MCS-388 Homework 5

Consider the following matrix multiplication program:

begin
  for i := 1 to n do
    for j := 1 to n do
      c[i,j] := 0
  for i := 1 to n do
    for j := 1 to n do
      for k := 1 to n do
        c[i,j] := c[i,j] + a[i,k] * b[k,j]

As you complete the numbered tasks found a little ways below, use the following bulleted guidelines.

Assume that the arrays a, b, and c are indexed from 1 to n in each dimension. Assume that you know at compile time that n is 300 and that the arrays a, b, and c start at addresses 10000, 4000000, and 6000000, respectively.
Because you know that n≥1, you know that each loop will execute at least once. Take advantage of this fact by having each loop execute its test only after it has made it through the loop body.
In your three address code, do not use the bracket notation, but rather do the address addition explicitly.
Create a new temporary for each value, rather than trying to be frugal and re-use temporaries. That is, do something like
```
    t1 := i * n
    t2 := t1 + j
```
not
```
    t1 := i * n
    t1 := t1 + j
```

Here are your tasks:

Assuming a, b, and c are allocated static storage and there are four bytes per word in a byte-addressed memory, produce three-address statements for the program.
Construct a flow graph from the three-address statements.
Eliminate the common subexpressions from each basic block.
Move the loop-invariant computations out of the loops.
Find the induction variables of each loop and eliminate them where possible.
Briefly describe any major opportunities for optimizing this routine that are not covered by the exercise. Keep in mind that on contemporary machines, it is memory accesses that are generally the most time consuming operations.