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When you embed loops within other loops, you create a loop nest . The loop or loops in the center are called the inner loops. The surrounding loops are called outer loops. Depending on the construction of the loop nest, we may have some flexibility in the ordering of the loops. At times, we can swap the outer and inner loops with great benefit. In the next sections we look at some common loop nestings and the optimizations that can be performed on these loop nests.

Often when we are working with nests of loops, we are working with multidimensional arrays. Computing in multidimensional arrays can lead to non-unit-stride memory access. Many of the optimizations we perform on loop nests are meant to improve the memory access patterns.

First, we examine the computation-related optimizations followed by the memory optimizations.

Outer loop unrolling

If you are faced with a loop nest, one simple 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; i<n; i++) for (j=0; j<n; j++) for (k=0; k<n; k++) a[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 [link] . 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; i<n; i++) for (j=0; j<n; j+=2) for (k=0; k<n; k++) { a[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; i<n; i++) for (j=0; j<n; j+=2) for (k=0; k<n; k+=2) { a[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,MA(J,I) = B(J,I) + C(J,I) * D ENDDOENDDO

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

II = IMOD (N,4) DO I=1,IIDO J=1,M A(J,I) = B(J,I) + C(J,I) * DENDDO ENDDODO I=II,N,4DO J=1,M A(J,I) = B(J,I) + C(J,I) * DA(J,I+1) = B(J,I+1) + C(J,I+1) * D A(J,I+2) = B(J,I+2) + C(J,I+2) * DA(J,I+3) = B(J,I+3) + C(J,I+3) * D ENDDOENDDO

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

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,JJDO I=2,N A(I,J) = A(I,J) + A(I-1,J) * BENDDO ENDDODO J=1+JJ,M,4DO I=2,N A(I,J) = A(I,J) + A(I-1,J) * BA(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) * BA(I,J+3) = A(I,J+3) + A(I-1,J+3) * B ENDDOENDDO

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.

Questions & Answers

Is there any normative that regulates the use of silver nanoparticles?
Damian Reply
what king of growth are you checking .?
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
Stoney Reply
why we need to study biomolecules, molecular biology in nanotechnology?
Adin Reply
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
what school?
biomolecules are e building blocks of every organics and inorganic materials.
anyone know any internet site where one can find nanotechnology papers?
Damian Reply
sciencedirect big data base
Introduction about quantum dots in nanotechnology
Praveena Reply
what does nano mean?
Anassong Reply
nano basically means 10^(-9). nanometer is a unit to measure length.
do you think it's worthwhile in the long term to study the effects and possibilities of nanotechnology on viral treatment?
Damian Reply
absolutely yes
how to know photocatalytic properties of tio2 nanoparticles...what to do now
Akash Reply
it is a goid question and i want to know the answer as well
characteristics of micro business
for teaching engĺish at school how nano technology help us
Do somebody tell me a best nano engineering book for beginners?
s. Reply
there is no specific books for beginners but there is book called principle of nanotechnology
what is fullerene does it is used to make bukky balls
Devang Reply
are you nano engineer ?
fullerene is a bucky ball aka Carbon 60 molecule. It was name by the architect Fuller. He design the geodesic dome. it resembles a soccer ball.
what is the actual application of fullerenes nowadays?
That is a great question Damian. best way to answer that question is to Google it. there are hundreds of applications for buck minister fullerenes, from medical to aerospace. you can also find plenty of research papers that will give you great detail on the potential applications of fullerenes.
what is the Synthesis, properties,and applications of carbon nano chemistry
Abhijith Reply
Mostly, they use nano carbon for electronics and for materials to be strengthened.
is Bucky paper clear?
carbon nanotubes has various application in fuel cells membrane, current research on cancer drug,and in electronics MEMS and NEMS etc
so some one know about replacing silicon atom with phosphorous in semiconductors device?
s. Reply
Yeah, it is a pain to say the least. You basically have to heat the substarte up to around 1000 degrees celcius then pass phosphene gas over top of it, which is explosive and toxic by the way, under very low pressure.
Do you know which machine is used to that process?
how to fabricate graphene ink ?
for screen printed electrodes ?
What is lattice structure?
s. Reply
of graphene you mean?
or in general
in general
Graphene has a hexagonal structure
On having this app for quite a bit time, Haven't realised there's a chat room in it.
what is biological synthesis of nanoparticles
Sanket Reply
how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
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Source:  OpenStax, High performance computing. OpenStax CNX. Aug 25, 2010 Download for free at http://cnx.org/content/col11136/1.5
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