# A brief introduction to parallel computing in Julia

In this session we will learn how to take advantage of a multi-core PC through the use of parallel computing in Julia. It is worth to mention that several of the exfmples provided here come from Julia’s website, that is why we have decided to preserve the structure of the contents developed there. The reader may appreciate that when using the same examples, we have detailed a little bit more the explanation such that the ideas exposed can be easily understood. As a result, this post turns out to be a mix of examples: ones coming from the official documentation, and others coming from our own.

All the code exposed here can be run either locally (downloading the Julia interpreter) or online (through JuliaBox). The code that we will use during the session can be download here in a Notebook file (right click and “Save as…”), and the only requirement will be to be familiar with the basics of Julia.

## First approach

The basic idea of parallel computing is: there are several tasks that, instead of being done sequentially, they could be carried out at the same time (improving performance) by different processes. Examples of tasks that can be parallelized are: generation of random numbers, matrix multiplication, the branch and bound algorithm, etc.

When using parallel computing, there are some commands that we will need most of the time:

Sys.CPU_CORES # Number of logical CPU cores available in the system.
nprocs()      # Number of available processes.
nworkers()    # Number of available worker processes. nworkers() = nprocs() - 1.
procs()       # An array of the processes.
workers()     # An array of the workers.
rmprocs(2)    # Removes the specified workers, in this case the 2nd.
# rmprocs(1) does not work because procs = 1 is not a worker.


The previous code brings some new notation that must be explained. Workers is the name given to the processes used for parallel operations. Each worker has an identifier that we will employ to refer to it. For instance, in the code above we have removed the second worker using its id. It is important to know that the process providing the interactive Julia prompt always has an id equal to $1$.

To clarify these commands let’s run the next code:

# Before adding workers.
nprocs()
nworkers() # when there is no extra workers, nprocs() = nworkers().

nprocs()
nworkers()
workers()


To carry out parallel operations, besides adding the required workers, we must learn how to communicate with them, i.e.: how to send them work to do. In our case, the communication flow will be established through remote references and remote calls. A remote reference is done for exchanging information between processes, while a remote call is used to make a request to execute a function. You can check more information about this topic at the official documentation. As for us, let’s see these ideas with examples.

First, imagine that we want to generate two matrices formed by random numbers in worker number $2$. One of the matrices is comprised of real numbers, while the other one contains integers between $1$ and $8$.

# addprocs(Sys.CPU_CORES - 1)  # Run this line if needed.
r1 = remotecall(rand, 2, 2, 2) # Function, id of the worker, args of the function.
r2 = remotecall(rand, 2, 1:8, 3, 4) # We obtain a 'Future'.


With the code above, we have told the second worker to do two different tasks. For both cases, the function remotecall has immediately returned a Future, i.e.: a special type of remote reference. With that reference, we will be able to get the result of the operation afterward using the function fetch:

fetch(r1) # We get the value.
fetch(r2)

# Notice what happens once the value is fetched:
r1
r1[2, 2]

# However, r1 is still a Future so some operations might not work:
typeof(r1)
sum(r1)         # This does not work because r1 is a Future, no an array.

r3 = fetch(r1); # We save the result in another variable.
typeof(r3)
sum(r3)


There are other ways, besides the function remotecall(), of sending work to the different processes. Check for instance the next code in which we create a random matrix and then we add $1$ to all its elements.

# Option 1: using remotecall and spawnat.
r1 = remotecall(rand, 2, 2, 2)
s1 = @spawnat 2 1 .+ fetch(r1) # id process, expression: 1 .+ fetch(r1)
fetch(s1)

# Option 2: using only spawnat.
s2 = @spawnat 2 rand(2, 2)
s3 = @spawnat 3 1 .+ fetch(s2)
fetch(s3) # The result changes because we are using random numbers.

# Option 3: Letting Julia to select the process for us.
s2 = @spawn rand(2, 2)     # Notice that there is no "at", so we do not specify the worker.
s3 = @spawn 1 .+ fetch(s2) # It is Julia that selects it.
fetch(s3)

# Tip: due to efficiency reasons, use
remotecall_fetch(rand, 2, 2, 2) # instead of fetch(remotecall())


In the same way that we use predefined functions from Julia (e.g.: rand), we can use our own functions to make computations in parallel. Let’s see how:

# We create a function that returns the sum of the eigenvalues.
function eig_sum(A)
autoVal, autoVec = eig(A);
return sum(autoVal)
end

# We test the function as usual... and it works fine.
eig_sum(rand(2, 2))

# We use it at process 1.
s1 = @spawnat 1 eig_sum(rand(2, 2))
fetch(s1)

# We use it at process 2.
s2 = @spawnat 2 eig_sum(rand(2, 2))
fetch(s2) # returns an error.


The error in the last piece of code is because the $2$nd process does not know about the function! It has only been declared at process $1$. To overcome this problem we have to use the macro @everywhere:

@everywhere function eig_sum(A) # Now all the processes know about the function.
autoVal, autoVec = eig(A);
return sum(autoVal)
end

s3 = @spawnat 2 eig_sum(rand(2, 2))
fetch(s3) # Now everything works fine.

# When having the functions in an external file:
@everywhere include("FileWithFunctions.jl")


## Data movements

Recall that the goal in parallel computing is to improve performance, and to that end, it is important to pay attention to all the steps involved. In particular, everything related to the quantity of information sent between processes and the scope of the variables is crucial. For instance, we will see later how parallel for loops are not efficient when the workload is not large, because the “large” amount of data movements required does not make up for it.

To better understand the idea of data movements, consider the following example:

A = rand(10, 10);   # We construct the matrix locally.
Bref1 = @spawn A^2; # We move the data to another process to perform the operation.

Bref2 = @spawn rand(10, 10)^2; # Less data movement since everything
# is done in the same process.


As it can be seen, the same operation can be done in two different ways. The use of one approach or another will depend on the necessities (e.g.: if the first process needs matrix A, the first approach might be better). We encourage the reader to check the official documentation about this example for more information.

Related to the scope of the variables, the use of global variables in remote calls can lead to a duplicity of them, consuming extra memory without necessity. In the next code we clearly expose that problem and a solution using “let blocks”.

A = rand(2, 2)
whos()            # We see that A is created locally.
@spawnat 2 whos() # We see that there is nothing in the second process.

s3 =  @spawnat 2 eig_sum(A) # We perform an operation in the second process.
fetch(s3)
@spawnat 2 whos()           # Now A is also in the second process!
# What can be really memory-consuming.

# Solution to the problem.
X = rand(4, 4)    # We create a new Matrix X.
whos()            # We see how it is in the first process,
@spawnat 2 whos() # but not in the second one.

# With this code, we will be able to use X in another process, but without duplicating it.
let B = X
s3 = @spawnat 2 eig_sum(B)
end
@spawnat 2 whos() # Now X has not been saved in the 2nd process!


## Parallel map and loops

So far we have seen how to communicate with different processes and which are the things to pay attention when doing parallel computing. Now we are going to learn two common constructions employed in parallel computing: parallel for loops and parallel mapping (the use of the map function in parallel).

We will be interested in using a parallel for loop when we have a really big number of easy and independent tasks to do. Let us explain why we emphasized the previous words:

1. We need a big number of operations to do because parallel for loops require more data/messages movements than ordinary for loops, so the number of operations must be big enough to be worthwhile.

2. We need easy operations because the idea is that the number of operations, and not the operations themselves, are the time-consuming factor. When this is the other way around (few operations, but really time-consuming) we will use parallel maps instead.

3. Independence is needed because iterations do not happen in a specified order.

To better understand all this, let’s see the next code.

n = 200000000;
nheads = @parallel (+) for i = 1:n
Int(rand(Bool))
end


In the code above, we are counting the number of successes when generating $200$ million independent realizations of a Bernoulli random variable with parameter $p=0.5$. Notice how all the requirements for the use of a parallel for loop are fulfilled: a really big number of easy and independent operations. In the code, (+) is called the reduction operator and it serves to merge the results of each iteration. Although the use of a reduction operator is not mandatory, it is commonly used. However, if for any reason you do not use it, you have to know that then the parallel for loop is executed asynchronously, so you will probably want to add the macro @sync at the beginning (we will explain this macro and the meaning of asynchronously later on with an example).

Let’s see another example where a parallel for loop is employed.

# Approximating pi through Monte Carlo method (area of the circle).
piAprox = 0.0; # Recall, pi = 3.1415926...
piAprox = @parallel (+) for i = 1:n
Int(rand()^2 + rand()^2 <= 1);
end
piAprox /= n/4 # Equivalent to piAprox = piAprox*4/n.


The important thing to remember when using parallel for loops is that they do not work as ordinary for loops. This implies that we can not modify arrays as it is exemplified in the next code:

a = zeros(100000);
@parallel for i = 1:100000
a[i] = i;
end
a # Nothing has changed.


The reason why the assignation operation did not take place is because the array belongs to the first process, while the iterations take place in different ones. Therefore, to modify an array we have to make it visible from all processes, something that we will learn to do at the end of this post.

Nevertheless, although we can not modify ordinary arrays, we can read from them without any problem:

# Another implementation to compute an approximation for pi.
# In this case, reading the data from local arrays.
n = 100000;
x = rand(n);
y = rand(n);

# We will also use an external function to exemplify
# how to use them in parallel for loops.
@everywhere function inside(x, y) return Int(x^2 + y^2 <= 1) end

piAprox = 0.0;
piAprox = @parallel (+) for i=1:n
inside(x[i], y[i]);
end

piAprox /= n/4


To conclude this part about the use of parallel for loops, let’s see what is the performance when the amount of work is not really big. We will compare it with an ordinary for loop and predefined functions.

# Ordinary for loop.
a = 0;
@time for i = 1:20000
a += Int(rand(Bool));
end

# Parallel for loop
@time @parallel (+) for i = 1:20000
Int(rand(Bool));
end

# Predefined function.
@time sum(rand(0:1, 20000))

# For a not so small amount of work:
n = 200000000;
@time @parallel (+) for i = 1:n
Int(rand(Bool))
end

# DO NOT run sum(rand(0:1, n)) or the ordinary for loop with 'n': You may run
# out of memory.


As we said before, parallel for loops makes sense under some conditions. When there are a few tasks to do, but really time-consuming (e.g.: computing the eigenvalues for different matrices), then it is better to use the function map, but in parallel, that is, to evaluate each input in a different process at the same time. Check the next code for an example.

# We create an array of matrices.
M = Matrix{Float64}[rand(1000, 1000) for i = 1:10];
pmap(svd, M); # Compute the svd for each of them.


## Dynamic scheduling

When parallelizing tasks, sometimes the time required by each of them might be different, so we will be interested in sending new work to each process as they finish (like in a FIFO system). Working like this is what is going to make parallel computation to become efficient. To better understand this, let’s go to discuss some code provided in the official documentation.

Consider the previous situation with the map function, but now with the following matrices:

M = Matrix{Float64}[rand(800, 800), rand(600, 600), rand(800, 800), rand(600, 600)];


It is clear that if we had two processes, it would not make any sense that one of them compute the Single Value Decomposition for the two big matrices ($800 \times 800$), and the other one for the two small ones. To see how to overcome this problem, let’s discuss the following “easy” implementation of a parallel map function that will also serve us to introduce new functions employed in parallel computing.

# The arguments are: 1) a function 'f' and 2) a list with the input.
function f_pmap(f, lst)
np = nprocs()            # Number of processes available.
n  = length(lst)         # Number of elements to apply the function.
results = Vector{Any}(n) # Where we will write the results. As we do not know
# the type (Integer, Tuple...) we write "Any"
i = 1
nextidx() = (idx = i; i += 1; idx) # Function to know which is the next work item.
# In this case it is just an index.
@sync begin # See below the discussion about all this part.
for p = 1:np
if p != myid() || np == 1
@async begin
while true
idx = nextidx()
if idx > n
break
end
results[idx] = remotecall_fetch(f, p, lst[idx])
end
end
end
end
end
results
end


The difficult and important part to understand in the function is the one with @sync and @async. Let’s analyze everything:

1. The @sync block is saying “it is forbidden to skip this block and continue executing code until all the jobs are done”. As we are parallelizing task that might need different times to be completed, we want to ensure that we do not continue running the program until everything is finished. You usually employ @sync when using an @async block.

2. The for loop is going through all the processes and the if conditional is establishing that the first process is only used if there are no others available. This is because the function myid() is returning $1$: the number of the process in which we are executing the function. If the function were executed in worker number $4$ the process would return $4$.

3. Once we are inside the condition, we start the @async block. The idea of this block is to launch several tasks at the same time, but not in a synchronous way, i.e.: we do not care which task starts or ends earlier, we just want to finish work as quick as we can. Thanks to the first block @sync we will achieve to synchronize everything at the end. Notice that @async is launching tasks, but not in different processes. We achieve to send work to each process $p$ through the use of the remotecall_fetch() function.

4. Everything that it is inside the @async block is the task to execute. Notice that in this case we have an infinite while loop from which we can escape thanks to the the break statement. The while loop works as follows: We get the position (using nextidx()) of the input to evaluate. Then we execute the function f in process $p$ and save the result (as the task are launched locally, there is no problem when working with an ordinary array). As we are in an infinite while loop we repeat again the operation. Imagine that the last input element is being evaluated at process $p=3$. Then, the task associated with process $p=2$ will leave the while loop because of the break statement. The block @async will also be abandoned, but @sync will not. For that, it will be necessary that the task associated with $p=3$ ends too.

## Channels

In the previous part we have seen how to launch different task simultaneously and how to manage them asynchronously. We did not have any problem related to sharing information because all the tasks were executed in process $1$, although the functions were evaluated in a different process. Sometimes this will not be so easy because each task will be also executed in a different process. To establish communication in that situation one of the options is to use a Channel. A channel can be thought a queue in a supermarket: you start adding elements in the back, but you take them from the front. Channel allows to easily implement parallelization when we need to read some data, process it, and write it.

Let’s check the code below (coming from the official documentation) for a better understanding. In there, it is simulated the following situation: There is a list of jobs to do and we want to initialize $4$ simultaneous task to complete them. Every single time a job is finished, we write the total time employ in another list (a Channel in our case).

# The first parameter of a channel is the Type (e.g.: Int) and the second one,
# the maximum number of elements allowed in the channel.
const jobs    = Channel{Int}(32); # Here we can save at maximum 32 integers.
const results = Channel{Tuple}(32);

function do_work()
for job_id in jobs
exec_time = rand()
sleep(exec_time)  # Simulates elapsed time doing actual work.
# Typically performed externally.
put!(results, (job_id, exec_time)) # To write elements in a channel we "put" them.
end
end;

function make_jobs(n)
for i in 1:n
put!(jobs, i)
end
end;

n = 12;

@schedule make_jobs(n); # Feed the jobs channel with "n" jobs.

for i in 1:4 # Start 4 tasks to process requests in parallel.
@schedule do_work()
end

@elapsed while n > 0 # Print out results.
job_id, exec_time = take!(results) # To get elements from a channel we "take" them.
println("$job_id finished in$(round(exec_time, 2)) seconds")
n = n - 1
end


1. We start declaring the channels and functions that we will use. The function do_work simulates the time required to complete the work and writes the result in the corresponding channel. The function make_jobs just initializes the channel jobs. In this case we will simulate $12$ different jobs.

2. The macro @schedule is transforming the function to a task and executing it. If we had written make_jobs(n), it would also have worked.

3. In the for loop we are launching $4$ tasks simultaneously. This implies that at the same moment, jobs $1$, $2$, $3$ and $4$ begin to be executed. Thanks to the macro @schedule, which is converting the function to a task (as @async does), we can keep track of the index in the for loop of the do_work function.

4. The last part of the code serves to see which job has finished before and which has been the time required for its execution. If you execute the code, you will notice that the total amount of time required by the jobs is larger than the time employed (because of parallelization). This is easy to see if you multiply by $5$ the rand time and run the code again.

To better understand this code we propose to the reader the following two exercises:

• Check what would be the difference if instead of initializing $4$ tasks, we just call the function as we do when not performing parallel computing.

• Instead of using less than a second for the sleep function, use larger amount of time (e.g.: a mean of $2$ seconds) and analyze what happens in the while loop when the task has not been completed and we take the output from channel results.

Notice that in the last example, all the tasks were running in the same process. However, we started motivating the use of channels saying that we might be interested in launching the tasks in different processes. To do that, we would need to communicate different processes in the reading/writing operations, so an especial type of channel is needed: RemoteChannel. Let’s see an implementation of the previous code using them. The difference, in this case, is that the jobs will be executed in different processes and not in the first one as before.

addprocs(4); # Add worker processes.

const jobs    = RemoteChannel(()->Channel{Int}(32));
const results = RemoteChannel(()->Channel{Tuple}(32));

@everywhere function do_work(jobs, results) # Define work function everywhere.
while true
job_id = take!(jobs)
exec_time = rand()
sleep(exec_time)
put!(results, (job_id, exec_time, myid()))
end
end

function make_jobs(n)
for i in 1:n
put!(jobs, i)
end
end;

n = 12;

@schedule make_jobs(n); # Feed the jobs channel with "n" jobs.

for p in workers() # Start tasks on the workers to process requests in parallel.
@async remote_do(do_work, p, jobs, results) # Similar to remotecall.
end

@elapsed while n > 0 # Print out results.
job_id, exec_time, where = take!(results)
println("$job_id finished in$(round(exec_time,2)) seconds on worker $where") n = n - 1 end  As you can see, the code is exactly the same as before, except that in this case we had to take care bout how we established communication between processes. ## Shared arrays In the previous examples, the reader might have appreciated that the use of channels, though helpful, might not be proper for all the situations. Sometimes we might be interested in working with usual arrays, but in parallel (e.g.: for a matrix multiplication algorithm). For those situations there exists a type name SharedArray. This turns out to be just an array, but that can be accessed from any process. Furthermore, although the type SharedArray is different from Array, functions that ask for an Array they usually also work with SharedArray. And if not, there exists a function call sdata() that transform a SharedArray to an Array. Let’s see a quick example of how SharedArray works. More information and examples can be found in the official documentation. First, let’s see how we create a SharedArray: SharedArray{T, N}(dims::NTuple, init = false, pids = Int[])  T stands for the type (e.g.: Float64) and N for the dimension of the array (e.g.:$2$for a$2$-dimensional array). Inside brackets, dims collects the number of elements in each dimension (e.g.: ($3$,$2$) for a matrix$3\times 2$), init is the function used to initialize the array, and pids is a vector with the workers that can access to the shared array. These last two arguments are voluntary. For instance, if nothing is specify in pids, all the workers can access to the shared array. Let’s see this with examples. S = SharedArray{Int, 2}((3, 4)) # We do not specify neither init nor pids. S = SharedArray{Int, 2}((3, 4), init = S -> S[Base.localindexes(S)] = 1) S = SharedArray{Int, 2}((4, 4), init = S -> S[collect(1:5:16)] = 1) S[3, 2] = 7 # We can work with a Shared Array as with ordinary ones. S  Recall that when we introduced parallel for loops we said that we could not modify the elements of the array. Now, using shared arrays this is not a problem anymore. Check for instance the following example, in which we are performing a dot product in parallel. x = rand(0:5, 1000); c = rand(0:1, 1000); output = SharedArray{Int, 1}(1000); @parallel (+) for i = 1:1000 output[i] = x[i]*c[i]; end output # We have been able to modify the array.  Notice that for the dimension of the problem a parallel for does not make sense, but it is okay for teaching purposes. Besides SharedArray, there also exist DistributedArray. These arrays also allow to work in parallel, but contrary to SharedArray, they chunk the array so each process has access to a limited portion of it. The advantage of this division is that it avoids the potential error of accessing to the same cell of the array by different workers, so one modify what the previous did (supposing this is not desired). To work with distributed arrays it is necessary to install a package, so we will not work with them in this occasion. Nevertheless, on this link can be found an extensive documentation about how to work with them. ## Last example To conclude this post dedicated to parallel computing, we provide a last example which is a parallel implementation of the$k$-nearest neighbors ($k$nn) algorithm. The performance of the algorithm will be tested using the famous Fisher’s iris dataset, so we will need to install some packages to execute it (DataFrames and RDatasets). Recall that in this dataset there are$3$groups of flowers: setosa, virginica and versicolor, and the goal is to classify new observations (from which we just know their characteristics) within a group. Be aware that this is an example for teaching purposes, so it is probably not the most efficient implementation. With all these ideas in mind, let’s start seeing the code we need. First, we will introduce everything related with the input employed in the functions. addprocs(5); # We have to add the processes before using @everywhere. @everywhere using DataFrames # We will need all the process to use this type. using RDatasets # We read the dataset and pick "npoints" to classify. df_data = dataset("datasets", "iris"); npoints = 10; # In this case we will classify only 10 points. sample = rand(1:size(df_data)[1], npoints); sample = unique(sample); # Just in case there are elements repeated. # We save the characteristics of the points to classify in a new dataset "df_clas" # and delete them from the original one. df_clas = df_data[sample, :]; deleterows!(df_data, sample);  So far we have a dataset named df_data with information to use for the classification process, and another dataset df_clas with the points we want to classify. Now, we are going to create a type Point so we can save all together information about: 1. To which group (setosa, virginica or versicolor) belongs a point p from df_data which is a candidate to be one of nearest neighbors. 2. The distance from the point p to the point x (belonging to df_clas) to classify. @everywhere type Point d::Float64; # distance. g::AbstractString; # group. end  Now, we create some auxiliary functions that we will need. # We need this function to sort Points. @everywhere function getdist(x::Point) return x.d end # We CAN NOT use a dot product because we have DataFrames NOT arrays! # In the function, 'n' is the number of the columns in the dataset. @everywhere function distance(df_data::DataFrame, df_clas::DataFrame, n::Int) d = 0.0; for k = 1:(n-1) d += (df_data[1, k] - df_clas[1, k])^2 end return d; # We return the square of the euclidean distance. end  Now we create the main function to perform the algorithm. In the input of the function, K stands for the number of neighbors to use in the classification process. Notice that, for simplicity, we assume in the function that the group labels are in the last column. @everywhere function knn(df_data::DataFrame, df_clas::DataFrame, K::Int) neighbors = Array{Point, 1}(K); # Where we save the K nearest neighbors. lastelem = ncol(df_data); dist = 0.0; # Variable to save the distances. maxdist = 0.0; # Variable to save the larger distance within array "neighbors". namegroup = ""; vnames = Array{AbstractString, 1}(K); # Vector of names to make the final count. groups = unique(df_data[:, lastelem]); # We collect which are the groups. gcount = zeros(Int, length(groups)); # Number of times each group appears. # To keep this simple, we assume that nrow(df_data) > K. # We select the first K neighbors. for j = 1:K dist = distance(df_data[j, :], df_clas[1, :], lastelem); namegroup = df_data[j, lastelem]; neighbors[j] = Point(dist, namegroup); end # We sort by distance and get the MAXIMUM value sort!(neighbors, by = getdist); maxdist = neighbors[K].d; # We compare with the rest of the points. for j = K:nrow(df_data) dist = distance(df_data[j, :], df_clas[1, :], lastelem); if dist < maxdist neighbors[K].d = dist; neighbors[K].g = df_data[j, lastelem]; sort!(neighbors, by = getdist); maxdist = neighbors[K].d; end end # We classify the point. # First, we get the names of the groups in the array "neighbors". for j = 1:K vnames[j] = neighbors[j].g; end # Second, we count how many times appears each of them. for j = 1:length(groups) gcount[j] = count(s->(s == groups[j]), vnames); end pos = find(gcount .== maximum(gcount)); pos = rand(pos, 1); # We sample vector pos in case of draws. # Third, we return the result. return groups[pos[1]]; end  So far, we can perform the$k$nn algorithm without using parallel computing: result = Array{AbstractString, 1}(nrow(df_clas)); # Vector with the solution. @time for i = 1:nrow(df_clas) result[i] = knn(df_data, df_clas[i, :], 5); # We use K = 5. end result df_clas  Now, let’s implement the parallelization: function knn_parallel(df_data::DataFrame, df_clas::DataFrame, counter::Int, output::Array{AbstractString, 1}) @sync begin for p in workers() @async begin while true idx = counter - 1; counter -= 1; # Why do not we need a Shared Array? if idx <= 0 break; end output[idx] = remotecall_fetch(knn, p, df_data, df_clas[idx,:], 5); end end end end end  # Input required. counter = nrow(df_clas) + 1; output = Array{AbstractString, 1}(nrow(df_clas)); # Function. @time knn_parallel(df_data, df_clas, counter, output) # The result. output df_clas  The first time you run the parallel implementation the execution time is not really good. Try to run it again and answer the following questions: 1. Why the first time you run the parallel implementation is slower than the second one? 2. Is it worth to use the parallel implementation to classify$10\$ points?

3. Can you modify the code so channels are used instead of arrays?

4. Why the use of other functions as remotecall will provide faster running times than remotecall_fetch?

## References

• The official Julia’s documentation on parallel computing.

• This Stack Overflow question has information about the use of the macros @sync and @async. You will also find there an answer to the last question proposed.