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{"documenter":{"julia_version":"1.10.4","generation_timestamp":"2024-08-26T11:01:02","documenter_version":"1.6.0"}} {"documenter":{"julia_version":"1.10.4","generation_timestamp":"2024-08-27T07:47:06","documenter_version":"1.6.0"}}

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</script></article><nav class="docs-footer"><p class="footer-message">Powered by <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> and the <a href="https://julialang.org/">Julia Programming Language</a>.</p></nav></div><div class="modal" id="documenter-settings"><div class="modal-background"></div><div class="modal-card"><header class="modal-card-head"><p class="modal-card-title">Settings</p><button class="delete"></button></header><section class="modal-card-body"><p><label class="label">Theme</label><div class="select"><select id="documenter-themepicker"><option value="auto">Automatic (OS)</option><option value="documenter-light">documenter-light</option><option value="documenter-dark">documenter-dark</option><option value="catppuccin-latte">catppuccin-latte</option><option value="catppuccin-frappe">catppuccin-frappe</option><option value="catppuccin-macchiato">catppuccin-macchiato</option><option value="catppuccin-mocha">catppuccin-mocha</option></select></div></p><hr/><p>This document was generated with <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> version 1.6.0 on <span class="colophon-date" title="Monday 26 August 2024 11:01">Monday 26 August 2024</span>. Using Julia version 1.10.4.</p></section><footer class="modal-card-foot"></footer></div></div></div></body></html> </script></article><nav class="docs-footer"><p class="footer-message">Powered by <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> and the <a href="https://julialang.org/">Julia Programming Language</a>.</p></nav></div><div class="modal" id="documenter-settings"><div class="modal-background"></div><div class="modal-card"><header class="modal-card-head"><p class="modal-card-title">Settings</p><button class="delete"></button></header><section class="modal-card-body"><p><label class="label">Theme</label><div class="select"><select id="documenter-themepicker"><option value="auto">Automatic (OS)</option><option value="documenter-light">documenter-light</option><option value="documenter-dark">documenter-dark</option><option value="catppuccin-latte">catppuccin-latte</option><option value="catppuccin-frappe">catppuccin-frappe</option><option value="catppuccin-macchiato">catppuccin-macchiato</option><option value="catppuccin-mocha">catppuccin-mocha</option></select></div></p><hr/><p>This document was generated with <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> version 1.6.0 on <span class="colophon-date" title="Tuesday 27 August 2024 07:47">Tuesday 27 August 2024</span>. Using Julia version 1.10.4.</p></section><footer class="modal-card-foot"></footer></div></div></div></body></html>

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@ -15,4 +15,4 @@ DataFrames = &quot;a93c6f00-e57d-5684-b7b6-d8193f3e46c0&quot;
MPI = &quot;da04e1cc-30fd-572f-bb4f-1f8673147195&quot;</code></pre><p>Copy the contents of previous code block into a file called <code>Project.toml</code> and place it in an empty folder named <code>newproject</code>. It is important that the file is named <code>Project.toml</code>. You can create a new folder from the REPL with</p><pre><code class="language-julia hljs">julia&gt; mkdir(&quot;newproject&quot;)</code></pre><p>To install all the packages registered in this file you need to activate the folder containing your <code>Project.toml</code> file</p><pre><code class="language-julia hljs">(@v1.10) pkg&gt; activate newproject</code></pre><p>and then <em>instantiating</em> it</p><pre><code class="language-julia hljs">(newproject) pkg&gt; instantiate</code></pre><p>The instantiate command will download and install all listed packages and their dependencies in just one click.</p><h3 id="Getting-help-in-package-mode"><a class="docs-heading-anchor" href="#Getting-help-in-package-mode">Getting help in package mode</a><a id="Getting-help-in-package-mode-1"></a><a class="docs-heading-anchor-permalink" href="#Getting-help-in-package-mode" title="Permalink"></a></h3><p>You can get help about a particular package operator by writing <code>help</code> in front of it</p><pre><code class="language-julia hljs">(@v1.10) pkg&gt; help activate</code></pre><p>You can get an overview of all package commands by typing <code>help</code> alone</p><pre><code class="language-julia hljs">(@v1.10) pkg&gt; help</code></pre><h3 id="Package-operations-in-Julia-code"><a class="docs-heading-anchor" href="#Package-operations-in-Julia-code">Package operations in Julia code</a><a id="Package-operations-in-Julia-code-1"></a><a class="docs-heading-anchor-permalink" href="#Package-operations-in-Julia-code" title="Permalink"></a></h3><p>In some situations it is required to use package commands in Julia code, e.g., to automatize installation and deployment of Julia applications. This can be done using the <code>Pkg</code> package. For instance</p><pre><code class="language-julia hljs">julia&gt; using Pkg MPI = &quot;da04e1cc-30fd-572f-bb4f-1f8673147195&quot;</code></pre><p>Copy the contents of previous code block into a file called <code>Project.toml</code> and place it in an empty folder named <code>newproject</code>. It is important that the file is named <code>Project.toml</code>. You can create a new folder from the REPL with</p><pre><code class="language-julia hljs">julia&gt; mkdir(&quot;newproject&quot;)</code></pre><p>To install all the packages registered in this file you need to activate the folder containing your <code>Project.toml</code> file</p><pre><code class="language-julia hljs">(@v1.10) pkg&gt; activate newproject</code></pre><p>and then <em>instantiating</em> it</p><pre><code class="language-julia hljs">(newproject) pkg&gt; instantiate</code></pre><p>The instantiate command will download and install all listed packages and their dependencies in just one click.</p><h3 id="Getting-help-in-package-mode"><a class="docs-heading-anchor" href="#Getting-help-in-package-mode">Getting help in package mode</a><a id="Getting-help-in-package-mode-1"></a><a class="docs-heading-anchor-permalink" href="#Getting-help-in-package-mode" title="Permalink"></a></h3><p>You can get help about a particular package operator by writing <code>help</code> in front of it</p><pre><code class="language-julia hljs">(@v1.10) pkg&gt; help activate</code></pre><p>You can get an overview of all package commands by typing <code>help</code> alone</p><pre><code class="language-julia hljs">(@v1.10) pkg&gt; help</code></pre><h3 id="Package-operations-in-Julia-code"><a class="docs-heading-anchor" href="#Package-operations-in-Julia-code">Package operations in Julia code</a><a id="Package-operations-in-Julia-code-1"></a><a class="docs-heading-anchor-permalink" href="#Package-operations-in-Julia-code" title="Permalink"></a></h3><p>In some situations it is required to use package commands in Julia code, e.g., to automatize installation and deployment of Julia applications. This can be done using the <code>Pkg</code> package. For instance</p><pre><code class="language-julia hljs">julia&gt; using Pkg
julia&gt; Pkg.status()</code></pre><p>is equivalent to calling <code>status</code> in package mode.</p><pre><code class="language-julia hljs">(@v1.10) pkg&gt; status</code></pre><h3 id="Creating-you-own-package"><a class="docs-heading-anchor" href="#Creating-you-own-package">Creating you own package</a><a id="Creating-you-own-package-1"></a><a class="docs-heading-anchor-permalink" href="#Creating-you-own-package" title="Permalink"></a></h3><p>In many situations is useful to create your own package, for instance, when working with a large code base, when you want to reduce compilation latency using <a href="https://github.com/timholy/Revise.jl"><code>Revise.jl</code></a>, or if you want to eventually <a href="https://github.com/JuliaRegistries/Registrator.jl">register your package</a> and share it with others.</p><p>The simplest way of generating a package (called <code>MyPackage</code>) is as follows. Open Julia go to package mode and type</p><pre><code class="language-julia hljs">(@v1.10) pkg&gt; generate MyPackage</code></pre><p>This will crate a minimal package consisting of a new folder <code>MyPackage</code> with two files:</p><ul><li><code>MyPackage/Project.toml</code>: Project file defining the direct dependencies of your package.</li><li><code>MyPackage/src/MyPackage.jl</code>: Main source file of your package. You can split your code in several files if needed, and include them in the package main file using function <code>include</code>.</li></ul><div class="admonition is-success"><header class="admonition-header">Tip</header><div class="admonition-body"><p>This approach only generates a very minimal package. To create a more sophisticated package skeleton (including unit testing, code coverage, readme file, licence, etc.) use <a href="https://github.com/JuliaCI/PkgTemplates.jl"><code>PkgTemplates.jl</code></a> or <a href="https://github.com/abelsiqueira/BestieTemplate.jl"><code>BestieTemplate.jl</code></a>. The later one is developed in Amsterdam at the <a href="https://www.esciencecenter.nl/">Netherlands eScience Center</a>.</p></div></div><p>You can add dependencies to the package by activating the <code>MyPackage</code> folder in package mode and adding new dependencies as always:</p><pre><code class="language-julia hljs">(@v1.10) pkg&gt; activate MyPackage julia&gt; Pkg.status()</code></pre><p>is equivalent to calling <code>status</code> in package mode.</p><pre><code class="language-julia hljs">(@v1.10) pkg&gt; status</code></pre><h3 id="Creating-you-own-package"><a class="docs-heading-anchor" href="#Creating-you-own-package">Creating you own package</a><a id="Creating-you-own-package-1"></a><a class="docs-heading-anchor-permalink" href="#Creating-you-own-package" title="Permalink"></a></h3><p>In many situations is useful to create your own package, for instance, when working with a large code base, when you want to reduce compilation latency using <a href="https://github.com/timholy/Revise.jl"><code>Revise.jl</code></a>, or if you want to eventually <a href="https://github.com/JuliaRegistries/Registrator.jl">register your package</a> and share it with others.</p><p>The simplest way of generating a package (called <code>MyPackage</code>) is as follows. Open Julia go to package mode and type</p><pre><code class="language-julia hljs">(@v1.10) pkg&gt; generate MyPackage</code></pre><p>This will crate a minimal package consisting of a new folder <code>MyPackage</code> with two files:</p><ul><li><code>MyPackage/Project.toml</code>: Project file defining the direct dependencies of your package.</li><li><code>MyPackage/src/MyPackage.jl</code>: Main source file of your package. You can split your code in several files if needed, and include them in the package main file using function <code>include</code>.</li></ul><div class="admonition is-success"><header class="admonition-header">Tip</header><div class="admonition-body"><p>This approach only generates a very minimal package. To create a more sophisticated package skeleton (including unit testing, code coverage, readme file, licence, etc.) use <a href="https://github.com/JuliaCI/PkgTemplates.jl"><code>PkgTemplates.jl</code></a> or <a href="https://github.com/abelsiqueira/BestieTemplate.jl"><code>BestieTemplate.jl</code></a>. The later one is developed in Amsterdam at the <a href="https://www.esciencecenter.nl/">Netherlands eScience Center</a>.</p></div></div><p>You can add dependencies to the package by activating the <code>MyPackage</code> folder in package mode and adding new dependencies as always:</p><pre><code class="language-julia hljs">(@v1.10) pkg&gt; activate MyPackage
(MyPackage) pkg&gt; add MPI</code></pre><p>This will add MPI to your package dependencies.</p><h3 id="Using-your-own-package"><a class="docs-heading-anchor" href="#Using-your-own-package">Using your own package</a><a id="Using-your-own-package-1"></a><a class="docs-heading-anchor-permalink" href="#Using-your-own-package" title="Permalink"></a></h3><p>To use your package you first need to add it to a package environment of your choice. This is done with <code>develop path/to/the/package/folder</code> in package mode. For instance:</p><pre><code class="language-julia hljs">(@v1.10) pkg&gt; develop MyPackage</code></pre><div class="admonition is-info"><header class="admonition-header">Note</header><div class="admonition-body"><p>You do not need to &quot;develop&quot; your package if you activated the package folder <code>MyPackage</code>.</p></div></div><p>Now, we can go back to standard Julia mode and use it as any other package:</p><pre><code class="language-julia hljs">using MyPackage (MyPackage) pkg&gt; add MPI</code></pre><p>This will add MPI to your package dependencies.</p><h3 id="Using-your-own-package"><a class="docs-heading-anchor" href="#Using-your-own-package">Using your own package</a><a id="Using-your-own-package-1"></a><a class="docs-heading-anchor-permalink" href="#Using-your-own-package" title="Permalink"></a></h3><p>To use your package you first need to add it to a package environment of your choice. This is done with <code>develop path/to/the/package/folder</code> in package mode. For instance:</p><pre><code class="language-julia hljs">(@v1.10) pkg&gt; develop MyPackage</code></pre><div class="admonition is-info"><header class="admonition-header">Note</header><div class="admonition-body"><p>You do not need to &quot;develop&quot; your package if you activated the package folder <code>MyPackage</code>.</p></div></div><p>Now, we can go back to standard Julia mode and use it as any other package:</p><pre><code class="language-julia hljs">using MyPackage
MyPackage.greet()</code></pre><p>Here, we just called the example function defined in <code>MyPackage/src/MyPackage.jl</code>.</p><h2 id="Conclusion"><a class="docs-heading-anchor" href="#Conclusion">Conclusion</a><a id="Conclusion-1"></a><a class="docs-heading-anchor-permalink" href="#Conclusion" title="Permalink"></a></h2><p>We have learned the basics of how to work with Julia, including how to run serial and parallel code, and how to manage, create, and use Julia packages. This knowledge will allow you to follow the course effectively! If you want to further dig into the topics we have covered here, you can take a look at the following links:</p><ul><li><a href="https://docs.julialang.org/en/v1/manual/getting-started/">Julia Manual</a></li><li><a href="https://pkgdocs.julialang.org/v1/getting-started/">Package manager</a></li></ul></article><nav class="docs-footer"><a class="docs-footer-prevpage" href="../">« Home</a><a class="docs-footer-nextpage" href="../julia_basics/">Julia Basics »</a><div class="flexbox-break"></div><p class="footer-message">Powered by <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> and the <a href="https://julialang.org/">Julia Programming Language</a>.</p></nav></div><div class="modal" id="documenter-settings"><div class="modal-background"></div><div class="modal-card"><header class="modal-card-head"><p class="modal-card-title">Settings</p><button class="delete"></button></header><section class="modal-card-body"><p><label class="label">Theme</label><div class="select"><select id="documenter-themepicker"><option value="auto">Automatic (OS)</option><option value="documenter-light">documenter-light</option><option value="documenter-dark">documenter-dark</option><option value="catppuccin-latte">catppuccin-latte</option><option value="catppuccin-frappe">catppuccin-frappe</option><option value="catppuccin-macchiato">catppuccin-macchiato</option><option value="catppuccin-mocha">catppuccin-mocha</option></select></div></p><hr/><p>This document was generated with <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> version 1.6.0 on <span class="colophon-date" title="Monday 26 August 2024 11:01">Monday 26 August 2024</span>. Using Julia version 1.10.4.</p></section><footer class="modal-card-foot"></footer></div></div></div></body></html> MyPackage.greet()</code></pre><p>Here, we just called the example function defined in <code>MyPackage/src/MyPackage.jl</code>.</p><h2 id="Conclusion"><a class="docs-heading-anchor" href="#Conclusion">Conclusion</a><a id="Conclusion-1"></a><a class="docs-heading-anchor-permalink" href="#Conclusion" title="Permalink"></a></h2><p>We have learned the basics of how to work with Julia, including how to run serial and parallel code, and how to manage, create, and use Julia packages. This knowledge will allow you to follow the course effectively! If you want to further dig into the topics we have covered here, you can take a look at the following links:</p><ul><li><a href="https://docs.julialang.org/en/v1/manual/getting-started/">Julia Manual</a></li><li><a href="https://pkgdocs.julialang.org/v1/getting-started/">Package manager</a></li></ul></article><nav class="docs-footer"><a class="docs-footer-prevpage" href="../">« Home</a><a class="docs-footer-nextpage" href="../julia_basics/">Julia Basics »</a><div class="flexbox-break"></div><p class="footer-message">Powered by <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> and the <a href="https://julialang.org/">Julia Programming Language</a>.</p></nav></div><div class="modal" id="documenter-settings"><div class="modal-background"></div><div class="modal-card"><header class="modal-card-head"><p class="modal-card-title">Settings</p><button class="delete"></button></header><section class="modal-card-body"><p><label class="label">Theme</label><div class="select"><select id="documenter-themepicker"><option value="auto">Automatic (OS)</option><option value="documenter-light">documenter-light</option><option value="documenter-dark">documenter-dark</option><option value="catppuccin-latte">catppuccin-latte</option><option value="catppuccin-frappe">catppuccin-frappe</option><option value="catppuccin-macchiato">catppuccin-macchiato</option><option value="catppuccin-mocha">catppuccin-mocha</option></select></div></p><hr/><p>This document was generated with <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> version 1.6.0 on <span class="colophon-date" title="Tuesday 27 August 2024 07:47">Tuesday 27 August 2024</span>. Using Julia version 1.10.4.</p></section><footer class="modal-card-foot"></footer></div></div></div></body></html>

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@ -55,18 +55,10 @@
}, },
{ {
"cell_type": "code", "cell_type": "code",
"execution_count": 8, "execution_count": null,
"id": "2f8ba040", "id": "2f8ba040",
"metadata": {}, "metadata": {},
"outputs": [ "outputs": [],
{
"name": "stdout",
"output_type": "stream",
"text": [
"🥳 Well done! \n"
]
}
],
"source": [ "source": [
"using Distributed\n", "using Distributed\n",
"using BenchmarkTools\n", "using BenchmarkTools\n",
@ -290,7 +282,7 @@
"\n", "\n",
"The matrix-matrix multiplication is an example of [embarrassingly parallel algorithm](https://en.wikipedia.org/wiki/Embarrassingly_parallel). An embarrassingly parallel (also known as trivially parallel) algorithm is an algorithm that can be split in parallel tasks with no (or very few) dependences between them. Such algorithms are typically easy to parallelize.\n", "The matrix-matrix multiplication is an example of [embarrassingly parallel algorithm](https://en.wikipedia.org/wiki/Embarrassingly_parallel). An embarrassingly parallel (also known as trivially parallel) algorithm is an algorithm that can be split in parallel tasks with no (or very few) dependences between them. Such algorithms are typically easy to parallelize.\n",
"\n", "\n",
"Which parts of an algorithm are completely independent and thus trivially parallel? To answer this question, it is useful to inspect the for loops, which are potential sources parallelism. If the iterations are independent of each other, then they are trivial to parallelize. An easy check to find out if the iterations are dependent or not is to change their order (for instance changing `for j in 1:n` by `for j in n:-1:1`, i.e. doing the loop in reverse). If the result changes, then the iterations are not independent.\n", "Which parts of an algorithm are completely independent and thus trivially parallel? To answer this question, it is useful to inspect the for loops, which are potential sources of parallelism. If the iterations are independent of each other, then they are trivial to parallelize. An easy check to find out if the iterations are dependent or not is to change their order (for instance changing `for j in 1:n` by `for j in n:-1:1`, i.e. doing the loop in reverse). If the result changes, then the iterations are not independent.\n",
"\n", "\n",
"Look at the three nested loops in the sequential implementation of the matrix-matrix product:\n", "Look at the three nested loops in the sequential implementation of the matrix-matrix product:\n",
"\n", "\n",
@ -320,7 +312,7 @@
"source": [ "source": [
"### Parallel algorithms\n", "### Parallel algorithms\n",
"\n", "\n",
"Parallelizing the loops over `i` and `j` means that all the entries of matrix C can be potentially computed in parallel. However, *which it the most efficient solution to solve all these entries in parallel in a distributed system?* To find this we will consider different parallelization strategies:\n", "The loops over `i` and `j` are trivially parallel implies that all the entries of matrix C can be potentially computed in parallel. However, *which it the most efficient solution to solve all these entries in parallel in a distributed system?* To find this we will consider different parallelization strategies:\n",
"\n", "\n",
"- Algorithm 1: each worker computes a single entry of C\n", "- Algorithm 1: each worker computes a single entry of C\n",
"- Algorithm 2: each worker computes a single row of C\n", "- Algorithm 2: each worker computes a single row of C\n",
@ -352,7 +344,7 @@
"source": [ "source": [
"### Data dependencies\n", "### Data dependencies\n",
"\n", "\n",
"Moving data through the network is expensive and reducing data movement is one of the key points in a distributed algorithm. To this end, we need to determine which is the minimum data needed by a worker to perform its computations. These are called the *data dependencies*. This will give us later information about the performance of the parallel algorithm.\n", "Moving data through the network is expensive and reducing data movement is one of the key points to design efficient distributed algorithms. To this end, we need to determine which is the minimum data needed by a worker to perform its computations. These are called the *data dependencies*. This will give us later information about the performance of the parallel algorithm.\n",
"\n", "\n",
"In algorithm 1, each worker computes only an entry of the result matrix C." "In algorithm 1, each worker computes only an entry of the result matrix C."
] ]
@ -403,7 +395,7 @@
"Taking into account the data dependencies, the parallel algorithm 1 can be efficiently implemented following these steps from the worker perspective:\n", "Taking into account the data dependencies, the parallel algorithm 1 can be efficiently implemented following these steps from the worker perspective:\n",
"\n", "\n",
"1. The worker receives the data dependencies, i.e., the corresponding row A[i,:] and column B[:,j] from the master process\n", "1. The worker receives the data dependencies, i.e., the corresponding row A[i,:] and column B[:,j] from the master process\n",
"2. The worker computes the dot product of A[i,:] and B[:,j]\n", "2. The worker computes the dot product of A[i,:] and B[:,j] locally\n",
"3. The worker sends back the result of C[i,j] to the master process" "3. The worker sends back the result of C[i,j] to the master process"
] ]
}, },
@ -423,7 +415,7 @@
"id": "9d22ccea", "id": "9d22ccea",
"metadata": {}, "metadata": {},
"source": [ "source": [
"A possible implementation of this algorithm in Julia is as follows:" "A possible implementation of this algorithm in Julia is as follows. Try to understand why `@sync` and `@async` are needed here."
] ]
}, },
{ {
@ -483,7 +475,8 @@
"A = rand(N,N)\n", "A = rand(N,N)\n",
"B = rand(N,N)\n", "B = rand(N,N)\n",
"C = similar(A)\n", "C = similar(A)\n",
"@test matmul_dist_1!(C,A,B) ≈ A*B" "matmul_dist_1!(C,A,B)\n",
"@test C ≈ A*B"
] ]
}, },
{ {

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<div class="highlight hl-julia"><pre><span></span><span class="k">using</span><span class="w"> </span><span class="n">Distributed</span> <div class="highlight hl-julia"><pre><span></span><span class="k">using</span><span class="w"> </span><span class="n">Distributed</span>
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<pre>🥳 Well done!
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@ -7859,7 +7846,7 @@ d) O(N³)</code></pre>
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<h2 id="Where-can-we-exploit-parallelism?">Where can we exploit parallelism?<a class="anchor-link" href="#Where-can-we-exploit-parallelism?"></a></h2><p>The matrix-matrix multiplication is an example of <a href="https://en.wikipedia.org/wiki/Embarrassingly_parallel">embarrassingly parallel algorithm</a>. An embarrassingly parallel (also known as trivially parallel) algorithm is an algorithm that can be split in parallel tasks with no (or very few) dependences between them. Such algorithms are typically easy to parallelize.</p> <h2 id="Where-can-we-exploit-parallelism?">Where can we exploit parallelism?<a class="anchor-link" href="#Where-can-we-exploit-parallelism?"></a></h2><p>The matrix-matrix multiplication is an example of <a href="https://en.wikipedia.org/wiki/Embarrassingly_parallel">embarrassingly parallel algorithm</a>. An embarrassingly parallel (also known as trivially parallel) algorithm is an algorithm that can be split in parallel tasks with no (or very few) dependences between them. Such algorithms are typically easy to parallelize.</p>
<p>Which parts of an algorithm are completely independent and thus trivially parallel? To answer this question, it is useful to inspect the for loops, which are potential sources parallelism. If the iterations are independent of each other, then they are trivial to parallelize. An easy check to find out if the iterations are dependent or not is to change their order (for instance changing <code>for j in 1:n</code> by <code>for j in n:-1:1</code>, i.e. doing the loop in reverse). If the result changes, then the iterations are not independent.</p> <p>Which parts of an algorithm are completely independent and thus trivially parallel? To answer this question, it is useful to inspect the for loops, which are potential sources of parallelism. If the iterations are independent of each other, then they are trivial to parallelize. An easy check to find out if the iterations are dependent or not is to change their order (for instance changing <code>for j in 1:n</code> by <code>for j in n:-1:1</code>, i.e. doing the loop in reverse). If the result changes, then the iterations are not independent.</p>
<p>Look at the three nested loops in the sequential implementation of the matrix-matrix product:</p> <p>Look at the three nested loops in the sequential implementation of the matrix-matrix product:</p>
<div class="highlight"><pre><span></span><span class="k">for</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="mi">1</span><span class="o">:</span><span class="n">n</span> <div class="highlight"><pre><span></span><span class="k">for</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="mi">1</span><span class="o">:</span><span class="n">n</span>
<span class="w"> </span><span class="k">for</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="mi">1</span><span class="o">:</span><span class="n">m</span> <span class="w"> </span><span class="k">for</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="mi">1</span><span class="o">:</span><span class="n">m</span>
@ -7886,7 +7873,7 @@ d) O(N³)</code></pre>
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<h3 id="Parallel-algorithms">Parallel algorithms<a class="anchor-link" href="#Parallel-algorithms"></a></h3><p>Parallelizing the loops over <code>i</code> and <code>j</code> means that all the entries of matrix C can be potentially computed in parallel. However, <em>which it the most efficient solution to solve all these entries in parallel in a distributed system?</em> To find this we will consider different parallelization strategies:</p> <h3 id="Parallel-algorithms">Parallel algorithms<a class="anchor-link" href="#Parallel-algorithms"></a></h3><p>The loops over <code>i</code> and <code>j</code> are trivially parallel implies that all the entries of matrix C can be potentially computed in parallel. However, <em>which it the most efficient solution to solve all these entries in parallel in a distributed system?</em> To find this we will consider different parallelization strategies:</p>
<ul> <ul>
<li>Algorithm 1: each worker computes a single entry of C</li> <li>Algorithm 1: each worker computes a single entry of C</li>
<li>Algorithm 2: each worker computes a single row of C</li> <li>Algorithm 2: each worker computes a single row of C</li>
@ -7926,7 +7913,7 @@ d) O(N³)</code></pre>
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<h3 id="Data-dependencies">Data dependencies<a class="anchor-link" href="#Data-dependencies"></a></h3><p>Moving data through the network is expensive and reducing data movement is one of the key points in a distributed algorithm. To this end, we need to determine which is the minimum data needed by a worker to perform its computations. These are called the <em>data dependencies</em>. This will give us later information about the performance of the parallel algorithm.</p> <h3 id="Data-dependencies">Data dependencies<a class="anchor-link" href="#Data-dependencies"></a></h3><p>Moving data through the network is expensive and reducing data movement is one of the key points to design efficient distributed algorithms. To this end, we need to determine which is the minimum data needed by a worker to perform its computations. These are called the <em>data dependencies</em>. This will give us later information about the performance of the parallel algorithm.</p>
<p>In algorithm 1, each worker computes only an entry of the result matrix C.</p> <p>In algorithm 1, each worker computes only an entry of the result matrix C.</p>
</div> </div>
</div> </div>
@ -7986,7 +7973,7 @@ d) row A[i,:] and the whole matrix B</code></pre>
<h3 id="Implementation">Implementation<a class="anchor-link" href="#Implementation"></a></h3><p>Taking into account the data dependencies, the parallel algorithm 1 can be efficiently implemented following these steps from the worker perspective:</p> <h3 id="Implementation">Implementation<a class="anchor-link" href="#Implementation"></a></h3><p>Taking into account the data dependencies, the parallel algorithm 1 can be efficiently implemented following these steps from the worker perspective:</p>
<ol> <ol>
<li>The worker receives the data dependencies, i.e., the corresponding row A[i,:] and column B[:,j] from the master process</li> <li>The worker receives the data dependencies, i.e., the corresponding row A[i,:] and column B[:,j] from the master process</li>
<li>The worker computes the dot product of A[i,:] and B[:,j]</li> <li>The worker computes the dot product of A[i,:] and B[:,j] locally</li>
<li>The worker sends back the result of C[i,j] to the master process</li> <li>The worker sends back the result of C[i,j] to the master process</li>
</ol> </ol>
</div> </div>
@ -8012,7 +7999,7 @@ d) row A[i,:] and the whole matrix B</code></pre>
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<p>A possible implementation of this algorithm in Julia is as follows:</p> <p>A possible implementation of this algorithm in Julia is as follows. Try to understand why <code>@sync</code> and <code>@async</code> are needed here.</p>
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@ -8081,7 +8068,8 @@ d) row A[i,:] and the whole matrix B</code></pre>
<span class="n">A</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">rand</span><span class="p">(</span><span class="n">N</span><span class="p">,</span><span class="n">N</span><span class="p">)</span> <span class="n">A</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">rand</span><span class="p">(</span><span class="n">N</span><span class="p">,</span><span class="n">N</span><span class="p">)</span>
<span class="n">B</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">rand</span><span class="p">(</span><span class="n">N</span><span class="p">,</span><span class="n">N</span><span class="p">)</span> <span class="n">B</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">rand</span><span class="p">(</span><span class="n">N</span><span class="p">,</span><span class="n">N</span><span class="p">)</span>
<span class="n">C</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">similar</span><span class="p">(</span><span class="n">A</span><span class="p">)</span> <span class="n">C</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">similar</span><span class="p">(</span><span class="n">A</span><span class="p">)</span>
<span class="nd">@test</span><span class="w"> </span><span class="n">matmul_dist_1!</span><span class="p">(</span><span class="n">C</span><span class="p">,</span><span class="n">A</span><span class="p">,</span><span class="n">B</span><span class="p">)</span><span class="w"> </span><span class="o">≈</span><span class="w"> </span><span class="n">A</span><span class="o">*</span><span class="n">B</span> <span class="n">matmul_dist_1!</span><span class="p">(</span><span class="n">C</span><span class="p">,</span><span class="n">A</span><span class="p">,</span><span class="n">B</span><span class="p">)</span>
<span class="nd">@test</span><span class="w"> </span><span class="n">C</span><span class="w"> </span><span class="o">≈</span><span class="w"> </span><span class="n">A</span><span class="o">*</span><span class="n">B</span>
</pre></div> </pre></div>
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@ -14,4 +14,4 @@
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@ -172,4 +172,4 @@ end</code></pre><h2 id="Jacobi-method"><a class="docs-heading-anchor" href="#Jac
end end
end end
return u_all return u_all
end</code></pre></article><nav class="docs-footer"><a class="docs-footer-prevpage" href="../jacobi_method/">« Jacobi method</a><div class="flexbox-break"></div><p class="footer-message">Powered by <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> and the <a href="https://julialang.org/">Julia Programming Language</a>.</p></nav></div><div class="modal" id="documenter-settings"><div class="modal-background"></div><div class="modal-card"><header class="modal-card-head"><p class="modal-card-title">Settings</p><button class="delete"></button></header><section class="modal-card-body"><p><label class="label">Theme</label><div class="select"><select id="documenter-themepicker"><option value="auto">Automatic (OS)</option><option value="documenter-light">documenter-light</option><option value="documenter-dark">documenter-dark</option><option value="catppuccin-latte">catppuccin-latte</option><option value="catppuccin-frappe">catppuccin-frappe</option><option value="catppuccin-macchiato">catppuccin-macchiato</option><option value="catppuccin-mocha">catppuccin-mocha</option></select></div></p><hr/><p>This document was generated with <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> version 1.6.0 on <span class="colophon-date" title="Monday 26 August 2024 11:01">Monday 26 August 2024</span>. Using Julia version 1.10.4.</p></section><footer class="modal-card-foot"></footer></div></div></div></body></html> end</code></pre></article><nav class="docs-footer"><a class="docs-footer-prevpage" href="../jacobi_method/">« Jacobi method</a><div class="flexbox-break"></div><p class="footer-message">Powered by <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> and the <a href="https://julialang.org/">Julia Programming Language</a>.</p></nav></div><div class="modal" id="documenter-settings"><div class="modal-background"></div><div class="modal-card"><header class="modal-card-head"><p class="modal-card-title">Settings</p><button class="delete"></button></header><section class="modal-card-body"><p><label class="label">Theme</label><div class="select"><select id="documenter-themepicker"><option value="auto">Automatic (OS)</option><option value="documenter-light">documenter-light</option><option value="documenter-dark">documenter-dark</option><option value="catppuccin-latte">catppuccin-latte</option><option value="catppuccin-frappe">catppuccin-frappe</option><option value="catppuccin-macchiato">catppuccin-macchiato</option><option value="catppuccin-mocha">catppuccin-mocha</option></select></div></p><hr/><p>This document was generated with <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> version 1.6.0 on <span class="colophon-date" title="Tuesday 27 August 2024 07:47">Tuesday 27 August 2024</span>. Using Julia version 1.10.4.</p></section><footer class="modal-card-foot"></footer></div></div></div></body></html>

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