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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

where we get a research paper on Nano chemistry....?
Maira Reply
nanopartical of organic/inorganic / physical chemistry , pdf / thesis / review
Ali
what are the products of Nano chemistry?
Maira Reply
There are lots of products of nano chemistry... Like nano coatings.....carbon fiber.. And lots of others..
learn
Even nanotechnology is pretty much all about chemistry... Its the chemistry on quantum or atomic level
learn
Google
da
no nanotechnology is also a part of physics and maths it requires angle formulas and some pressure regarding concepts
Bhagvanji
hey
Giriraj
Preparation and Applications of Nanomaterial for Drug Delivery
Hafiz Reply
revolt
da
Application of nanotechnology in medicine
what is variations in raman spectra for nanomaterials
Jyoti Reply
ya I also want to know the raman spectra
Bhagvanji
I only see partial conversation and what's the question here!
Crow Reply
what about nanotechnology for water purification
RAW Reply
please someone correct me if I'm wrong but I think one can use nanoparticles, specially silver nanoparticles for water treatment.
Damian
yes that's correct
Professor
I think
Professor
Nasa has use it in the 60's, copper as water purification in the moon travel.
Alexandre
nanocopper obvius
Alexandre
what is the stm
Brian Reply
is there industrial application of fullrenes. What is the method to prepare fullrene on large scale.?
Rafiq
industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
Damian
How we are making nano material?
LITNING Reply
what is a peer
LITNING Reply
What is meant by 'nano scale'?
LITNING Reply
What is STMs full form?
LITNING
scanning tunneling microscope
Sahil
how nano science is used for hydrophobicity
Santosh
Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
Rafiq
what is differents between GO and RGO?
Mahi
what is simplest way to understand the applications of nano robots used to detect the cancer affected cell of human body.? How this robot is carried to required site of body cell.? what will be the carrier material and how can be detected that correct delivery of drug is done Rafiq
Rafiq
if virus is killing to make ARTIFICIAL DNA OF GRAPHENE FOR KILLED THE VIRUS .THIS IS OUR ASSUMPTION
Anam
analytical skills graphene is prepared to kill any type viruses .
Anam
Any one who tell me about Preparation and application of Nanomaterial for drug Delivery
Hafiz
what is Nano technology ?
Bob Reply
write examples of Nano molecule?
Bob
The nanotechnology is as new science, to scale nanometric
brayan
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Damian
Is there any normative that regulates the use of silver nanoparticles?
Damian Reply
what king of growth are you checking .?
Renato
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
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
Adin
why?
Adin
what school?
Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
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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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