mirror of
https://github.com/fverdugo/XM_40017.git
synced 2025-12-30 02:38:31 +01:00
build based on e6474b1
This commit is contained in:
Documenter.jl
@@ -7333,11 +7333,12 @@ a.anchor-link {
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if (!diagrams.length) {
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return;
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}
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const mermaid = (await import("https://cdnjs.cloudflare.com/ajax/libs/mermaid/10.5.0/mermaid.esm.min.mjs")).default;
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const mermaid = (await import("https://cdnjs.cloudflare.com/ajax/libs/mermaid/10.7.0/mermaid.esm.min.mjs")).default;
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const parser = new DOMParser();
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mermaid.initialize({
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maxTextSize: 100000,
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maxEdges: 100000,
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startOnLoad: false,
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fontFamily: window
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.getComputedStyle(document.body)
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@@ -7408,7 +7409,8 @@ a.anchor-link {
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let results = null;
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let output = null;
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try {
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const { svg } = await mermaid.render(id, raw, el);
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let { svg } = await mermaid.render(id, raw, el);
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svg = cleanMermaidSvg(svg);
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results = makeMermaidImage(svg);
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output = document.createElement("figure");
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results.map(output.appendChild, output);
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@@ -7423,6 +7425,38 @@ a.anchor-link {
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parent.appendChild(output);
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}
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/**
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* Post-process to ensure mermaid diagrams contain only valid SVG and XHTML.
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*/
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function cleanMermaidSvg(svg) {
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return svg.replace(RE_VOID_ELEMENT, replaceVoidElement);
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}
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/**
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* A regular expression for all void elements, which may include attributes and
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* a slash.
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*
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* @see https://developer.mozilla.org/en-US/docs/Glossary/Void_element
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*
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* Of these, only `<br>` is generated by Mermaid in place of `\n`,
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* but _any_ "malformed" tag will break the SVG rendering entirely.
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*/
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const RE_VOID_ELEMENT =
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/<\s*(area|base|br|col|embed|hr|img|input|link|meta|param|source|track|wbr)\s*([^>]*?)\s*>/gi;
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/**
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* Ensure a void element is closed with a slash, preserving any attributes.
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*/
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function replaceVoidElement(match, tag, rest) {
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rest = rest.trim();
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if (!rest.endsWith('/')) {
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rest = `${rest} /`;
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}
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return `<${tag} ${rest}>`;
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}
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void Promise.all([...diagrams].map(renderOneMarmaid));
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});
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</script>
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@@ -7742,9 +7776,9 @@ a.anchor-link {
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<div class="jp-InputArea jp-Cell-inputArea"><div class="jp-InputPrompt jp-InputArea-prompt">
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</div><div class="jp-RenderedHTMLCommon jp-RenderedMarkdown jp-MarkdownOutput" data-mime-type="text/markdown">
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<div class="alert alert-block alert-info">
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<b>Note:</b> The matrix-matrix multiplication naively implemented with 3 nested loops as above is known to be very inefficient (memory bound). Libraries such as BLAS provide much more efficient implementations, which are the ones used in practice (e.g., by the `*` operator in Julia). We consider, our hand-written implementation as a simple way of expressing the algorithm we are interested in.
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<b>Note:</b> The matrix-matrix multiplication naively implemented with 3 nested loops as above is known to be very inefficient (memory bound). Libraries such as BLAS provide much more efficient implementations, which are the ones used in practice (e.g., by the `*` operator in Julia). We consider our hand-written implementation as a simple way of expressing the algorithm we are interested in.
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</div>
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<p>Run the following cell to compare the performance of our hand-written function with respect to the built in function <code>mul!</code></p>
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<p>Run the following cell to compare the performance of our hand-written function with respect to the built in function <code>mul!</code>.</p>
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</div>
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</div>
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</div>
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@@ -8757,131 +8791,6 @@ d) O(N²/P) communication and O(N³/P) computation</code></pre>
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</div>
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</div>
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<div class="jp-Cell jp-MarkdownCell jp-Notebook-cell" id="cell-id=fa8d7f40">
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<div class="jp-Cell-inputWrapper" tabindex="0">
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<div class="jp-InputArea jp-Cell-inputArea"><div class="jp-InputPrompt jp-InputArea-prompt">
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</div><div class="jp-RenderedHTMLCommon jp-RenderedMarkdown jp-MarkdownOutput" data-mime-type="text/markdown">
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<h3 id="Exercise-2">Exercise 2<a class="anchor-link" href="#Exercise-2">¶</a></h3>
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</div>
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</div>
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</div>
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</div>
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<div class="jp-Cell jp-MarkdownCell jp-Notebook-cell" id="cell-id=0e7c607e">
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<div class="jp-Cell-inputWrapper" tabindex="0">
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<div class="jp-Collapser jp-InputCollapser jp-Cell-inputCollapser">
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</div>
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<div class="jp-InputArea jp-Cell-inputArea"><div class="jp-InputPrompt jp-InputArea-prompt">
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</div><div class="jp-RenderedHTMLCommon jp-RenderedMarkdown jp-MarkdownOutput" data-mime-type="text/markdown">
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<p>The implementation of algorithm 1 is very impractical. One needs as many processors as entries in the result matrix C. For 1000 times 1000 matrix one would need a supercomputer with one million processes! We can easily fix this problem by using less processors and spawning the computation of an entry in any of the available processes.
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See the following code:</p>
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</div>
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</div>
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</div>
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</div><div class="jp-Cell jp-CodeCell jp-Notebook-cell jp-mod-noOutputs" id="cell-id=023b20d1">
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<div class="jp-Cell-inputWrapper" tabindex="0">
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<div class="jp-Collapser jp-InputCollapser jp-Cell-inputCollapser">
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</div>
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<div class="jp-InputArea jp-Cell-inputArea">
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<div class="jp-InputPrompt jp-InputArea-prompt">In [ ]:</div>
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<div class="jp-CodeMirrorEditor jp-Editor jp-InputArea-editor" data-type="inline">
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<div class="cm-editor cm-s-jupyter">
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<div class="highlight hl-julia"><pre><span></span><span class="k">function</span><span class="w"> </span><span class="n">matmul_dist_1_v2!</span><span class="p">(</span><span class="n">C</span><span class="p">,</span><span class="w"> </span><span class="n">A</span><span class="p">,</span><span class="w"> </span><span class="n">B</span><span class="p">)</span>
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<span class="w"> </span><span class="n">m</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">size</span><span class="p">(</span><span class="n">C</span><span class="p">,</span><span class="mi">1</span><span class="p">)</span>
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<span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">size</span><span class="p">(</span><span class="n">C</span><span class="p">,</span><span class="mi">2</span><span class="p">)</span>
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<span class="w"> </span><span class="n">l</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">size</span><span class="p">(</span><span class="n">A</span><span class="p">,</span><span class="mi">2</span><span class="p">)</span>
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<span class="w"> </span><span class="nd">@assert</span><span class="w"> </span><span class="n">size</span><span class="p">(</span><span class="n">A</span><span class="p">,</span><span class="mi">1</span><span class="p">)</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="n">m</span>
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<span class="w"> </span><span class="nd">@assert</span><span class="w"> </span><span class="n">size</span><span class="p">(</span><span class="n">B</span><span class="p">,</span><span class="mi">2</span><span class="p">)</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="n">n</span>
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<span class="w"> </span><span class="nd">@assert</span><span class="w"> </span><span class="n">size</span><span class="p">(</span><span class="n">B</span><span class="p">,</span><span class="mi">1</span><span class="p">)</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="n">l</span>
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<span class="w"> </span><span class="n">z</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">zero</span><span class="p">(</span><span class="n">eltype</span><span class="p">(</span><span class="n">C</span><span class="p">))</span>
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<span class="w"> </span><span class="nd">@sync</span><span class="w"> </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>
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<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>
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<span class="w"> </span><span class="n">Ai</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">A</span><span class="p">[</span><span class="n">i</span><span class="p">,</span><span class="o">:</span><span class="p">]</span>
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<span class="w"> </span><span class="n">Bj</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">B</span><span class="p">[</span><span class="o">:</span><span class="p">,</span><span class="n">j</span><span class="p">]</span>
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<span class="w"> </span><span class="n">ftr</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nd">@spawnat</span><span class="w"> </span><span class="ss">:any</span><span class="w"> </span><span class="k">begin</span>
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<span class="w"> </span><span class="n">Cij</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">z</span>
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<span class="w"> </span><span class="k">for</span><span class="w"> </span><span class="n">k</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">l</span>
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<span class="w"> </span><span class="nd">@inbounds</span><span class="w"> </span><span class="n">Cij</span><span class="w"> </span><span class="o">+=</span><span class="w"> </span><span class="n">Ai</span><span class="p">[</span><span class="n">k</span><span class="p">]</span><span class="o">*</span><span class="n">Bj</span><span class="p">[</span><span class="n">k</span><span class="p">]</span>
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<span class="w"> </span><span class="k">end</span>
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<span class="w"> </span><span class="n">Cij</span>
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<span class="w"> </span><span class="k">end</span>
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<span class="w"> </span><span class="nd">@async</span><span class="w"> </span><span class="n">C</span><span class="p">[</span><span class="n">i</span><span class="p">,</span><span class="n">j</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">fetch</span><span class="p">(</span><span class="n">ftr</span><span class="p">)</span>
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<span class="w"> </span><span class="k">end</span>
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<span class="w"> </span><span class="k">end</span>
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<span class="w"> </span><span class="n">C</span>
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<span class="k">end</span>
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</pre></div>
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</div>
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</div>
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</div>
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</div>
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<div class="jp-Cell jp-MarkdownCell jp-Notebook-cell" id="cell-id=52005ca1">
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<div class="jp-Cell-inputWrapper" tabindex="0">
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<div class="jp-InputArea jp-Cell-inputArea"><div class="jp-InputPrompt jp-InputArea-prompt">
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</div><div class="jp-RenderedHTMLCommon jp-RenderedMarkdown jp-MarkdownOutput" data-mime-type="text/markdown">
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<p>With this new implementation, we can multiply matrices of arbitrary size with a fixed number of workers. Test it:</p>
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</div>
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</div>
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</div>
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</div><div class="jp-Cell jp-CodeCell jp-Notebook-cell jp-mod-noOutputs" id="cell-id=c1d3595b">
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<div class="jp-Cell-inputWrapper" tabindex="0">
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<div class="jp-Collapser jp-InputCollapser jp-Cell-inputCollapser">
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</div>
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<div class="jp-InputArea jp-Cell-inputArea">
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<div class="jp-InputPrompt jp-InputArea-prompt">In [ ]:</div>
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<div class="jp-CodeMirrorEditor jp-Editor jp-InputArea-editor" data-type="inline">
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<div class="cm-editor cm-s-jupyter">
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<div class="highlight hl-julia"><pre><span></span><span class="k">using</span><span class="w"> </span><span class="n">Test</span>
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<span class="n">N</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">50</span>
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<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>
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<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>
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<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>
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<span class="nd">@test</span><span class="w"> </span><span class="n">matmul_dist_1_v2!</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>
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</pre></div>
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</div>
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</div>
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</div>
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</div>
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<div class="jp-Cell jp-MarkdownCell jp-Notebook-cell" id="cell-id=ab609c18">
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<div class="jp-InputArea jp-Cell-inputArea"><div class="jp-InputPrompt jp-InputArea-prompt">
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</div><div class="jp-RenderedHTMLCommon jp-RenderedMarkdown jp-MarkdownOutput" data-mime-type="text/markdown">
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<p>Run the next cell to check the performance of this implementation. Note that we are far away from the optimal speed up. Why? To answer this question compute the theoretical communication over computation ratio for this implementation and reason about the obtained result. Hint: the number of times a worker is spawned in this implementation is N^2/P on average.</p>
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</div>
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</div>
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</div>
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</div><div class="jp-Cell jp-CodeCell jp-Notebook-cell jp-mod-noOutputs" id="cell-id=d7d31710">
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</div>
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<div class="jp-InputArea jp-Cell-inputArea">
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<div class="jp-InputPrompt jp-InputArea-prompt">In [ ]:</div>
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<div class="jp-CodeMirrorEditor jp-Editor jp-InputArea-editor" data-type="inline">
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<div class="cm-editor cm-s-jupyter">
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<div class="highlight hl-julia"><pre><span></span><span class="n">N</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">100</span>
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<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>
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<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>
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<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>
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<span class="n">P</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">nworkers</span><span class="p">()</span>
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<span class="n">T1</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nd">@belapsed</span><span class="w"> </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>
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<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>
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<span class="n">TP</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nd">@belapsed</span><span class="w"> </span><span class="n">matmul_dist_1_v2!</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>
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<span class="n">println</span><span class="p">(</span><span class="s">"Speedup = "</span><span class="p">,</span><span class="w"> </span><span class="n">T1</span><span class="o">/</span><span class="n">TP</span><span class="p">)</span>
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<span class="n">println</span><span class="p">(</span><span class="s">"Optimal speedup = "</span><span class="p">,</span><span class="w"> </span><span class="n">P</span><span class="p">)</span>
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<span class="n">println</span><span class="p">(</span><span class="s">"Efficiency = "</span><span class="p">,</span><span class="w"> </span><span class="mi">100</span><span class="o">*</span><span class="p">(</span><span class="n">T1</span><span class="o">/</span><span class="n">TP</span><span class="p">)</span><span class="o">/</span><span class="n">P</span><span class="p">,</span><span class="w"> </span><span class="s">"%"</span><span class="p">)</span>
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</pre></div>
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</div>
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</div>
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Block a user