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    <title>Forem: TinyAlg</title>
    <description>The latest articles on Forem by TinyAlg (@tinyalg).</description>
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      <title>2.78 TFLOPS on a Fanless MacBook Air? Benchmarking Apple's M4 with MLX</title>
      <dc:creator>lwgena</dc:creator>
      <pubDate>Thu, 19 Mar 2026 05:58:37 +0000</pubDate>
      <link>https://forem.com/tinyalg/278-tflops-on-a-fanless-macbook-air-benchmarking-apples-m4-with-mlx-38n4</link>
      <guid>https://forem.com/tinyalg/278-tflops-on-a-fanless-macbook-air-benchmarking-apples-m4-with-mlx-38n4</guid>
      <description>&lt;p&gt;&lt;em&gt;(This article is an English translation of a post originally published in Japanese on my blog. You can read the original Japanese version &lt;a href="https://tinyalg.be/ja/blog/m4-macbook-air-tflops-performance/" rel="noopener noreferrer"&gt;here&lt;/a&gt;)&lt;/em&gt;.&lt;/p&gt;

&lt;p&gt;My fanless M4 MacBook Air hit &lt;strong&gt;2.78 TFLOPS&lt;/strong&gt; in a matrix multiplication benchmark using Apple's MLX framework. &lt;/p&gt;

&lt;p&gt;Matrix multiplication (GEMM) isn't just a basic math problem; it's the beating heart of modern Machine Learning and Large Language Models (LLMs). By measuring how fast a machine can multiply huge matrices, we are essentially measuring its raw capability to run AI locally. Let's see what the M4 chip can do.&lt;/p&gt;

&lt;p&gt;&lt;strong&gt;&amp;lt; Test Environment &amp;gt;&lt;/strong&gt;&lt;/p&gt;

&lt;ul&gt;
&lt;li&gt;
&lt;strong&gt;Machine:&lt;/strong&gt; M4 Macbook Air&lt;/li&gt;
&lt;li&gt;
&lt;strong&gt;Memory:&lt;/strong&gt; 16GB&lt;/li&gt;
&lt;li&gt;
&lt;strong&gt;Python:&lt;/strong&gt; 3.10.11&lt;/li&gt;
&lt;li&gt;
&lt;strong&gt;Framework:&lt;/strong&gt; &lt;a href="https://github.com/ml-explore/mlx" rel="noopener noreferrer"&gt;MLX&lt;/a&gt; v0.28.0&lt;/li&gt;
&lt;/ul&gt;

&lt;h2&gt;
  
  
  1. Measuring Execution Time
&lt;/h2&gt;

&lt;p&gt;To measure the execution time, I used a simple matrix multiplication operation.&lt;/p&gt;

&lt;h3&gt;
  
  
  1.1 The Benchmark Script
&lt;/h3&gt;

&lt;p&gt;I've published the &lt;a href="https://gist.github.com/lwgena/4423759b7262cdc72fcf5e94b4b69414" rel="noopener noreferrer"&gt;measurement script on GitHub Gist&lt;/a&gt;. Feel free to download it and test it on your own Apple Silicon Mac. &lt;/p&gt;

&lt;blockquote&gt;
&lt;p&gt;&lt;strong&gt;Note:&lt;/strong&gt; I ran the benchmark last summer. I have recently verified that the script still runs on &lt;strong&gt;Python 3.12.12&lt;/strong&gt; and &lt;strong&gt;MLX v0.31.1&lt;/strong&gt; without any modifications.&lt;/p&gt;
&lt;/blockquote&gt;

&lt;p&gt;Save the script as &lt;code&gt;matmul_benchmark.py&lt;/code&gt; and run the following commands in your terminal:&lt;br&gt;
&lt;/p&gt;

&lt;div class="highlight js-code-highlight"&gt;
&lt;pre class="highlight shell"&gt;&lt;code&gt;% python3 &lt;span class="nt"&gt;-m&lt;/span&gt; venv venv
% &lt;span class="nb"&gt;.&lt;/span&gt; venv/bin/activate
% pip &lt;span class="nb"&gt;install &lt;/span&gt;mlx
% caffeinate python matmul_benchmark.py
&lt;/code&gt;&lt;/pre&gt;

&lt;/div&gt;

&lt;h3&gt;
  
  
  1.2 Results
&lt;/h3&gt;

&lt;p&gt;I measured the execution time for various matrix sizes. The average time below is based on 5 consecutive runs for each size. All computations were done using the &lt;code&gt;bfloat16&lt;/code&gt; data type.&lt;/p&gt;

&lt;div class="table-wrapper-paragraph"&gt;&lt;table&gt;
&lt;thead&gt;
&lt;tr&gt;
&lt;th&gt;Matrix Size&lt;/th&gt;
&lt;th&gt;Avg Time (s)&lt;/th&gt;
&lt;th&gt;Total Ops&lt;/th&gt;
&lt;th&gt;Real TFLOPS&lt;/th&gt;
&lt;th&gt;Run 1 (s)&lt;/th&gt;
&lt;th&gt;Run 2 (s)&lt;/th&gt;
&lt;th&gt;Run 3 (s)&lt;/th&gt;
&lt;th&gt;Run 4 (s)&lt;/th&gt;
&lt;th&gt;Run 5 (s)&lt;/th&gt;
&lt;/tr&gt;
&lt;/thead&gt;
&lt;tbody&gt;
&lt;tr&gt;
&lt;td&gt;&lt;code&gt;10000x10000&lt;/code&gt;&lt;/td&gt;
&lt;td&gt;0.740&lt;/td&gt;
&lt;td&gt;2 Trillion&lt;/td&gt;
&lt;td&gt;&lt;strong&gt;2.70&lt;/strong&gt;&lt;/td&gt;
&lt;td&gt;0.738&lt;/td&gt;
&lt;td&gt;0.738&lt;/td&gt;
&lt;td&gt;0.742&lt;/td&gt;
&lt;td&gt;0.740&lt;/td&gt;
&lt;td&gt;0.743&lt;/td&gt;
&lt;/tr&gt;
&lt;tr&gt;
&lt;td&gt;&lt;code&gt;20000x20000&lt;/code&gt;&lt;/td&gt;
&lt;td&gt;5.754&lt;/td&gt;
&lt;td&gt;16 Trillion&lt;/td&gt;
&lt;td&gt;&lt;strong&gt;2.78&lt;/strong&gt;&lt;/td&gt;
&lt;td&gt;5.667&lt;/td&gt;
&lt;td&gt;5.698&lt;/td&gt;
&lt;td&gt;5.750&lt;/td&gt;
&lt;td&gt;5.810&lt;/td&gt;
&lt;td&gt;5.847&lt;/td&gt;
&lt;/tr&gt;
&lt;tr&gt;
&lt;td&gt;&lt;code&gt;30000x30000&lt;/code&gt;&lt;/td&gt;
&lt;td&gt;21.624&lt;/td&gt;
&lt;td&gt;54 Trillion&lt;/td&gt;
&lt;td&gt;&lt;strong&gt;2.50&lt;/strong&gt;&lt;/td&gt;
&lt;td&gt;20.341&lt;/td&gt;
&lt;td&gt;21.076&lt;/td&gt;
&lt;td&gt;22.252&lt;/td&gt;
&lt;td&gt;21.897&lt;/td&gt;
&lt;td&gt;22.555&lt;/td&gt;
&lt;/tr&gt;
&lt;tr&gt;
&lt;td&gt;&lt;code&gt;40000x40000&lt;/code&gt;&lt;/td&gt;
&lt;td&gt;75.872&lt;/td&gt;
&lt;td&gt;128 Trillion&lt;/td&gt;
&lt;td&gt;&lt;strong&gt;1.69&lt;/strong&gt;&lt;/td&gt;
&lt;td&gt;62.925&lt;/td&gt;
&lt;td&gt;77.502&lt;/td&gt;
&lt;td&gt;72.921&lt;/td&gt;
&lt;td&gt;85.509&lt;/td&gt;
&lt;td&gt;80.505&lt;/td&gt;
&lt;/tr&gt;
&lt;/tbody&gt;
&lt;/table&gt;&lt;/div&gt;
&lt;h3&gt;
  
  
  1.3 Observations
&lt;/h3&gt;

&lt;ul&gt;
&lt;li&gt;
&lt;strong&gt;Peak Performance:&lt;/strong&gt; The highest performance recorded was &lt;strong&gt;2.78 TFLOPS&lt;/strong&gt; at a matrix size of &lt;code&gt;20000x20000&lt;/code&gt;.&lt;/li&gt;
&lt;li&gt;
&lt;strong&gt;Thermal Behavior:&lt;/strong&gt; For the &lt;code&gt;20000x20000&lt;/code&gt; and &lt;code&gt;30000x30000&lt;/code&gt; sizes, execution times creep up slightly with each run. This hints that the fanless chassis might be starting to manage heat.&lt;/li&gt;
&lt;li&gt;
&lt;strong&gt;The Impact of Swap Memory:&lt;/strong&gt; At &lt;code&gt;40000x40000&lt;/code&gt;, the execution time spiked, and the variance between runs became larger. This is because the 16GB of unified memory ran out, forcing the system to frequently swap to the slower SSD. I confirmed this by watching the swap space usage shoot up in Activity Monitor during the run.&lt;/li&gt;
&lt;/ul&gt;
&lt;h2&gt;
  
  
  2. Calculating Total Operations and TFLOPS
&lt;/h2&gt;

&lt;p&gt;&lt;strong&gt;TFLOPS&lt;/strong&gt; (Tera Floating-point Operations Per Second) is a performance metric indicating how many trillions of floating-point calculations a processor can execute in one second. &lt;/p&gt;

&lt;p&gt;We can calculate this using the total number of operations and the execution time.&lt;/p&gt;
&lt;h3&gt;
  
  
  2.1 Total Operations
&lt;/h3&gt;

&lt;p&gt;First, let's calculate the total operations. A matrix multiplication 

&lt;span class="katex-element"&gt;
  &lt;span class="katex"&gt;&lt;span class="katex-mathml"&gt;C=A⋅BC = A \cdot B&lt;/span&gt;&lt;span class="katex-html"&gt;&lt;span class="base"&gt;&lt;span class="strut"&gt;&lt;/span&gt;&lt;span class="mord mathnormal"&gt;C&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;span class="mrel"&gt;=&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="base"&gt;&lt;span class="strut"&gt;&lt;/span&gt;&lt;span class="mord mathnormal"&gt;A&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;span class="mbin"&gt;⋅&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="base"&gt;&lt;span class="strut"&gt;&lt;/span&gt;&lt;span class="mord mathnormal"&gt;B&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;
&lt;/span&gt;
 for matrices 
&lt;span class="katex-element"&gt;
  &lt;span class="katex"&gt;&lt;span class="katex-mathml"&gt;AA&lt;/span&gt;&lt;span class="katex-html"&gt;&lt;span class="base"&gt;&lt;span class="strut"&gt;&lt;/span&gt;&lt;span class="mord mathnormal"&gt;A&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;
&lt;/span&gt;
 and 
&lt;span class="katex-element"&gt;
  &lt;span class="katex"&gt;&lt;span class="katex-mathml"&gt;BB&lt;/span&gt;&lt;span class="katex-html"&gt;&lt;span class="base"&gt;&lt;span class="strut"&gt;&lt;/span&gt;&lt;span class="mord mathnormal"&gt;B&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;
&lt;/span&gt;
 of size 
&lt;span class="katex-element"&gt;
  &lt;span class="katex"&gt;&lt;span class="katex-mathml"&gt;N×NN \times N&lt;/span&gt;&lt;span class="katex-html"&gt;&lt;span class="base"&gt;&lt;span class="strut"&gt;&lt;/span&gt;&lt;span class="mord mathnormal"&gt;N&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;span class="mbin"&gt;×&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="base"&gt;&lt;span class="strut"&gt;&lt;/span&gt;&lt;span class="mord mathnormal"&gt;N&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;
&lt;/span&gt;
 can be expressed as:&lt;/p&gt;


&lt;div class="katex-element"&gt;
  &lt;span class="katex-display"&gt;&lt;span class="katex"&gt;&lt;span class="katex-mathml"&gt;(c11…c1N⋮⋱⋮cN1…cNN)=(a11…a1N⋮⋱⋮aN1…aNN)⋅(b11…b1N⋮⋱⋮bN1…bNN)
\left(
\begin{array}{rrr}
c_{11} &amp;amp; \dots &amp;amp; c_{1N} \\
\vdots &amp;amp; \ddots &amp;amp; \vdots \\
c_{N1} &amp;amp; \dots &amp;amp; c_{NN}
\end{array}
\right)=
\left(
\begin{array}{rrr}
a_{11} &amp;amp; \dots &amp;amp; a_{1N} \\
\vdots &amp;amp; \ddots &amp;amp; \vdots \\
a_{N1} &amp;amp; \dots &amp;amp; a_{NN}
\end{array}
\right)\cdot
\left(
\begin{array}{rrr}
b_{11} &amp;amp; \dots &amp;amp; b_{1N} \\
\vdots &amp;amp; \ddots &amp;amp; \vdots \\
b_{N1} &amp;amp; \dots &amp;amp; b_{NN}
\end{array}
\right)
&lt;/span&gt;&lt;span class="katex-html"&gt;&lt;span class="base"&gt;&lt;span class="strut"&gt;&lt;/span&gt;&lt;span class="minner"&gt;&lt;span class="mopen"&gt;&lt;span class="delimsizing mult"&gt;&lt;span class="vlist-t vlist-t2"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="delimsizinginner delim-size4"&gt;&lt;span&gt;⎝&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="delimsizinginner delim-size4"&gt;&lt;span&gt;⎛&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-s"&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mtable"&gt;&lt;span class="arraycolsep"&gt;&lt;/span&gt;&lt;span class="col-align-r"&gt;&lt;span class="vlist-t vlist-t2"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mord"&gt;&lt;span class="mord mathnormal"&gt;c&lt;/span&gt;&lt;span class="msupsub"&gt;&lt;span class="vlist-t vlist-t2"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mtight"&gt;&lt;span class="mord mtight"&gt;11&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-s"&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mord"&gt;&lt;span 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class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="arraycolsep"&gt;&lt;/span&gt;&lt;span class="arraycolsep"&gt;&lt;/span&gt;&lt;span class="col-align-r"&gt;&lt;span class="vlist-t vlist-t2"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="minner"&gt;…&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="minner"&gt;⋱&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="minner"&gt;…&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-s"&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="arraycolsep"&gt;&lt;/span&gt;&lt;span class="arraycolsep"&gt;&lt;/span&gt;&lt;span class="col-align-r"&gt;&lt;span class="vlist-t vlist-t2"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mord"&gt;&lt;span class="mord mathnormal"&gt;c&lt;/span&gt;&lt;span class="msupsub"&gt;&lt;span class="vlist-t vlist-t2"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mtight"&gt;&lt;span class="mord mtight"&gt;1&lt;/span&gt;&lt;span class="mord mathnormal mtight"&gt;N&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-s"&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mord"&gt;&lt;span class="mord"&gt;⋮&lt;/span&gt;&lt;span class="mord rule"&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mord"&gt;&lt;span class="mord mathnormal"&gt;c&lt;/span&gt;&lt;span class="msupsub"&gt;&lt;span class="vlist-t vlist-t2"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mtight"&gt;&lt;span class="mord mathnormal mtight"&gt;NN&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-s"&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-s"&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="arraycolsep"&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="mclose"&gt;&lt;span class="delimsizing mult"&gt;&lt;span class="vlist-t vlist-t2"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="delimsizinginner delim-size4"&gt;&lt;span&gt;⎠&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="delimsizinginner delim-size4"&gt;&lt;span&gt;⎞&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-s"&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;span class="mrel"&gt;=&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="base"&gt;&lt;span class="strut"&gt;&lt;/span&gt;&lt;span class="minner"&gt;&lt;span class="mopen"&gt;&lt;span class="delimsizing mult"&gt;&lt;span class="vlist-t vlist-t2"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="delimsizinginner delim-size4"&gt;&lt;span&gt;⎝&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="delimsizinginner delim-size4"&gt;&lt;span&gt;⎛&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-s"&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mtable"&gt;&lt;span class="arraycolsep"&gt;&lt;/span&gt;&lt;span class="col-align-r"&gt;&lt;span class="vlist-t vlist-t2"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mord"&gt;&lt;span class="mord mathnormal"&gt;a&lt;/span&gt;&lt;span class="msupsub"&gt;&lt;span class="vlist-t vlist-t2"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mtight"&gt;&lt;span class="mord mtight"&gt;11&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-s"&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mord"&gt;&lt;span class="mord"&gt;⋮&lt;/span&gt;&lt;span class="mord rule"&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mord"&gt;&lt;span class="mord mathnormal"&gt;a&lt;/span&gt;&lt;span class="msupsub"&gt;&lt;span class="vlist-t vlist-t2"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mtight"&gt;&lt;span class="mord mathnormal mtight"&gt;N&lt;/span&gt;&lt;span class="mord mtight"&gt;1&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-s"&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-s"&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="arraycolsep"&gt;&lt;/span&gt;&lt;span class="arraycolsep"&gt;&lt;/span&gt;&lt;span class="col-align-r"&gt;&lt;span class="vlist-t vlist-t2"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="minner"&gt;…&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="minner"&gt;⋱&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="minner"&gt;…&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-s"&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="arraycolsep"&gt;&lt;/span&gt;&lt;span class="arraycolsep"&gt;&lt;/span&gt;&lt;span class="col-align-r"&gt;&lt;span class="vlist-t vlist-t2"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mord"&gt;&lt;span class="mord mathnormal"&gt;a&lt;/span&gt;&lt;span class="msupsub"&gt;&lt;span class="vlist-t vlist-t2"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mtight"&gt;&lt;span class="mord mtight"&gt;1&lt;/span&gt;&lt;span class="mord mathnormal mtight"&gt;N&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-s"&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mord"&gt;&lt;span class="mord"&gt;⋮&lt;/span&gt;&lt;span class="mord rule"&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mord"&gt;&lt;span class="mord mathnormal"&gt;a&lt;/span&gt;&lt;span class="msupsub"&gt;&lt;span class="vlist-t vlist-t2"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mtight"&gt;&lt;span class="mord mathnormal mtight"&gt;NN&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-s"&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-s"&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="arraycolsep"&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="mclose"&gt;&lt;span class="delimsizing mult"&gt;&lt;span class="vlist-t vlist-t2"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="delimsizinginner delim-size4"&gt;&lt;span&gt;⎠&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="delimsizinginner delim-size4"&gt;&lt;span&gt;⎞&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-s"&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;span class="mbin"&gt;⋅&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="base"&gt;&lt;span class="strut"&gt;&lt;/span&gt;&lt;span class="minner"&gt;&lt;span class="mopen"&gt;&lt;span class="delimsizing mult"&gt;&lt;span class="vlist-t vlist-t2"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="delimsizinginner delim-size4"&gt;&lt;span&gt;⎝&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="delimsizinginner delim-size4"&gt;&lt;span&gt;⎛&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-s"&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mtable"&gt;&lt;span class="arraycolsep"&gt;&lt;/span&gt;&lt;span class="col-align-r"&gt;&lt;span class="vlist-t vlist-t2"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mord"&gt;&lt;span class="mord mathnormal"&gt;b&lt;/span&gt;&lt;span class="msupsub"&gt;&lt;span class="vlist-t vlist-t2"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mtight"&gt;&lt;span class="mord mtight"&gt;11&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-s"&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mord"&gt;&lt;span class="mord"&gt;⋮&lt;/span&gt;&lt;span class="mord rule"&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mord"&gt;&lt;span class="mord mathnormal"&gt;b&lt;/span&gt;&lt;span class="msupsub"&gt;&lt;span class="vlist-t vlist-t2"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mtight"&gt;&lt;span class="mord mathnormal mtight"&gt;N&lt;/span&gt;&lt;span class="mord mtight"&gt;1&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-s"&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-s"&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="arraycolsep"&gt;&lt;/span&gt;&lt;span class="arraycolsep"&gt;&lt;/span&gt;&lt;span class="col-align-r"&gt;&lt;span class="vlist-t vlist-t2"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="minner"&gt;…&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="minner"&gt;⋱&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="minner"&gt;…&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-s"&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="arraycolsep"&gt;&lt;/span&gt;&lt;span class="arraycolsep"&gt;&lt;/span&gt;&lt;span class="col-align-r"&gt;&lt;span class="vlist-t vlist-t2"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mord"&gt;&lt;span class="mord mathnormal"&gt;b&lt;/span&gt;&lt;span class="msupsub"&gt;&lt;span class="vlist-t vlist-t2"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mtight"&gt;&lt;span class="mord mtight"&gt;1&lt;/span&gt;&lt;span class="mord mathnormal mtight"&gt;N&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-s"&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mord"&gt;&lt;span class="mord"&gt;⋮&lt;/span&gt;&lt;span class="mord rule"&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mord"&gt;&lt;span class="mord mathnormal"&gt;b&lt;/span&gt;&lt;span class="msupsub"&gt;&lt;span class="vlist-t vlist-t2"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mtight"&gt;&lt;span class="mord mathnormal mtight"&gt;NN&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-s"&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-s"&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="arraycolsep"&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="mclose"&gt;&lt;span class="delimsizing mult"&gt;&lt;span class="vlist-t vlist-t2"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="delimsizinginner delim-size4"&gt;&lt;span&gt;⎠&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="delimsizinginner delim-size4"&gt;&lt;span&gt;⎞&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-s"&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;
&lt;/div&gt;



&lt;p&gt;A single element 
&lt;span class="katex-element"&gt;
  &lt;span class="katex"&gt;&lt;span class="katex-mathml"&gt;cijc_{ij}&lt;/span&gt;&lt;span class="katex-html"&gt;&lt;span class="base"&gt;&lt;span class="strut"&gt;&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mord mathnormal"&gt;c&lt;/span&gt;&lt;span class="msupsub"&gt;&lt;span class="vlist-t vlist-t2"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mtight"&gt;&lt;span class="mord mathnormal mtight"&gt;ij&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-s"&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;
&lt;/span&gt;
 in matrix 
&lt;span class="katex-element"&gt;
  &lt;span class="katex"&gt;&lt;span class="katex-mathml"&gt;CC&lt;/span&gt;&lt;span class="katex-html"&gt;&lt;span class="base"&gt;&lt;span class="strut"&gt;&lt;/span&gt;&lt;span class="mord mathnormal"&gt;C&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;
&lt;/span&gt;
 is calculated as:&lt;/p&gt;


&lt;div class="katex-element"&gt;
  &lt;span class="katex-display"&gt;&lt;span class="katex"&gt;&lt;span class="katex-mathml"&gt;cij=∑k=1Naikbkj=ai1b1j+ai2b2j+⋯+aiNbNj
c_{ij} = \sum_{k=1}^{N} a_{ik}b_{kj} = a_{i1}b_{1j} + a_{i2}b_{2j} + \cdots + a_{iN}b_{Nj}
&lt;/span&gt;&lt;span class="katex-html"&gt;&lt;span class="base"&gt;&lt;span class="strut"&gt;&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mord mathnormal"&gt;c&lt;/span&gt;&lt;span class="msupsub"&gt;&lt;span class="vlist-t vlist-t2"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mtight"&gt;&lt;span class="mord mathnormal mtight"&gt;ij&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-s"&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;span class="mrel"&gt;=&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="base"&gt;&lt;span class="strut"&gt;&lt;/span&gt;&lt;span class="mop op-limits"&gt;&lt;span class="vlist-t vlist-t2"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mtight"&gt;&lt;span class="mord mathnormal mtight"&gt;k&lt;/span&gt;&lt;span class="mrel mtight"&gt;=&lt;/span&gt;&lt;span class="mord mtight"&gt;1&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span&gt;&lt;span class="mop op-symbol large-op"&gt;∑&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mtight"&gt;&lt;span class="mord mathnormal mtight"&gt;N&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-s"&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mord mathnormal"&gt;a&lt;/span&gt;&lt;span class="msupsub"&gt;&lt;span class="vlist-t vlist-t2"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mtight"&gt;&lt;span class="mord mathnormal mtight"&gt;ik&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-s"&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mord mathnormal"&gt;b&lt;/span&gt;&lt;span class="msupsub"&gt;&lt;span class="vlist-t vlist-t2"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mtight"&gt;&lt;span class="mord mathnormal mtight"&gt;kj&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-s"&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;span class="mrel"&gt;=&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="base"&gt;&lt;span class="strut"&gt;&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mord mathnormal"&gt;a&lt;/span&gt;&lt;span class="msupsub"&gt;&lt;span class="vlist-t vlist-t2"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mtight"&gt;&lt;span class="mord mathnormal mtight"&gt;i&lt;/span&gt;&lt;span class="mord mtight"&gt;1&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-s"&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mord mathnormal"&gt;b&lt;/span&gt;&lt;span class="msupsub"&gt;&lt;span class="vlist-t vlist-t2"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mtight"&gt;&lt;span class="mord mtight"&gt;1&lt;/span&gt;&lt;span class="mord mathnormal mtight"&gt;j&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-s"&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;span class="mbin"&gt;+&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="base"&gt;&lt;span class="strut"&gt;&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mord mathnormal"&gt;a&lt;/span&gt;&lt;span class="msupsub"&gt;&lt;span class="vlist-t vlist-t2"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mtight"&gt;&lt;span class="mord mathnormal mtight"&gt;i&lt;/span&gt;&lt;span class="mord mtight"&gt;2&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-s"&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mord mathnormal"&gt;b&lt;/span&gt;&lt;span class="msupsub"&gt;&lt;span class="vlist-t vlist-t2"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mtight"&gt;&lt;span class="mord mtight"&gt;2&lt;/span&gt;&lt;span class="mord mathnormal mtight"&gt;j&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-s"&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;span class="mbin"&gt;+&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="base"&gt;&lt;span class="strut"&gt;&lt;/span&gt;&lt;span class="minner"&gt;⋯&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;span class="mbin"&gt;+&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="base"&gt;&lt;span class="strut"&gt;&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mord mathnormal"&gt;a&lt;/span&gt;&lt;span class="msupsub"&gt;&lt;span class="vlist-t vlist-t2"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mtight"&gt;&lt;span class="mord mathnormal mtight"&gt;i&lt;/span&gt;&lt;span class="mord mathnormal mtight"&gt;N&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-s"&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mord mathnormal"&gt;b&lt;/span&gt;&lt;span class="msupsub"&gt;&lt;span class="vlist-t vlist-t2"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mtight"&gt;&lt;span class="mord mathnormal mtight"&gt;N&lt;/span&gt;&lt;span class="mord mathnormal mtight"&gt;j&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-s"&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;
&lt;/div&gt;


&lt;p&gt;This requires 
&lt;span class="katex-element"&gt;
  &lt;span class="katex"&gt;&lt;span class="katex-mathml"&gt;NN&lt;/span&gt;&lt;span class="katex-html"&gt;&lt;span class="base"&gt;&lt;span class="strut"&gt;&lt;/span&gt;&lt;span class="mord mathnormal"&gt;N&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;
&lt;/span&gt;
 multiplications and 
&lt;span class="katex-element"&gt;
  &lt;span class="katex"&gt;&lt;span class="katex-mathml"&gt;N−1N-1&lt;/span&gt;&lt;span class="katex-html"&gt;&lt;span class="base"&gt;&lt;span class="strut"&gt;&lt;/span&gt;&lt;span class="mord mathnormal"&gt;N&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;span class="mbin"&gt;−&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="base"&gt;&lt;span class="strut"&gt;&lt;/span&gt;&lt;span class="mord"&gt;1&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;
&lt;/span&gt;
 additions, totaling 
&lt;span class="katex-element"&gt;
  &lt;span class="katex"&gt;&lt;span class="katex-mathml"&gt;2N−12N-1&lt;/span&gt;&lt;span class="katex-html"&gt;&lt;span class="base"&gt;&lt;span class="strut"&gt;&lt;/span&gt;&lt;span class="mord"&gt;2&lt;/span&gt;&lt;span class="mord mathnormal"&gt;N&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;span class="mbin"&gt;−&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="base"&gt;&lt;span class="strut"&gt;&lt;/span&gt;&lt;span class="mord"&gt;1&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;
&lt;/span&gt;
 operations. In our experiment, since 
&lt;span class="katex-element"&gt;
  &lt;span class="katex"&gt;&lt;span class="katex-mathml"&gt;NN&lt;/span&gt;&lt;span class="katex-html"&gt;&lt;span class="base"&gt;&lt;span class="strut"&gt;&lt;/span&gt;&lt;span class="mord mathnormal"&gt;N&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;
&lt;/span&gt;
 is large, we can approximate this to 
&lt;span class="katex-element"&gt;
  &lt;span class="katex"&gt;&lt;span class="katex-mathml"&gt;2N2N&lt;/span&gt;&lt;span class="katex-html"&gt;&lt;span class="base"&gt;&lt;span class="strut"&gt;&lt;/span&gt;&lt;span class="mord"&gt;2&lt;/span&gt;&lt;span class="mord mathnormal"&gt;N&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;
&lt;/span&gt;
 operations per element.&lt;/p&gt;

&lt;p&gt;Since the matrix 
&lt;span class="katex-element"&gt;
  &lt;span class="katex"&gt;&lt;span class="katex-mathml"&gt;CC&lt;/span&gt;&lt;span class="katex-html"&gt;&lt;span class="base"&gt;&lt;span class="strut"&gt;&lt;/span&gt;&lt;span class="mord mathnormal"&gt;C&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;
&lt;/span&gt;
 has 
&lt;span class="katex-element"&gt;
  &lt;span class="katex"&gt;&lt;span class="katex-mathml"&gt;N2N^2&lt;/span&gt;&lt;span class="katex-html"&gt;&lt;span class="base"&gt;&lt;span class="strut"&gt;&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mord mathnormal"&gt;N&lt;/span&gt;&lt;span class="msupsub"&gt;&lt;span class="vlist-t"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mtight"&gt;2&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;
&lt;/span&gt;
 elements, the total number of operations is:&lt;/p&gt;


&lt;div class="katex-element"&gt;
  &lt;span class="katex-display"&gt;&lt;span class="katex"&gt;&lt;span class="katex-mathml"&gt;Total Operations≈(2N)×N2=2N3
\text{Total Operations} \approx (2N) \times N^2 = 2N^3
&lt;/span&gt;&lt;span class="katex-html"&gt;&lt;span class="base"&gt;&lt;span class="strut"&gt;&lt;/span&gt;&lt;span class="mord text"&gt;&lt;span class="mord"&gt;Total Operations&lt;/span&gt;&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;span class="mrel"&gt;≈&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="base"&gt;&lt;span class="strut"&gt;&lt;/span&gt;&lt;span class="mopen"&gt;(&lt;/span&gt;&lt;span class="mord"&gt;2&lt;/span&gt;&lt;span class="mord mathnormal"&gt;N&lt;/span&gt;&lt;span class="mclose"&gt;)&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;span class="mbin"&gt;×&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="base"&gt;&lt;span class="strut"&gt;&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mord mathnormal"&gt;N&lt;/span&gt;&lt;span class="msupsub"&gt;&lt;span class="vlist-t"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mtight"&gt;2&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;span class="mrel"&gt;=&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="base"&gt;&lt;span class="strut"&gt;&lt;/span&gt;&lt;span class="mord"&gt;2&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mord mathnormal"&gt;N&lt;/span&gt;&lt;span class="msupsub"&gt;&lt;span class="vlist-t"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mtight"&gt;3&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;
&lt;/div&gt;


&lt;p&gt;For example, when 
&lt;span class="katex-element"&gt;
  &lt;span class="katex"&gt;&lt;span class="katex-mathml"&gt;N=20000N=20000&lt;/span&gt;&lt;span class="katex-html"&gt;&lt;span class="base"&gt;&lt;span class="strut"&gt;&lt;/span&gt;&lt;span class="mord mathnormal"&gt;N&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;span class="mrel"&gt;=&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="base"&gt;&lt;span class="strut"&gt;&lt;/span&gt;&lt;span class="mord"&gt;20000&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;
&lt;/span&gt;
:&lt;/p&gt;


&lt;div class="katex-element"&gt;
  &lt;span class="katex-display"&gt;&lt;span class="katex"&gt;&lt;span class="katex-mathml"&gt;Total Operations≈2×200003=16×1012 (16 Trillion Operations)
\text{Total Operations} \approx 2 \times 20000^3 = 16 \times 10^{12} \text{ (16 Trillion Operations)}
&lt;/span&gt;&lt;span class="katex-html"&gt;&lt;span class="base"&gt;&lt;span class="strut"&gt;&lt;/span&gt;&lt;span class="mord text"&gt;&lt;span class="mord"&gt;Total Operations&lt;/span&gt;&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;span class="mrel"&gt;≈&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="base"&gt;&lt;span class="strut"&gt;&lt;/span&gt;&lt;span class="mord"&gt;2&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;span class="mbin"&gt;×&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="base"&gt;&lt;span class="strut"&gt;&lt;/span&gt;&lt;span class="mord"&gt;2000&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mord"&gt;0&lt;/span&gt;&lt;span class="msupsub"&gt;&lt;span class="vlist-t"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mtight"&gt;3&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;span class="mrel"&gt;=&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="base"&gt;&lt;span class="strut"&gt;&lt;/span&gt;&lt;span class="mord"&gt;16&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;span class="mbin"&gt;×&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="base"&gt;&lt;span class="strut"&gt;&lt;/span&gt;&lt;span class="mord"&gt;1&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mord"&gt;0&lt;/span&gt;&lt;span class="msupsub"&gt;&lt;span class="vlist-t"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mtight"&gt;&lt;span class="mord mtight"&gt;12&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="mord text"&gt;&lt;span class="mord"&gt; (16 Trillion Operations)&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;
&lt;/div&gt;


&lt;h3&gt;
  
  
  2.2 Calculating the TFLOPS Performance
&lt;/h3&gt;

&lt;p&gt;Finally, we use the total operations and the execution time to calculate the TFLOPS using this formula:&lt;/p&gt;


&lt;div class="katex-element"&gt;
  &lt;span class="katex-display"&gt;&lt;span class="katex"&gt;&lt;span class="katex-mathml"&gt;TFLOPS=Total OperationsExecution Time [s]×1012
\text{TFLOPS} = \frac{\text{Total Operations}}{\text{Execution Time [s]} \times 10^{12}}
&lt;/span&gt;&lt;span class="katex-html"&gt;&lt;span class="base"&gt;&lt;span class="strut"&gt;&lt;/span&gt;&lt;span class="mord text"&gt;&lt;span class="mord"&gt;TFLOPS&lt;/span&gt;&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;span class="mrel"&gt;=&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="base"&gt;&lt;span class="strut"&gt;&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mopen nulldelimiter"&gt;&lt;/span&gt;&lt;span class="mfrac"&gt;&lt;span class="vlist-t vlist-t2"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mord text"&gt;&lt;span class="mord"&gt;Execution Time [s]&lt;/span&gt;&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;span class="mbin"&gt;×&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;span class="mord"&gt;1&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mord"&gt;0&lt;/span&gt;&lt;span class="msupsub"&gt;&lt;span class="vlist-t"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mtight"&gt;&lt;span class="mord mtight"&gt;12&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="frac-line"&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mord text"&gt;&lt;span class="mord"&gt;Total Operations&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-s"&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="mclose nulldelimiter"&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;
&lt;/div&gt;


&lt;p&gt;For 
&lt;span class="katex-element"&gt;
  &lt;span class="katex"&gt;&lt;span class="katex-mathml"&gt;N=20000N=20000&lt;/span&gt;&lt;span class="katex-html"&gt;&lt;span class="base"&gt;&lt;span class="strut"&gt;&lt;/span&gt;&lt;span class="mord mathnormal"&gt;N&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;span class="mrel"&gt;=&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="base"&gt;&lt;span class="strut"&gt;&lt;/span&gt;&lt;span class="mord"&gt;20000&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;
&lt;/span&gt;
:&lt;/p&gt;


&lt;div class="katex-element"&gt;
  &lt;span class="katex-display"&gt;&lt;span class="katex"&gt;&lt;span class="katex-mathml"&gt;TFLOPS=16×10125.754×1012≈2.78
\text{TFLOPS} = \frac{16 \times 10^{12}}{5.754 \times 10^{12}} \approx 2.78
&lt;/span&gt;&lt;span class="katex-html"&gt;&lt;span class="base"&gt;&lt;span class="strut"&gt;&lt;/span&gt;&lt;span class="mord text"&gt;&lt;span class="mord"&gt;TFLOPS&lt;/span&gt;&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;span class="mrel"&gt;=&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="base"&gt;&lt;span class="strut"&gt;&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mopen nulldelimiter"&gt;&lt;/span&gt;&lt;span class="mfrac"&gt;&lt;span class="vlist-t vlist-t2"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mord"&gt;5.754&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;span class="mbin"&gt;×&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;span class="mord"&gt;1&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mord"&gt;0&lt;/span&gt;&lt;span class="msupsub"&gt;&lt;span class="vlist-t"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mtight"&gt;&lt;span class="mord mtight"&gt;12&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="frac-line"&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mord"&gt;16&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;span class="mbin"&gt;×&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;span class="mord"&gt;1&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mord"&gt;0&lt;/span&gt;&lt;span class="msupsub"&gt;&lt;span class="vlist-t"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mtight"&gt;&lt;span class="mord mtight"&gt;12&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-s"&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="mclose nulldelimiter"&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;span class="mrel"&gt;≈&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="base"&gt;&lt;span class="strut"&gt;&lt;/span&gt;&lt;span class="mord"&gt;2.78&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;
&lt;/div&gt;


&lt;h2&gt;
  
  
  **
&lt;/h2&gt;

&lt;p&gt;Did you feel the sheer scale of TFLOPS calculations? It's pretty mind-blowing what a fanless laptop can achieve locally these days.&lt;/p&gt;

</description>
      <category>benchmark</category>
      <category>python</category>
      <category>mlx</category>
      <category>machinelearning</category>
    </item>
    <item>
      <title>I Fed My Tech Book into NotebookLM, and It Generated Spot-On Infographics</title>
      <dc:creator>lwgena</dc:creator>
      <pubDate>Sat, 07 Mar 2026 07:48:07 +0000</pubDate>
      <link>https://forem.com/tinyalg/i-fed-my-tech-book-into-notebooklm-and-it-generated-spot-on-infographics-3668</link>
      <guid>https://forem.com/tinyalg/i-fed-my-tech-book-into-notebooklm-and-it-generated-spot-on-infographics-3668</guid>
      <description>&lt;p&gt;Recently, I published a technical book (in Japanese) titled "&lt;em&gt;&lt;a href="https://zenn.dev/lwgena/books/youtube-timer-quest" rel="noopener noreferrer"&gt;The YouTube Timer Quest: Mastering the Engineer's Mindset via Chrome Extensions.&lt;/a&gt;&lt;/em&gt;"&lt;/p&gt;

&lt;p&gt;Out of curiosity, I decided to feed my entire Japanese book into Google's &lt;strong&gt;NotebookLM&lt;/strong&gt; to see how it would summarize the content. The result? It generated surprisingly accurate, professional-grade infographics—&lt;strong&gt;in natural English!&lt;/strong&gt;&lt;/p&gt;

&lt;p&gt;Even if you haven't read my book, I want to share two of these diagrams and explain why this blew my mind as a developer and author.&lt;/p&gt;

&lt;h2&gt;
  
  
  How AI Visualized the Author's Intent
&lt;/h2&gt;

&lt;p&gt;First, take a look at the initial infographic it generated.&lt;/p&gt;

&lt;p&gt;&lt;a href="https://media2.dev.to/dynamic/image/width=800%2Cheight=%2Cfit=scale-down%2Cgravity=auto%2Cformat=auto/https%3A%2F%2Fdev-to-uploads.s3.amazonaws.com%2Fuploads%2Farticles%2F8j9c371nvg6oulrkq5fq.jpg" class="article-body-image-wrapper"&gt;&lt;img src="https://media2.dev.to/dynamic/image/width=800%2Cheight=%2Cfit=scale-down%2Cgravity=auto%2Cformat=auto/https%3A%2F%2Fdev-to-uploads.s3.amazonaws.com%2Fuploads%2Farticles%2F8j9c371nvg6oulrkq5fq.jpg" alt="NotebookLM Generated Infographic 1" width="800" height="446"&gt;&lt;/a&gt;&lt;/p&gt;

&lt;p&gt;As the author, I was astonished by how well it captured both the core philosophy and the specific technical challenges discussed in the book.&lt;/p&gt;

&lt;p&gt;My book uses the development of a real-world Chrome extension (&lt;a href="https://tinyalg.be/watchtime-badge/" rel="noopener noreferrer"&gt;Watchtime Badge&lt;/a&gt;) to teach readers how to move beyond simple "copy-paste" coding and adopt an actual "Engineer's Mindset." It focuses heavily on solving real-world browser constraints. &lt;/p&gt;

&lt;p&gt;Here is why this diagram is so impressive:&lt;/p&gt;

&lt;ul&gt;
&lt;li&gt;
&lt;strong&gt;Extracting the Core Philosophy:&lt;/strong&gt; The diagram clearly illustrates the "Define, Decompose, Build" pillars. This isn't just a specific chapter; it's the foundational theme of the entire book. The AI understood the underlying message remarkably well.&lt;/li&gt;
&lt;li&gt;
&lt;strong&gt;Pinpointing Technical Solutions:&lt;/strong&gt; In the middle right, you'll see "Delta Time for Precision". This directly corresponds to my chapter explaining why JavaScript's &lt;code&gt;setInterval&lt;/code&gt; drifts over time and how to fix it using timestamp differences.&lt;/li&gt;
&lt;li&gt;
&lt;strong&gt;Understanding API Context:&lt;/strong&gt; It highlighted the use of the &lt;code&gt;chrome.alarms&lt;/code&gt; API as "The Heartbeat for Survival". This reflects my chapter on how to revive Chrome Service Workers that are automatically terminated by the browser's energy-saving features.&lt;/li&gt;
&lt;/ul&gt;

&lt;h2&gt;
  
  
  The Secret: NotebookLM Prompt Settings
&lt;/h2&gt;

&lt;p&gt;To generate these high-quality diagrams, I used the following settings in NotebookLM:&lt;/p&gt;

&lt;ul&gt;
&lt;li&gt;
&lt;strong&gt;Choose Language:&lt;/strong&gt; &lt;code&gt;English&lt;/code&gt;
&lt;/li&gt;
&lt;li&gt;
&lt;strong&gt;Choose Orientation:&lt;/strong&gt; &lt;code&gt;Landscape&lt;/code&gt;
&lt;/li&gt;
&lt;li&gt;
&lt;strong&gt;Level of Detail:&lt;/strong&gt; &lt;code&gt;Standard&lt;/code&gt;
&lt;/li&gt;
&lt;/ul&gt;

&lt;p&gt;&lt;strong&gt;💡 Author's Tip:&lt;/strong&gt; Even though my source material was completely in Japanese, I intentionally set the output language to &lt;strong&gt;English&lt;/strong&gt;. Why? Because for technical diagrams, English text tends to fit the layout much better visually, resulting in a cleaner, more standard, and stylish look. NotebookLM handled the cross-lingual summarization seamlessly.&lt;/p&gt;

&lt;p&gt;Here is the second infographic it generated:&lt;/p&gt;

&lt;p&gt;&lt;a href="https://media2.dev.to/dynamic/image/width=800%2Cheight=%2Cfit=scale-down%2Cgravity=auto%2Cformat=auto/https%3A%2F%2Fdev-to-uploads.s3.amazonaws.com%2Fuploads%2Farticles%2Fzaw72afa3rtn929bket8.jpg" class="article-body-image-wrapper"&gt;&lt;img src="https://media2.dev.to/dynamic/image/width=800%2Cheight=%2Cfit=scale-down%2Cgravity=auto%2Cformat=auto/https%3A%2F%2Fdev-to-uploads.s3.amazonaws.com%2Fuploads%2Farticles%2Fzaw72afa3rtn929bket8.jpg" alt="NotebookLM Generated Infographic 2" width="800" height="446"&gt;&lt;/a&gt;&lt;/p&gt;

&lt;p&gt;While the first infographic visualizes the development process as a dynamic journey (a "quest"), this second one organizes the same knowledge into a structured, static overview.&lt;/p&gt;

&lt;p&gt;It clearly categorizes the abstract "Engineer Thinking" on the left and concrete "System Constraints" on the right. It is fascinating to see how the exact same prompt generated two completely different visual approaches—process versus structure.&lt;/p&gt;

&lt;h2&gt;
  
  
  Conclusion: What Engineers Should Focus On
&lt;/h2&gt;

&lt;p&gt;Seeing these images made me realize something profound: &lt;strong&gt;We no longer need to spend hours manually formatting our ideas into visual diagrams.&lt;/strong&gt;&lt;/p&gt;

&lt;p&gt;AI can now effectively grasp technical context, understand system constraints, and translate complex concepts into beautiful visuals. This frees us engineers to let go of tedious layout adjustments and focus on what truly matters: defining the right problems and designing the core system.&lt;/p&gt;

&lt;p&gt;Have you tried generating diagrams with NotebookLM for your projects? Let me know your thoughts in the comments!&lt;/p&gt;

</description>
      <category>ai</category>
      <category>notebooklm</category>
      <category>chromeextension</category>
      <category>infographics</category>
    </item>
    <item>
      <title>Introducing 'Publish to Dev.to Organization' Action: How to Safely Test Your Workflow</title>
      <dc:creator>lwgena</dc:creator>
      <pubDate>Fri, 06 Mar 2026 01:38:29 +0000</pubDate>
      <link>https://forem.com/tinyalg/introducing-publish-to-devto-organization-action-how-to-safely-test-your-workflow-12gb</link>
      <guid>https://forem.com/tinyalg/introducing-publish-to-devto-organization-action-how-to-safely-test-your-workflow-12gb</guid>
      <description>&lt;p&gt;I recently joined Dev.to and wanted to automate my article publishing directly from GitHub. However, I quickly hit a roadblock: I wanted to publish posts under an &lt;strong&gt;Organization&lt;/strong&gt;, but I realized I had to write a custom script using the Forem API to make that happen, as existing tools didn't fully support it.&lt;/p&gt;

&lt;p&gt;I started writing a custom GitHub Action just for my own workflow. But halfway through, I thought, &lt;em&gt;"Why not just make this available for everyone on the GitHub Marketplace?"&lt;/em&gt; And so, &lt;strong&gt;&lt;a href="https://github.com/marketplace/actions/publish-to-dev-to-organization" rel="noopener noreferrer"&gt;Publish to Dev.to Organization Action&lt;/a&gt;&lt;/strong&gt; was born! &lt;/p&gt;

&lt;p&gt;But of course, before publishing a tool that interacts with a live API, I needed a way to test it thoroughly. If you test a publishing workflow live, you risk spamming your followers or hitting rate limits. &lt;/p&gt;

&lt;p&gt;To solve this, I implemented a &lt;code&gt;dry_run&lt;/code&gt; feature in this action. In this post, I'll show you how to use this feature to test all &lt;strong&gt;8 possible publishing patterns&lt;/strong&gt; without making any actual publishing API calls (&lt;code&gt;POST&lt;/code&gt; or &lt;code&gt;PUT&lt;/code&gt;) to Dev.to. &lt;em&gt;(Note: The action only makes safe &lt;code&gt;GET&lt;/code&gt; requests to resolve Organization IDs when necessary.)&lt;/em&gt;&lt;/p&gt;

&lt;h2&gt;
  
  
  The 8-Pattern Matrix
&lt;/h2&gt;

&lt;p&gt;When managing Dev.to articles via front matter, there are 3 variables that create 8 unique scenarios:&lt;/p&gt;

&lt;ol&gt;
&lt;li&gt;
&lt;strong&gt;Target:&lt;/strong&gt; Personal Account vs. Organization&lt;/li&gt;
&lt;li&gt;
&lt;strong&gt;State:&lt;/strong&gt; Draft (&lt;code&gt;published: false&lt;/code&gt;) vs. Public (&lt;code&gt;published: true&lt;/code&gt;)&lt;/li&gt;
&lt;li&gt;
&lt;strong&gt;Action:&lt;/strong&gt; Create (&lt;code&gt;POST&lt;/code&gt;) vs. Update (&lt;code&gt;PUT&lt;/code&gt;)&lt;/li&gt;
&lt;/ol&gt;

&lt;p&gt;You can test all of these in two simple rounds using GitHub Actions.&lt;/p&gt;

&lt;h3&gt;
  
  
  Prerequisites: The Workflow Setup
&lt;/h3&gt;

&lt;p&gt;First, set up your workflow using &lt;code&gt;tj-actions/changed-files&lt;/code&gt; to only process modified files, and enable &lt;code&gt;dry_run: 'true'&lt;/code&gt; in this custom action.&lt;br&gt;
&lt;/p&gt;

&lt;div class="highlight js-code-highlight"&gt;
&lt;pre class="highlight yaml"&gt;&lt;code&gt;&lt;span class="c1"&gt;# .github/workflows/publish.yml&lt;/span&gt;
&lt;span class="na"&gt;name&lt;/span&gt;&lt;span class="pi"&gt;:&lt;/span&gt; &lt;span class="s"&gt;Publish to Dev.to&lt;/span&gt;

&lt;span class="na"&gt;on&lt;/span&gt;&lt;span class="pi"&gt;:&lt;/span&gt;
  &lt;span class="na"&gt;push&lt;/span&gt;&lt;span class="pi"&gt;:&lt;/span&gt;
    &lt;span class="na"&gt;branches&lt;/span&gt;&lt;span class="pi"&gt;:&lt;/span&gt;
      &lt;span class="pi"&gt;-&lt;/span&gt; &lt;span class="s"&gt;main&lt;/span&gt;
    &lt;span class="na"&gt;paths&lt;/span&gt;&lt;span class="pi"&gt;:&lt;/span&gt;
      &lt;span class="pi"&gt;-&lt;/span&gt; &lt;span class="s1"&gt;'&lt;/span&gt;&lt;span class="s"&gt;posts/**.md'&lt;/span&gt;

&lt;span class="na"&gt;jobs&lt;/span&gt;&lt;span class="pi"&gt;:&lt;/span&gt;
  &lt;span class="na"&gt;publish&lt;/span&gt;&lt;span class="pi"&gt;:&lt;/span&gt;
    &lt;span class="na"&gt;runs-on&lt;/span&gt;&lt;span class="pi"&gt;:&lt;/span&gt; &lt;span class="s"&gt;ubuntu-latest&lt;/span&gt;
    &lt;span class="na"&gt;permissions&lt;/span&gt;&lt;span class="pi"&gt;:&lt;/span&gt;
      &lt;span class="na"&gt;contents&lt;/span&gt;&lt;span class="pi"&gt;:&lt;/span&gt; &lt;span class="s"&gt;write&lt;/span&gt;

    &lt;span class="na"&gt;steps&lt;/span&gt;&lt;span class="pi"&gt;:&lt;/span&gt;
      &lt;span class="pi"&gt;-&lt;/span&gt; &lt;span class="na"&gt;uses&lt;/span&gt;&lt;span class="pi"&gt;:&lt;/span&gt; &lt;span class="s"&gt;actions/checkout@v4&lt;/span&gt;
        &lt;span class="na"&gt;with&lt;/span&gt;&lt;span class="pi"&gt;:&lt;/span&gt;
          &lt;span class="na"&gt;fetch-depth&lt;/span&gt;&lt;span class="pi"&gt;:&lt;/span&gt; &lt;span class="m"&gt;0&lt;/span&gt;

      &lt;span class="pi"&gt;-&lt;/span&gt; &lt;span class="na"&gt;name&lt;/span&gt;&lt;span class="pi"&gt;:&lt;/span&gt; &lt;span class="s"&gt;Get changed files&lt;/span&gt;
        &lt;span class="na"&gt;id&lt;/span&gt;&lt;span class="pi"&gt;:&lt;/span&gt; &lt;span class="s"&gt;changed-files&lt;/span&gt;
        &lt;span class="na"&gt;uses&lt;/span&gt;&lt;span class="pi"&gt;:&lt;/span&gt; &lt;span class="s"&gt;tj-actions/changed-files@v47&lt;/span&gt;
        &lt;span class="na"&gt;with&lt;/span&gt;&lt;span class="pi"&gt;:&lt;/span&gt;
          &lt;span class="na"&gt;files&lt;/span&gt;&lt;span class="pi"&gt;:&lt;/span&gt; &lt;span class="s"&gt;posts/*.md&lt;/span&gt;

      &lt;span class="pi"&gt;-&lt;/span&gt; &lt;span class="na"&gt;name&lt;/span&gt;&lt;span class="pi"&gt;:&lt;/span&gt; &lt;span class="s"&gt;Publish Articles&lt;/span&gt;
        &lt;span class="na"&gt;if&lt;/span&gt;&lt;span class="pi"&gt;:&lt;/span&gt; &lt;span class="s"&gt;steps.changed-files.outputs.any_changed == 'true'&lt;/span&gt;
        &lt;span class="na"&gt;uses&lt;/span&gt;&lt;span class="pi"&gt;:&lt;/span&gt; &lt;span class="s"&gt;tinyalg/publish-devto-org-action@v1&lt;/span&gt;
        &lt;span class="na"&gt;with&lt;/span&gt;&lt;span class="pi"&gt;:&lt;/span&gt;
          &lt;span class="na"&gt;devto_api_key&lt;/span&gt;&lt;span class="pi"&gt;:&lt;/span&gt; &lt;span class="s"&gt;${{ secrets.DEVTO_API_KEY }}&lt;/span&gt;
          &lt;span class="na"&gt;file_path&lt;/span&gt;&lt;span class="pi"&gt;:&lt;/span&gt; &lt;span class="s"&gt;${{ steps.changed-files.outputs.all_changed_files }}&lt;/span&gt;
          &lt;span class="na"&gt;dry_run&lt;/span&gt;&lt;span class="pi"&gt;:&lt;/span&gt; &lt;span class="s1"&gt;'&lt;/span&gt;&lt;span class="s"&gt;true'&lt;/span&gt; &lt;span class="c1"&gt;# This prevents actual POST/PUT API calls!&lt;/span&gt;

      &lt;span class="pi"&gt;-&lt;/span&gt; &lt;span class="na"&gt;name&lt;/span&gt;&lt;span class="pi"&gt;:&lt;/span&gt; &lt;span class="s"&gt;Commit Article IDs&lt;/span&gt;
        &lt;span class="na"&gt;if&lt;/span&gt;&lt;span class="pi"&gt;:&lt;/span&gt; &lt;span class="s"&gt;steps.changed-files.outputs.any_changed == 'true'&lt;/span&gt;
        &lt;span class="na"&gt;uses&lt;/span&gt;&lt;span class="pi"&gt;:&lt;/span&gt; &lt;span class="s"&gt;stefanzweifel/git-auto-commit-action@v7&lt;/span&gt;
        &lt;span class="na"&gt;with&lt;/span&gt;&lt;span class="pi"&gt;:&lt;/span&gt;
          &lt;span class="na"&gt;commit_message&lt;/span&gt;&lt;span class="pi"&gt;:&lt;/span&gt; &lt;span class="s2"&gt;"&lt;/span&gt;&lt;span class="s"&gt;chore:&lt;/span&gt;&lt;span class="nv"&gt; &lt;/span&gt;&lt;span class="s"&gt;update&lt;/span&gt;&lt;span class="nv"&gt; &lt;/span&gt;&lt;span class="s"&gt;Dev.to&lt;/span&gt;&lt;span class="nv"&gt; &lt;/span&gt;&lt;span class="s"&gt;article&lt;/span&gt;&lt;span class="nv"&gt; &lt;/span&gt;&lt;span class="s"&gt;IDs&lt;/span&gt;&lt;span class="nv"&gt; &lt;/span&gt;&lt;span class="s"&gt;[skip&lt;/span&gt;&lt;span class="nv"&gt; &lt;/span&gt;&lt;span class="s"&gt;ci]"&lt;/span&gt;
          &lt;span class="na"&gt;file_pattern&lt;/span&gt;&lt;span class="pi"&gt;:&lt;/span&gt; &lt;span class="s1"&gt;'&lt;/span&gt;&lt;span class="s"&gt;posts/*.md'&lt;/span&gt;

&lt;/code&gt;&lt;/pre&gt;

&lt;/div&gt;



&lt;h3&gt;
  
  
  Round 1: Testing "Create" (POST)
&lt;/h3&gt;

&lt;p&gt;To test the 4 creation patterns, create the following four markdown files in your &lt;code&gt;posts/&lt;/code&gt; directory.&lt;/p&gt;

&lt;p&gt;&lt;strong&gt;1. Personal &amp;amp; Draft&lt;/strong&gt; (&lt;code&gt;pattern1.md&lt;/code&gt;)&lt;br&gt;
&lt;/p&gt;

&lt;div class="highlight js-code-highlight"&gt;
&lt;pre class="highlight markdown"&gt;&lt;code&gt;&lt;span class="nn"&gt;---&lt;/span&gt;
&lt;span class="na"&gt;title&lt;/span&gt;&lt;span class="pi"&gt;:&lt;/span&gt; &lt;span class="s"&gt;Pattern 1 Personal Draft&lt;/span&gt;
&lt;span class="na"&gt;published&lt;/span&gt;&lt;span class="pi"&gt;:&lt;/span&gt; &lt;span class="kc"&gt;false&lt;/span&gt;
&lt;span class="nn"&gt;---&lt;/span&gt;
Test content here.

&lt;/code&gt;&lt;/pre&gt;

&lt;/div&gt;



&lt;p&gt;&lt;strong&gt;2. Personal &amp;amp; Published&lt;/strong&gt; (&lt;code&gt;pattern2.md&lt;/code&gt;)&lt;br&gt;
&lt;/p&gt;

&lt;div class="highlight js-code-highlight"&gt;
&lt;pre class="highlight markdown"&gt;&lt;code&gt;&lt;span class="nn"&gt;---&lt;/span&gt;
&lt;span class="na"&gt;title&lt;/span&gt;&lt;span class="pi"&gt;:&lt;/span&gt; &lt;span class="s"&gt;Pattern 2 Personal Published&lt;/span&gt;
&lt;span class="na"&gt;published&lt;/span&gt;&lt;span class="pi"&gt;:&lt;/span&gt; &lt;span class="kc"&gt;true&lt;/span&gt;
&lt;span class="nn"&gt;---&lt;/span&gt;
Test content here.

&lt;/code&gt;&lt;/pre&gt;

&lt;/div&gt;



&lt;p&gt;&lt;strong&gt;3. Organization &amp;amp; Draft&lt;/strong&gt; (&lt;code&gt;pattern3.md&lt;/code&gt;)&lt;br&gt;
&lt;/p&gt;

&lt;div class="highlight js-code-highlight"&gt;
&lt;pre class="highlight markdown"&gt;&lt;code&gt;&lt;span class="nn"&gt;---&lt;/span&gt;
&lt;span class="na"&gt;title&lt;/span&gt;&lt;span class="pi"&gt;:&lt;/span&gt; &lt;span class="s"&gt;Pattern 3 Org Draft&lt;/span&gt;
&lt;span class="na"&gt;published&lt;/span&gt;&lt;span class="pi"&gt;:&lt;/span&gt; &lt;span class="kc"&gt;false&lt;/span&gt;
&lt;span class="na"&gt;organization_username&lt;/span&gt;&lt;span class="pi"&gt;:&lt;/span&gt; &lt;span class="s"&gt;tinyalg&lt;/span&gt;
&lt;span class="nn"&gt;---&lt;/span&gt;
Test content here.

&lt;/code&gt;&lt;/pre&gt;

&lt;/div&gt;



&lt;p&gt;&lt;strong&gt;4. Organization &amp;amp; Published&lt;/strong&gt; (&lt;code&gt;pattern4.md&lt;/code&gt;)&lt;br&gt;
&lt;/p&gt;

&lt;div class="highlight js-code-highlight"&gt;
&lt;pre class="highlight markdown"&gt;&lt;code&gt;&lt;span class="nn"&gt;---&lt;/span&gt;
&lt;span class="na"&gt;title&lt;/span&gt;&lt;span class="pi"&gt;:&lt;/span&gt; &lt;span class="s"&gt;Pattern 4 Org Published&lt;/span&gt;
&lt;span class="na"&gt;published&lt;/span&gt;&lt;span class="pi"&gt;:&lt;/span&gt; &lt;span class="kc"&gt;true&lt;/span&gt;
&lt;span class="na"&gt;organization_username&lt;/span&gt;&lt;span class="pi"&gt;:&lt;/span&gt; &lt;span class="s"&gt;tinyalg&lt;/span&gt;
&lt;span class="nn"&gt;---&lt;/span&gt;
Test content here.

&lt;/code&gt;&lt;/pre&gt;

&lt;/div&gt;



&lt;p&gt;After committing and pushing these files, check your GitHub Actions logs. You will see the &lt;code&gt;dry_run&lt;/code&gt; output. The action simulates the API, confirms it &lt;em&gt;would&lt;/em&gt; have sent a &lt;code&gt;POST&lt;/code&gt; request, and injects a mock &lt;code&gt;id&lt;/code&gt; (e.g., &lt;code&gt;id: 9123456&lt;/code&gt;) into each file. The &lt;code&gt;git-auto-commit&lt;/code&gt; step then saves this back to your repo.&lt;/p&gt;

&lt;h3&gt;
  
  
  Round 2: Testing "Update" (PUT)
&lt;/h3&gt;

&lt;p&gt;Now that the files have an &lt;code&gt;id&lt;/code&gt; assigned to them, you can test the update patterns.&lt;/p&gt;

&lt;ol&gt;
&lt;li&gt;Run &lt;code&gt;git pull&lt;/code&gt; on your local machine to fetch the mock IDs added by the GitHub Action.&lt;/li&gt;
&lt;li&gt;Open the 4 files and make a tiny change (e.g., add a new line at the bottom).&lt;/li&gt;
&lt;li&gt;Commit and push again.&lt;/li&gt;
&lt;/ol&gt;

&lt;p&gt;Check the GitHub Actions logs once more. Because the files now contain an &lt;code&gt;id&lt;/code&gt; in their front matter, the action correctly identifies them as existing articles. You will see that it &lt;em&gt;would&lt;/em&gt; have executed a &lt;code&gt;PUT&lt;/code&gt; request to &lt;code&gt;https://dev.to/api/articles/&amp;lt;mock-id&amp;gt;&lt;/code&gt;.&lt;/p&gt;

&lt;h2&gt;
  
  
  Conclusion
&lt;/h2&gt;

&lt;p&gt;And just like that, you've tested your entire CI/CD publishing pipeline—including file diffing, ID injection, and payload generation—without publishing a single test post to your actual Dev.to feed!&lt;/p&gt;

&lt;p&gt;Once you are confident, simply remove &lt;code&gt;dry_run: 'true'&lt;/code&gt; from your workflow, and you are ready to automate your Dev.to blog.&lt;/p&gt;

&lt;p&gt;Feel free to check out the action here: &lt;a href="https://github.com/marketplace/actions/publish-to-dev-to-organization" rel="noopener noreferrer"&gt;Publish to Dev.to Organization Action&lt;/a&gt;&lt;/p&gt;

&lt;p&gt;Happy automating! 🚀&lt;/p&gt;

</description>
      <category>githubactions</category>
      <category>testing</category>
      <category>tutorial</category>
      <category>devto</category>
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