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By Root 1299 0

approach is to unroll the inner loop. Unrolling the innermost loop in a nest isn’t any different from what we saw above. You just pretend the rest of the loop nest doesn’t exist and approach it in the nor- mal way. However, there are times when you want to apply loop unrolling not just to the inner loop, but to outer loops as well — or perhaps only to the outer loops. Here’s a typical loop nest:

for (i=0; ifor (j=0; jfor (k=0; ka[i][j][k] = a[i][j][k] + b[i][j][k] * c;

To unroll an outer loop, you pick one of the outer loop index variables and replicate the innermost loop body so that several iterations are performed at the same time, just like we saw in the The Section Called “Qualifying Candidates For Loop Unrolling Up One Level ”. The difference is in the index variable for which you unroll. In the code below, we have unrolled the middle (j) loop twice:

for (i=0; ifor (j=0; jfor (k=0; ka[i][j][k] = a[i][j][k] + b[i][k][j] * c;

a[i][j+1][k] = a[i][j+1][k] + b[i][k][j+1] * c;

}

We left the k loop untouched; however, we could unroll that one, too. That would give us outer and inner loop unrolling at the same time:

for (i=0; ifor (j=0; jfor (k=0; ka[i][j][k] = a[i][j][k] + b[i][k][j] * c;

a[i][j+1][k] = a[i][j+1][k] + b[i][k][j+1] * c;

a[i][j][k+1] = a[i][j][k+1] + b[i][k+1][j] * c;

a[i][j+1][k+1] = a[i][j+1][k+1] + b[i][k+1][j+1] * c;

}

We could even unroll the i loop too, leaving eight copies of the loop innards. (Notice that we completely ignored preconditioning; in a real application, of course, we couldn’t.)

Outer Loop Unrolling to Expose Computations

Say that you have a doubly nested loop and that the inner loop trip count is low — perhaps 4 or 5 on average. Inner loop unrolling doesn’t make sense in this case because there won’t be enough iterations to justify the cost of the preconditioning loop. However, you may be able to unroll an outer loop. Consider this loop, assuming that M is small and N is large:

DO I=1,N

DO J=1,M

A(J,I) = B(J,I) + C(J,I) * D

ENDDO

ENDDO

Unrolling the I loop gives you lots of floating-point operations that can be overlapped:

II = IMOD (N,4)

DO I=1,II

DO J=1,M

A(J,I) = B(J,I) + C(J,I) * D

ENDDO

ENDDO

DO I=II,N,4

DO J=1,M

A(J,I) = B(J,I) + C(J,I) * D

A(J,I+1) = B(J,I+1) + C(J,I+1) * D

A(J,I+2) = B(J,I+2) + C(J,I+2) * D

A(J,I+3) = B(J,I+3) + C(J,I+3) * D

ENDDO

ENDDO

In this particular case, there is bad news to go with the good news: unrolling the outer loop causes strided memory references on A, B, and C. However, it probably won’t be too much of a problem because the inner loop trip count is small, so it naturally groups references to conserve cache entries.

Outer loop unrolling can also be helpful when you have a nest with recursion in the inner loop, but not in the outer loops. In this next example, there is a first- order linear recursion in the inner loop:

DO J=1,M

DO I=2,N

A(I,J) = A(I,J) + A(I-1,J) * B

ENDDO

ENDDO

Because of the recursion, we can’t unroll the inner loop, but we can work on several copies of the outer loop at the same time. When unrolled, it looks like this:

JJ = IMOD (M,4)

DO J=1,JJ

DO I=2,N

A(I,J) = A(I,J) + A(I-1,J) * B

ENDDO

ENDDO

DO J=1+JJ,M,4

DO I=2,N

A(I,J) = A(I,J) + A(I-1,J) * B

A(I,J+1) = A(I,J+1) + A(I-1,J+1) * B

A(I,J+2) = A(I,J+2) + A(I-1,J+2) * B

A(I,J+3) = A(I,J+3) + A(I-1,J+3) * B

ENDDO

ENDDO

You can see the recursion still exists in the I loop, but we have succeeded in finding lots of work to do anyway.

Sometimes the reason for unrolling the outer loop is to get a hold of much larger chunks of things that can be done in parallel. If the outer loop iterations are independent, and the inner loop trip count is high, then each outer loop iteration represents a significant, parallel chunk of work. On a single CPU that doesn’t matter much, but on a tightly coupled multiprocessor, it can translate into a tremendous increase in speeds.

Loop Interchange*

Loop