be done on a X86-based computer.  Find out
what processor the machine you are using contains -- find the exact
model/part number of the x86 processor you use.  Once that is found,
find out how much cache memory it has and also find out how many
levels of cache it has.