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{"documenter":{"julia_version":"1.9.3","generation_timestamp":"2023-09-19T17:13:09","documenter_version":"1.0.1"}} {"documenter":{"julia_version":"1.9.3","generation_timestamp":"2023-09-20T07:14:16","documenter_version":"1.0.1"}}

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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="documenter-light">documenter-light</option><option value="documenter-dark">documenter-dark</option><option value="auto">Automatic (OS)</option></select></div></p><hr/><p>This document was generated with <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> version 1.0.1 on <span class="colophon-date" title="Tuesday 19 September 2023 17:02">Tuesday 19 September 2023</span>. Using Julia version 1.9.3.</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="documenter-light">documenter-light</option><option value="documenter-dark">documenter-dark</option><option value="auto">Automatic (OS)</option></select></div></p><hr/><p>This document was generated with <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> version 1.0.1 on <span class="colophon-date" title="Wednesday 20 September 2023 07:14">Wednesday 20 September 2023</span>. Using Julia version 1.9.3.</p></section><footer class="modal-card-foot"></footer></div></div></div></body></html>

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var myIframe = document.getElementById("notebook"); var myIframe = document.getElementById("notebook");
iFrameResize({log:true}, myIframe); iFrameResize({log:true}, myIframe);
}); });
</script></article><nav class="docs-footer"><a class="docs-footer-prevpage" href="../jacobi_method/">« Jacobi method</a><a class="docs-footer-nextpage" href="../solutions_for_all_notebooks/">Solutions »</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="documenter-light">documenter-light</option><option value="documenter-dark">documenter-dark</option><option value="auto">Automatic (OS)</option></select></div></p><hr/><p>This document was generated with <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> version 1.0.1 on <span class="colophon-date" title="Tuesday 19 September 2023 17:11">Tuesday 19 September 2023</span>. Using Julia version 1.9.3.</p></section><footer class="modal-card-foot"></footer></div></div></div></body></html> </script></article><nav class="docs-footer"><a class="docs-footer-prevpage" href="../jacobi_method/">« Jacobi method</a><a class="docs-footer-nextpage" href="../solutions_for_all_notebooks/">Solutions »</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="documenter-light">documenter-light</option><option value="documenter-dark">documenter-dark</option><option value="auto">Automatic (OS)</option></select></div></p><hr/><p>This document was generated with <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> version 1.0.1 on <span class="colophon-date" title="Wednesday 20 September 2023 07:14">Wednesday 20 September 2023</span>. Using Julia version 1.9.3.</p></section><footer class="modal-card-foot"></footer></div></div></div></body></html>

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@ -13,4 +13,4 @@ julia&gt; DataFrame(a=[1,2],b=[3,4])</code></pre><p>You should get an error or a
BenchmarkTools = &quot;6e4b80f9-dd63-53aa-95a3-0cdb28fa8baf&quot; BenchmarkTools = &quot;6e4b80f9-dd63-53aa-95a3-0cdb28fa8baf&quot;
DataFrames = &quot;a93c6f00-e57d-5684-b7b6-d8193f3e46c0&quot; 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">(@v1.8) 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">(@v1.8) 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">(@v1.8) 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">(@v1.8) 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">(@v1.8) 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">(@v1.8) 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">(@v1.8) pkg&gt; status</code></pre><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. 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="documenter-light">documenter-light</option><option value="documenter-dark">documenter-dark</option><option value="auto">Automatic (OS)</option></select></div></p><hr/><p>This document was generated with <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> version 1.0.1 on <span class="colophon-date" title="Tuesday 19 September 2023 17:00">Tuesday 19 September 2023</span>. Using Julia version 1.9.3.</p></section><footer class="modal-card-foot"></footer></div></div></div></body></html> julia&gt; Pkg.status()</code></pre><p>is equivalent to calling <code>status</code> in package mode.</p><pre><code class="language-julia hljs">(@v1.8) pkg&gt; status</code></pre><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. 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="documenter-light">documenter-light</option><option value="documenter-dark">documenter-dark</option><option value="auto">Automatic (OS)</option></select></div></p><hr/><p>This document was generated with <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> version 1.0.1 on <span class="colophon-date" title="Wednesday 20 September 2023 07:14">Wednesday 20 September 2023</span>. Using Julia version 1.9.3.</p></section><footer class="modal-card-foot"></footer></div></div></div></body></html>

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</script></article><nav class="docs-footer"><a class="docs-footer-prevpage" href="../matrix_matrix/">« Matrix-matrix multiplication</a><a class="docs-footer-nextpage" href="../asp/">All pairs of shortest paths »</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="documenter-light">documenter-light</option><option value="documenter-dark">documenter-dark</option><option value="auto">Automatic (OS)</option></select></div></p><hr/><p>This document was generated with <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> version 1.0.1 on <span class="colophon-date" title="Tuesday 19 September 2023 17:07">Tuesday 19 September 2023</span>. Using Julia version 1.9.3.</p></section><footer class="modal-card-foot"></footer></div></div></div></body></html> </script></article><nav class="docs-footer"><a class="docs-footer-prevpage" href="../matrix_matrix/">« Matrix-matrix multiplication</a><a class="docs-footer-nextpage" href="../asp/">All pairs of shortest paths »</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="documenter-light">documenter-light</option><option value="documenter-dark">documenter-dark</option><option value="auto">Automatic (OS)</option></select></div></p><hr/><p>This document was generated with <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> version 1.0.1 on <span class="colophon-date" title="Wednesday 20 September 2023 07:14">Wednesday 20 September 2023</span>. Using Julia version 1.9.3.</p></section><footer class="modal-card-foot"></footer></div></div></div></body></html>

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@ -14,4 +14,4 @@
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iFrameResize({log:true}, myIframe); iFrameResize({log:true}, myIframe);
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@ -14,4 +14,4 @@
var myIframe = document.getElementById("notebook"); var myIframe = document.getElementById("notebook");
iFrameResize({log:true}, myIframe); iFrameResize({log:true}, myIframe);
}); });
</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="documenter-light">documenter-light</option><option value="documenter-dark">documenter-dark</option><option value="auto">Automatic (OS)</option></select></div></p><hr/><p>This document was generated with <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> version 1.0.1 on <span class="colophon-date" title="Tuesday 19 September 2023 17:04">Tuesday 19 September 2023</span>. Using Julia version 1.9.3.</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="documenter-light">documenter-light</option><option value="documenter-dark">documenter-dark</option><option value="auto">Automatic (OS)</option></select></div></p><hr/><p>This document was generated with <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> version 1.0.1 on <span class="colophon-date" title="Wednesday 20 September 2023 07:14">Wednesday 20 September 2023</span>. Using Julia version 1.9.3.</p></section><footer class="modal-card-foot"></footer></div></div></div></body></html>

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@ -72,9 +72,10 @@
" \"It's not correct. Keep trying! 💪\"\n", " \"It's not correct. Keep trying! 💪\"\n",
" end |> println\n", " end |> println\n",
"end\n", "end\n",
"alg_0_comp_check(answer) = answer_checker(answer, \"d\")\n",
"alg_1_deps_check(answer) = answer_checker(answer,\"b\")\n", "alg_1_deps_check(answer) = answer_checker(answer,\"b\")\n",
"alg_1_comm_overhead_check(answer) = answer_checker(answer, \"c\")\n", "alg_1_comm_overhead_check(answer) = answer_checker(answer, \"b\")\n",
"alg_1_comp_check(answer) = answer_checker(answer, \"a\")\n", "alg_1_comp_check(answer) = answer_checker(answer, \"b\")\n",
"alg_2_complex_check(answer) = answer_checker(answer, \"b\")\n", "alg_2_complex_check(answer) = answer_checker(answer, \"b\")\n",
"alg_2_deps_check(answer) = answer_checker(answer,\"d\")\n", "alg_2_deps_check(answer) = answer_checker(answer,\"d\")\n",
"alg_3_deps_check(answer) = answer_checker(answer, \"c\")\n", "alg_3_deps_check(answer) = answer_checker(answer, \"c\")\n",
@ -88,7 +89,7 @@
"source": [ "source": [
"## Problem Statement\n", "## Problem Statement\n",
"\n", "\n",
"Let us consider the (dense) matrix-matrix product `C=A*B`." "Given $A$ and $B$ two $N$-by-$N$ matrices, compute the matrix-matrix product $C=AB$. Compute it in parallel and efficiently."
] ]
}, },
{ {
@ -157,7 +158,7 @@
"source": [ "source": [
"## Serial implementation\n", "## Serial implementation\n",
"\n", "\n",
"We start by considering the (naive) sequential algorithm:" "We start by considering the (naive) sequential algorithm, which is based on the math definition of the matrix-matrix product $C_{ij} = \\sum_k A_{ik} B_{kj}$"
] ]
}, },
{ {
@ -188,6 +189,30 @@
"end" "end"
] ]
}, },
{
"cell_type": "markdown",
"id": "e3b86457",
"metadata": {},
"source": [
"Run next cell to test the implementation."
]
},
{
"cell_type": "code",
"execution_count": null,
"id": "c5caf799",
"metadata": {},
"outputs": [],
"source": [
"using Test\n",
"N = 10\n",
"A = rand(N,N)\n",
"B = rand(N,N)\n",
"C = similar(A)\n",
"matmul_seq!(C,A,B)\n",
"@test C ≈ A*B"
]
},
{ {
"cell_type": "markdown", "cell_type": "markdown",
"id": "f967d2ea", "id": "f967d2ea",
@ -216,6 +241,32 @@
"@btime mul!(C,A,B);" "@btime mul!(C,A,B);"
] ]
}, },
{
"cell_type": "markdown",
"id": "0ca2fbd4",
"metadata": {},
"source": [
"<div class=\"alert alert-block alert-success\">\n",
"<b>Question:</b> Which is the complexity (number of operations) of the serial algorithm? Assume that all matrices are $N$-by-$N$ matrices. \n",
"</div>\n",
"\n",
" a) O(1)\n",
" b) O(N)\n",
" c) O(N²)\n",
" d) O(N³)"
]
},
{
"cell_type": "code",
"execution_count": null,
"id": "078e974e",
"metadata": {},
"outputs": [],
"source": [
"answer = \"x\" # replace x with a, b, c, or d \n",
"alg_0_comp_check(answer)"
]
},
{ {
"cell_type": "markdown", "cell_type": "markdown",
"id": "0eedd28a", "id": "0eedd28a",
@ -489,10 +540,10 @@
"<b>Question:</b> How many scalars are communicated from and to a worker? Assume that matrices A, B, and C are N by N matrices.\n", "<b>Question:</b> How many scalars are communicated from and to a worker? Assume that matrices A, B, and C are N by N matrices.\n",
"</div>\n", "</div>\n",
"\n", "\n",
" a) 3N\n", " a) O(1)\n",
" b) 2N + 2\n", " b) O(N)\n",
" c) 2N + 1\n", " c) O(N²)\n",
" d) N² + 1" " d) O(N³)"
] ]
}, },
{ {
@ -515,9 +566,10 @@
"<b>Question:</b> How many operations are done in a worker? \n", "<b>Question:</b> How many operations are done in a worker? \n",
"</div>\n", "</div>\n",
"\n", "\n",
" a) O(N)\n", " a) O(1)\n",
" b) O(N²)\n", " b) O(N)\n",
" c) O(N³)" " c) O(N²)\n",
" d) O(N³)"
] ]
}, },
{ {
@ -905,9 +957,9 @@
"\n", "\n",
"| Algorithm | Parallelism <br>(#workers) | Communication <br>per worker | Computation <br>per worker | Ratio communication/<br>computation |\n", "| Algorithm | Parallelism <br>(#workers) | Communication <br>per worker | Computation <br>per worker | Ratio communication/<br>computation |\n",
"|---|---|---|---|---|\n", "|---|---|---|---|---|\n",
"| 1 | N² | 2N + 1 | N | O(1) |\n", "| 1 | N² | O(N) | O(N) | O(1) |\n",
"| 2 | N | 2N + N² | N² | O(1) |\n", "| 2 | N | O(N²) | O(N²) | O(1) |\n",
"| 3 | P | N² + 2N²/P | N³/P | O(P/N) |\n", "| 3 | P | O(N²) | O(N³/P) | O(P/N) |\n",
"\n", "\n",
"\n", "\n",
"- Matrix-matrix multiplication is trivially parallelizable (all entries in the result matrix can be computed in parallel, at least in theory)\n", "- Matrix-matrix multiplication is trivially parallelizable (all entries in the result matrix can be computed in parallel, at least in theory)\n",

2
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@ -14,4 +14,4 @@
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iFrameResize({log:true}, myIframe); iFrameResize({log:true}, myIframe);
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99
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@ -7564,9 +7564,10 @@ a.anchor-link {
<span class="w"> </span><span class="s">"It's not correct. Keep trying! 💪"</span> <span class="w"> </span><span class="s">"It's not correct. Keep trying! 💪"</span>
<span class="w"> </span><span class="k">end</span><span class="w"> </span><span class="o">|&gt;</span><span class="w"> </span><span class="n">println</span> <span class="w"> </span><span class="k">end</span><span class="w"> </span><span class="o">|&gt;</span><span class="w"> </span><span class="n">println</span>
<span class="k">end</span> <span class="k">end</span>
<span class="n">alg_0_comp_check</span><span class="p">(</span><span class="n">answer</span><span class="p">)</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">answer_checker</span><span class="p">(</span><span class="n">answer</span><span class="p">,</span><span class="w"> </span><span class="s">"d"</span><span class="p">)</span>
<span class="n">alg_1_deps_check</span><span class="p">(</span><span class="n">answer</span><span class="p">)</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">answer_checker</span><span class="p">(</span><span class="n">answer</span><span class="p">,</span><span class="s">"b"</span><span class="p">)</span> <span class="n">alg_1_deps_check</span><span class="p">(</span><span class="n">answer</span><span class="p">)</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">answer_checker</span><span class="p">(</span><span class="n">answer</span><span class="p">,</span><span class="s">"b"</span><span class="p">)</span>
<span class="n">alg_1_comm_overhead_check</span><span class="p">(</span><span class="n">answer</span><span class="p">)</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">answer_checker</span><span class="p">(</span><span class="n">answer</span><span class="p">,</span><span class="w"> </span><span class="s">"c"</span><span class="p">)</span> <span class="n">alg_1_comm_overhead_check</span><span class="p">(</span><span class="n">answer</span><span class="p">)</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">answer_checker</span><span class="p">(</span><span class="n">answer</span><span class="p">,</span><span class="w"> </span><span class="s">"b"</span><span class="p">)</span>
<span class="n">alg_1_comp_check</span><span class="p">(</span><span class="n">answer</span><span class="p">)</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">answer_checker</span><span class="p">(</span><span class="n">answer</span><span class="p">,</span><span class="w"> </span><span class="s">"a"</span><span class="p">)</span> <span class="n">alg_1_comp_check</span><span class="p">(</span><span class="n">answer</span><span class="p">)</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">answer_checker</span><span class="p">(</span><span class="n">answer</span><span class="p">,</span><span class="w"> </span><span class="s">"b"</span><span class="p">)</span>
<span class="n">alg_2_complex_check</span><span class="p">(</span><span class="n">answer</span><span class="p">)</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">answer_checker</span><span class="p">(</span><span class="n">answer</span><span class="p">,</span><span class="w"> </span><span class="s">"b"</span><span class="p">)</span> <span class="n">alg_2_complex_check</span><span class="p">(</span><span class="n">answer</span><span class="p">)</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">answer_checker</span><span class="p">(</span><span class="n">answer</span><span class="p">,</span><span class="w"> </span><span class="s">"b"</span><span class="p">)</span>
<span class="n">alg_2_deps_check</span><span class="p">(</span><span class="n">answer</span><span class="p">)</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">answer_checker</span><span class="p">(</span><span class="n">answer</span><span class="p">,</span><span class="s">"d"</span><span class="p">)</span> <span class="n">alg_2_deps_check</span><span class="p">(</span><span class="n">answer</span><span class="p">)</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">answer_checker</span><span class="p">(</span><span class="n">answer</span><span class="p">,</span><span class="s">"d"</span><span class="p">)</span>
<span class="n">alg_3_deps_check</span><span class="p">(</span><span class="n">answer</span><span class="p">)</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">answer_checker</span><span class="p">(</span><span class="n">answer</span><span class="p">,</span><span class="w"> </span><span class="s">"c"</span><span class="p">)</span> <span class="n">alg_3_deps_check</span><span class="p">(</span><span class="n">answer</span><span class="p">)</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">answer_checker</span><span class="p">(</span><span class="n">answer</span><span class="p">,</span><span class="w"> </span><span class="s">"c"</span><span class="p">)</span>
@ -7583,7 +7584,7 @@ a.anchor-link {
</div> </div>
<div class="jp-InputArea jp-Cell-inputArea"><div class="jp-InputPrompt jp-InputArea-prompt"> <div class="jp-InputArea jp-Cell-inputArea"><div class="jp-InputPrompt jp-InputArea-prompt">
</div><div class="jp-RenderedHTMLCommon jp-RenderedMarkdown jp-MarkdownOutput" data-mime-type="text/markdown"> </div><div class="jp-RenderedHTMLCommon jp-RenderedMarkdown jp-MarkdownOutput" data-mime-type="text/markdown">
<h2 id="Problem-Statement">Problem Statement<a class="anchor-link" href="#Problem-Statement"></a></h2><p>Let us consider the (dense) matrix-matrix product <code>C=A*B</code>.</p> <h2 id="Problem-Statement">Problem Statement<a class="anchor-link" href="#Problem-Statement"></a></h2><p>Given $A$ and $B$ two $N$-by-$N$ matrices, compute the matrix-matrix product $C=AB$. Compute it in parallel and efficiently.</p>
</div> </div>
</div> </div>
</div> </div>
@ -7666,7 +7667,7 @@ a.anchor-link {
</div> </div>
<div class="jp-InputArea jp-Cell-inputArea"><div class="jp-InputPrompt jp-InputArea-prompt"> <div class="jp-InputArea jp-Cell-inputArea"><div class="jp-InputPrompt jp-InputArea-prompt">
</div><div class="jp-RenderedHTMLCommon jp-RenderedMarkdown jp-MarkdownOutput" data-mime-type="text/markdown"> </div><div class="jp-RenderedHTMLCommon jp-RenderedMarkdown jp-MarkdownOutput" data-mime-type="text/markdown">
<h2 id="Serial-implementation">Serial implementation<a class="anchor-link" href="#Serial-implementation"></a></h2><p>We start by considering the (naive) sequential algorithm:</p> <h2 id="Serial-implementation">Serial implementation<a class="anchor-link" href="#Serial-implementation"></a></h2><p>We start by considering the (naive) sequential algorithm, which is based on the math definition of the matrix-matrix product $C_{ij} = \sum_k A_{ik} B_{kj}$</p>
</div> </div>
</div> </div>
</div> </div>
@ -7703,6 +7704,37 @@ a.anchor-link {
</div> </div>
</div> </div>
</div> </div>
<div class="jp-Cell jp-MarkdownCell jp-Notebook-cell" id="cell-id=e3b86457">
<div class="jp-Cell-inputWrapper" tabindex="0">
<div class="jp-Collapser jp-InputCollapser jp-Cell-inputCollapser">
</div>
<div class="jp-InputArea jp-Cell-inputArea"><div class="jp-InputPrompt jp-InputArea-prompt">
</div><div class="jp-RenderedHTMLCommon jp-RenderedMarkdown jp-MarkdownOutput" data-mime-type="text/markdown">
<p>Run next cell to test the implementation.</p>
</div>
</div>
</div>
</div><div class="jp-Cell jp-CodeCell jp-Notebook-cell jp-mod-noOutputs" id="cell-id=c5caf799">
<div class="jp-Cell-inputWrapper" tabindex="0">
<div class="jp-Collapser jp-InputCollapser jp-Cell-inputCollapser">
</div>
<div class="jp-InputArea jp-Cell-inputArea">
<div class="jp-InputPrompt jp-InputArea-prompt">In [ ]:</div>
<div class="jp-CodeMirrorEditor jp-Editor jp-InputArea-editor" data-type="inline">
<div class="cm-editor cm-s-jupyter">
<div class="highlight hl-julia"><pre><span></span><span class="k">using</span><span class="w"> </span><span class="n">Test</span>
<span class="n">N</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">10</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">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">matmul_seq!</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>
</div>
</div>
</div>
</div>
</div>
<div class="jp-Cell jp-MarkdownCell jp-Notebook-cell" id="cell-id=f967d2ea"> <div class="jp-Cell jp-MarkdownCell jp-Notebook-cell" id="cell-id=f967d2ea">
<div class="jp-Cell-inputWrapper" tabindex="0"> <div class="jp-Cell-inputWrapper" tabindex="0">
<div class="jp-Collapser jp-InputCollapser jp-Cell-inputCollapser"> <div class="jp-Collapser jp-InputCollapser jp-Cell-inputCollapser">
@ -7737,6 +7769,38 @@ a.anchor-link {
</div> </div>
</div> </div>
</div> </div>
<div class="jp-Cell jp-MarkdownCell jp-Notebook-cell" id="cell-id=0ca2fbd4">
<div class="jp-Cell-inputWrapper" tabindex="0">
<div class="jp-Collapser jp-InputCollapser jp-Cell-inputCollapser">
</div>
<div class="jp-InputArea jp-Cell-inputArea"><div class="jp-InputPrompt jp-InputArea-prompt">
</div><div class="jp-RenderedHTMLCommon jp-RenderedMarkdown jp-MarkdownOutput" data-mime-type="text/markdown">
<div class="alert alert-block alert-success">
<b>Question:</b> Which is the complexity (number of operations) of the serial algorithm? Assume that all matrices are $N$-by-$N$ matrices.
</div>
<pre><code>a) O(1)
b) O(N)
c) O(N²)
d) O(N³)</code></pre>
</div>
</div>
</div>
</div><div class="jp-Cell jp-CodeCell jp-Notebook-cell jp-mod-noOutputs" id="cell-id=078e974e">
<div class="jp-Cell-inputWrapper" tabindex="0">
<div class="jp-Collapser jp-InputCollapser jp-Cell-inputCollapser">
</div>
<div class="jp-InputArea jp-Cell-inputArea">
<div class="jp-InputPrompt jp-InputArea-prompt">In [ ]:</div>
<div class="jp-CodeMirrorEditor jp-Editor jp-InputArea-editor" data-type="inline">
<div class="cm-editor cm-s-jupyter">
<div class="highlight hl-julia"><pre><span></span><span class="n">answer</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="s">"x"</span><span class="w"> </span><span class="c"># replace x with a, b, c, or d </span>
<span class="n">alg_0_comp_check</span><span class="p">(</span><span class="n">answer</span><span class="p">)</span>
</pre></div>
</div>
</div>
</div>
</div>
</div>
<div class="jp-Cell jp-MarkdownCell jp-Notebook-cell" id="cell-id=0eedd28a"> <div class="jp-Cell jp-MarkdownCell jp-Notebook-cell" id="cell-id=0eedd28a">
<div class="jp-Cell-inputWrapper" tabindex="0"> <div class="jp-Cell-inputWrapper" tabindex="0">
<div class="jp-Collapser jp-InputCollapser jp-Cell-inputCollapser"> <div class="jp-Collapser jp-InputCollapser jp-Cell-inputCollapser">
@ -8043,10 +8107,10 @@ $$</p>
<div class="alert alert-block alert-success"> <div class="alert alert-block alert-success">
<b>Question:</b> How many scalars are communicated from and to a worker? Assume that matrices A, B, and C are N by N matrices. <b>Question:</b> How many scalars are communicated from and to a worker? Assume that matrices A, B, and C are N by N matrices.
</div> </div>
<pre><code>a) 3N <pre><code>a) O(1)
b) 2N + 2 b) O(N)
c) 2N + 1 c) O(N²)
d) N² + 1</code></pre> d) O(N³)</code></pre>
</div> </div>
</div> </div>
</div> </div>
@ -8075,9 +8139,10 @@ d) N² + 1</code></pre>
<div class="alert alert-block alert-success"> <div class="alert alert-block alert-success">
<b>Question:</b> How many operations are done in a worker? <b>Question:</b> How many operations are done in a worker?
</div> </div>
<pre><code>a) O(N) <pre><code>a) O(1)
b) O(N²) b) O(N)
c) O(N³)</code></pre> c) O(N²)
d) O(N³)</code></pre>
</div> </div>
</div> </div>
</div> </div>
@ -8548,22 +8613,22 @@ d) O(N²/P) communication and O(N³/P) computation</code></pre>
<tr> <tr>
<td>1</td> <td>1</td>
<td>N²</td> <td>N²</td>
<td>2N + 1</td> <td>O(N)</td>
<td>N</td> <td>O(N)</td>
<td>O(1)</td> <td>O(1)</td>
</tr> </tr>
<tr> <tr>
<td>2</td> <td>2</td>
<td>N</td> <td>N</td>
<td>2N + N²</td> <td>O(N²)</td>
<td>N²</td> <td>O(N²)</td>
<td>O(1)</td> <td>O(1)</td>
</tr> </tr>
<tr> <tr>
<td>3</td> <td>3</td>
<td>P</td> <td>P</td>
<td>N² + 2N²/P</td> <td>O(N²)</td>
<td>N³/P</td> <td>O(N³/P)</td>
<td>O(P/N)</td> <td>O(P/N)</td>
</tr> </tr>
</tbody> </tbody>

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