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    <title>Forem: Preston Vasquez</title>
    <description>The latest articles on Forem by Preston Vasquez (@prestonvasquez).</description>
    <link>https://forem.com/prestonvasquez</link>
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      <title>Forem: Preston Vasquez</title>
      <link>https://forem.com/prestonvasquez</link>
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      <title>Mocking MongoDB Go Driver V2: How to Do It (and Why You Probably Shouldn't)</title>
      <dc:creator>Preston Vasquez</dc:creator>
      <pubDate>Tue, 17 Jun 2025 20:18:08 +0000</pubDate>
      <link>https://forem.com/prestonvasquez/mocking-mongodb-go-driver-v2-how-to-do-it-and-why-you-probably-shouldnt-4f30</link>
      <guid>https://forem.com/prestonvasquez/mocking-mongodb-go-driver-v2-how-to-do-it-and-why-you-probably-shouldnt-4f30</guid>
      <description>&lt;p&gt;MongoDB Go Driver V1 included the &lt;code&gt;mtest&lt;/code&gt; package allowing developers to mock driver responses in memory. A few popular blog posts since emerged detailing how to do this, including &lt;a href="https://liza.io/how-im-testing-mongodb-calls-in-go-for-now/" rel="noopener noreferrer"&gt;"How I'm testing MongoDB calls in Go (for now)"&lt;/a&gt; by Liza Shulyayeva. In V1, &lt;a href="https://jira.mongodb.org/browse/GODRIVER-1767" rel="noopener noreferrer"&gt;GODRIVER-1767&lt;/a&gt; marked &lt;code&gt;mtest&lt;/code&gt; unstable so that the driver team could alter the API to accommodate new testing requirements. In V2, &lt;a href="https://dev.to4"&gt;GODRIVER-2724&lt;/a&gt; internalized &lt;code&gt;mtest&lt;/code&gt; making the V1 solution for mocking using &lt;code&gt;mtest&lt;/code&gt; unavailable.&lt;/p&gt;

&lt;p&gt;Mocking can be overly complex and is brittle (AKA “Mock Hell”). Tests should be focused on your code, not the internals of the Go Driver. Since mocks are not identical to your live environment, problems can stay hidden longer. By the time an issue shows up in production, fixing it can be far more costly than if you’d caught it earlier in a more realistic test. Instead, it's better to use a container-based integration testing framework like &lt;a href="https://testcontainers.com/guides/introducing-testcontainers/" rel="noopener noreferrer"&gt;testcontainers&lt;/a&gt;:&lt;br&gt;
&lt;/p&gt;

&lt;div class="highlight js-code-highlight"&gt;
&lt;pre class="highlight go"&gt;&lt;code&gt;&lt;span class="k"&gt;func&lt;/span&gt; &lt;span class="n"&gt;TestMyRepositoryFunction&lt;/span&gt;&lt;span class="p"&gt;(&lt;/span&gt;&lt;span class="n"&gt;t&lt;/span&gt; &lt;span class="o"&gt;*&lt;/span&gt;&lt;span class="n"&gt;testing&lt;/span&gt;&lt;span class="o"&gt;.&lt;/span&gt;&lt;span class="n"&gt;T&lt;/span&gt;&lt;span class="p"&gt;)&lt;/span&gt; &lt;span class="p"&gt;{&lt;/span&gt;
    &lt;span class="n"&gt;ctx&lt;/span&gt; &lt;span class="o"&gt;:=&lt;/span&gt; &lt;span class="n"&gt;context&lt;/span&gt;&lt;span class="o"&gt;.&lt;/span&gt;&lt;span class="n"&gt;Background&lt;/span&gt;&lt;span class="p"&gt;()&lt;/span&gt;

    &lt;span class="c"&gt;//  Start up a MongoDB container&lt;/span&gt;
    &lt;span class="n"&gt;req&lt;/span&gt; &lt;span class="o"&gt;:=&lt;/span&gt; &lt;span class="n"&gt;testcontainers&lt;/span&gt;&lt;span class="o"&gt;.&lt;/span&gt;&lt;span class="n"&gt;ContainerRequest&lt;/span&gt;&lt;span class="p"&gt;{&lt;/span&gt;
        &lt;span class="n"&gt;Image&lt;/span&gt;&lt;span class="o"&gt;:&lt;/span&gt;        &lt;span class="s"&gt;"mongo:latest"&lt;/span&gt;&lt;span class="p"&gt;,&lt;/span&gt;
        &lt;span class="n"&gt;ExposedPorts&lt;/span&gt;&lt;span class="o"&gt;:&lt;/span&gt; &lt;span class="p"&gt;[]&lt;/span&gt;&lt;span class="kt"&gt;string&lt;/span&gt;&lt;span class="p"&gt;{&lt;/span&gt;&lt;span class="s"&gt;"27017/tcp"&lt;/span&gt;&lt;span class="p"&gt;},&lt;/span&gt;
        &lt;span class="n"&gt;WaitingFor&lt;/span&gt;&lt;span class="o"&gt;:&lt;/span&gt;   &lt;span class="n"&gt;wait&lt;/span&gt;&lt;span class="o"&gt;.&lt;/span&gt;&lt;span class="n"&gt;ForLog&lt;/span&gt;&lt;span class="p"&gt;(&lt;/span&gt;&lt;span class="s"&gt;"Waiting for connections"&lt;/span&gt;&lt;span class="p"&gt;)&lt;/span&gt;&lt;span class="o"&gt;.&lt;/span&gt;&lt;span class="n"&gt;WithStartupTimeout&lt;/span&gt;&lt;span class="p"&gt;(&lt;/span&gt;&lt;span class="m"&gt;30&lt;/span&gt; &lt;span class="o"&gt;*&lt;/span&gt; &lt;span class="n"&gt;time&lt;/span&gt;&lt;span class="o"&gt;.&lt;/span&gt;&lt;span class="n"&gt;Second&lt;/span&gt;&lt;span class="p"&gt;),&lt;/span&gt;
    &lt;span class="p"&gt;}&lt;/span&gt;
    &lt;span class="n"&gt;mongoC&lt;/span&gt;&lt;span class="p"&gt;,&lt;/span&gt; &lt;span class="n"&gt;err&lt;/span&gt; &lt;span class="o"&gt;:=&lt;/span&gt; &lt;span class="n"&gt;testcontainers&lt;/span&gt;&lt;span class="o"&gt;.&lt;/span&gt;&lt;span class="n"&gt;GenericContainer&lt;/span&gt;&lt;span class="p"&gt;(&lt;/span&gt;&lt;span class="n"&gt;ctx&lt;/span&gt;&lt;span class="p"&gt;,&lt;/span&gt; &lt;span class="n"&gt;testcontainers&lt;/span&gt;&lt;span class="o"&gt;.&lt;/span&gt;&lt;span class="n"&gt;GenericContainerRequest&lt;/span&gt;&lt;span class="p"&gt;{&lt;/span&gt;
        &lt;span class="n"&gt;ContainerRequest&lt;/span&gt;&lt;span class="o"&gt;:&lt;/span&gt; &lt;span class="n"&gt;req&lt;/span&gt;&lt;span class="p"&gt;,&lt;/span&gt;
        &lt;span class="n"&gt;Started&lt;/span&gt;&lt;span class="o"&gt;:&lt;/span&gt;          &lt;span class="no"&gt;true&lt;/span&gt;&lt;span class="p"&gt;,&lt;/span&gt;
    &lt;span class="p"&gt;})&lt;/span&gt;
    &lt;span class="n"&gt;require&lt;/span&gt;&lt;span class="o"&gt;.&lt;/span&gt;&lt;span class="n"&gt;NoError&lt;/span&gt;&lt;span class="p"&gt;(&lt;/span&gt;&lt;span class="n"&gt;t&lt;/span&gt;&lt;span class="p"&gt;,&lt;/span&gt; &lt;span class="n"&gt;err&lt;/span&gt;&lt;span class="p"&gt;)&lt;/span&gt;
    &lt;span class="k"&gt;defer&lt;/span&gt; &lt;span class="n"&gt;mongoC&lt;/span&gt;&lt;span class="o"&gt;.&lt;/span&gt;&lt;span class="n"&gt;Terminate&lt;/span&gt;&lt;span class="p"&gt;(&lt;/span&gt;&lt;span class="n"&gt;ctx&lt;/span&gt;&lt;span class="p"&gt;)&lt;/span&gt;

    &lt;span class="c"&gt;// Get mapped host &amp;amp; port&lt;/span&gt;
    &lt;span class="n"&gt;host&lt;/span&gt;&lt;span class="p"&gt;,&lt;/span&gt; &lt;span class="n"&gt;err&lt;/span&gt; &lt;span class="o"&gt;:=&lt;/span&gt; &lt;span class="n"&gt;mongoC&lt;/span&gt;&lt;span class="o"&gt;.&lt;/span&gt;&lt;span class="n"&gt;Host&lt;/span&gt;&lt;span class="p"&gt;(&lt;/span&gt;&lt;span class="n"&gt;ctx&lt;/span&gt;&lt;span class="p"&gt;)&lt;/span&gt;
    &lt;span class="n"&gt;require&lt;/span&gt;&lt;span class="o"&gt;.&lt;/span&gt;&lt;span class="n"&gt;NoError&lt;/span&gt;&lt;span class="p"&gt;(&lt;/span&gt;&lt;span class="n"&gt;t&lt;/span&gt;&lt;span class="p"&gt;,&lt;/span&gt; &lt;span class="n"&gt;err&lt;/span&gt;&lt;span class="p"&gt;)&lt;/span&gt;
    &lt;span class="n"&gt;port&lt;/span&gt;&lt;span class="p"&gt;,&lt;/span&gt; &lt;span class="n"&gt;err&lt;/span&gt; &lt;span class="o"&gt;:=&lt;/span&gt; &lt;span class="n"&gt;mongoC&lt;/span&gt;&lt;span class="o"&gt;.&lt;/span&gt;&lt;span class="n"&gt;MappedPort&lt;/span&gt;&lt;span class="p"&gt;(&lt;/span&gt;&lt;span class="n"&gt;ctx&lt;/span&gt;&lt;span class="p"&gt;,&lt;/span&gt; &lt;span class="s"&gt;"27017"&lt;/span&gt;&lt;span class="p"&gt;)&lt;/span&gt;
    &lt;span class="n"&gt;require&lt;/span&gt;&lt;span class="o"&gt;.&lt;/span&gt;&lt;span class="n"&gt;NoError&lt;/span&gt;&lt;span class="p"&gt;(&lt;/span&gt;&lt;span class="n"&gt;t&lt;/span&gt;&lt;span class="p"&gt;,&lt;/span&gt; &lt;span class="n"&gt;err&lt;/span&gt;&lt;span class="p"&gt;)&lt;/span&gt;

    &lt;span class="c"&gt;// Connect with the Go driver&lt;/span&gt;
    &lt;span class="n"&gt;uri&lt;/span&gt; &lt;span class="o"&gt;:=&lt;/span&gt; &lt;span class="n"&gt;fmt&lt;/span&gt;&lt;span class="o"&gt;.&lt;/span&gt;&lt;span class="n"&gt;Sprintf&lt;/span&gt;&lt;span class="p"&gt;(&lt;/span&gt;&lt;span class="s"&gt;"mongodb://%s:%s"&lt;/span&gt;&lt;span class="p"&gt;,&lt;/span&gt; &lt;span class="n"&gt;host&lt;/span&gt;&lt;span class="p"&gt;,&lt;/span&gt; &lt;span class="n"&gt;port&lt;/span&gt;&lt;span class="o"&gt;.&lt;/span&gt;&lt;span class="n"&gt;Port&lt;/span&gt;&lt;span class="p"&gt;())&lt;/span&gt;
    &lt;span class="n"&gt;client&lt;/span&gt;&lt;span class="p"&gt;,&lt;/span&gt; &lt;span class="n"&gt;err&lt;/span&gt; &lt;span class="o"&gt;:=&lt;/span&gt; &lt;span class="n"&gt;mongo&lt;/span&gt;&lt;span class="o"&gt;.&lt;/span&gt;&lt;span class="n"&gt;Connect&lt;/span&gt;&lt;span class="p"&gt;(&lt;/span&gt;&lt;span class="n"&gt;options&lt;/span&gt;&lt;span class="o"&gt;.&lt;/span&gt;&lt;span class="n"&gt;Client&lt;/span&gt;&lt;span class="p"&gt;()&lt;/span&gt;&lt;span class="o"&gt;.&lt;/span&gt;&lt;span class="n"&gt;ApplyURI&lt;/span&gt;&lt;span class="p"&gt;(&lt;/span&gt;&lt;span class="n"&gt;uri&lt;/span&gt;&lt;span class="p"&gt;))&lt;/span&gt;
    &lt;span class="n"&gt;require&lt;/span&gt;&lt;span class="o"&gt;.&lt;/span&gt;&lt;span class="n"&gt;NoError&lt;/span&gt;&lt;span class="p"&gt;(&lt;/span&gt;&lt;span class="n"&gt;t&lt;/span&gt;&lt;span class="p"&gt;,&lt;/span&gt; &lt;span class="n"&gt;err&lt;/span&gt;&lt;span class="p"&gt;)&lt;/span&gt;
    &lt;span class="k"&gt;defer&lt;/span&gt; &lt;span class="n"&gt;client&lt;/span&gt;&lt;span class="o"&gt;.&lt;/span&gt;&lt;span class="n"&gt;Disconnect&lt;/span&gt;&lt;span class="p"&gt;(&lt;/span&gt;&lt;span class="n"&gt;ctx&lt;/span&gt;&lt;span class="p"&gt;)&lt;/span&gt;

    &lt;span class="n"&gt;err&lt;/span&gt; &lt;span class="o"&gt;:=&lt;/span&gt; &lt;span class="n"&gt;MyRepositoryFunction&lt;/span&gt;&lt;span class="p"&gt;(&lt;/span&gt;&lt;span class="n"&gt;context&lt;/span&gt;&lt;span class="o"&gt;.&lt;/span&gt;&lt;span class="n"&gt;TODO&lt;/span&gt;&lt;span class="p"&gt;(),&lt;/span&gt; &lt;span class="n"&gt;client&lt;/span&gt;&lt;span class="o"&gt;.&lt;/span&gt;&lt;span class="n"&gt;Database&lt;/span&gt;&lt;span class="p"&gt;(&lt;/span&gt;&lt;span class="s"&gt;"mydb"&lt;/span&gt;&lt;span class="p"&gt;)&lt;/span&gt;&lt;span class="o"&gt;.&lt;/span&gt;&lt;span class="n"&gt;Collection&lt;/span&gt;&lt;span class="p"&gt;(&lt;/span&gt;&lt;span class="s"&gt;"mycoll"&lt;/span&gt;&lt;span class="p"&gt;))&lt;/span&gt;
    &lt;span class="n"&gt;assert&lt;/span&gt;&lt;span class="o"&gt;.&lt;/span&gt;&lt;span class="n"&gt;NoError&lt;/span&gt;&lt;span class="p"&gt;(&lt;/span&gt;&lt;span class="n"&gt;t&lt;/span&gt;&lt;span class="p"&gt;,&lt;/span&gt; &lt;span class="n"&gt;err&lt;/span&gt;&lt;span class="p"&gt;,&lt;/span&gt; &lt;span class="s"&gt;"Should have successfully run"&lt;/span&gt;&lt;span class="p"&gt;)&lt;/span&gt;
&lt;span class="p"&gt;}&lt;/span&gt;
&lt;/code&gt;&lt;/pre&gt;

&lt;/div&gt;



&lt;p&gt;For necessary edge cases V2 provides a &lt;a href="https://pkg.go.dev/go.mongodb.org/mongo-driver/v2/x/mongo/driver/drivertest#MockDeployment" rel="noopener noreferrer"&gt;&lt;code&gt;MockDeployment&lt;/code&gt;&lt;/a&gt; option in the experimental API which &lt;strong&gt;may be modified or removed without notice&lt;/strong&gt;. Here is a re-worked example from &lt;a href="https://stackoverflow.com/questions/79240512/how-to-mock-mongodb-driver-in-golang/79240725#79240725" rel="noopener noreferrer"&gt;this&lt;/a&gt; StackOverflow answer for V2:&lt;br&gt;
&lt;/p&gt;

&lt;div class="highlight js-code-highlight"&gt;
&lt;pre class="highlight go"&gt;&lt;code&gt;&lt;span class="k"&gt;func&lt;/span&gt; &lt;span class="n"&gt;TestMyRepositoryFunction&lt;/span&gt;&lt;span class="p"&gt;(&lt;/span&gt;&lt;span class="n"&gt;t&lt;/span&gt; &lt;span class="o"&gt;*&lt;/span&gt;&lt;span class="n"&gt;testing&lt;/span&gt;&lt;span class="o"&gt;.&lt;/span&gt;&lt;span class="n"&gt;T&lt;/span&gt;&lt;span class="p"&gt;)&lt;/span&gt; &lt;span class="p"&gt;{&lt;/span&gt;
    &lt;span class="n"&gt;mockDep&lt;/span&gt; &lt;span class="o"&gt;:=&lt;/span&gt; &lt;span class="n"&gt;drivertest&lt;/span&gt;&lt;span class="o"&gt;.&lt;/span&gt;&lt;span class="n"&gt;NewMockDeployment&lt;/span&gt;&lt;span class="p"&gt;()&lt;/span&gt;

    &lt;span class="n"&gt;mockDep&lt;/span&gt;&lt;span class="o"&gt;.&lt;/span&gt;&lt;span class="n"&gt;AddResponses&lt;/span&gt;&lt;span class="p"&gt;(&lt;/span&gt;
        &lt;span class="n"&gt;bson&lt;/span&gt;&lt;span class="o"&gt;.&lt;/span&gt;&lt;span class="n"&gt;D&lt;/span&gt;&lt;span class="p"&gt;{{&lt;/span&gt;&lt;span class="n"&gt;Key&lt;/span&gt;&lt;span class="o"&gt;:&lt;/span&gt; &lt;span class="s"&gt;"ok"&lt;/span&gt;&lt;span class="p"&gt;,&lt;/span&gt; &lt;span class="n"&gt;Value&lt;/span&gt;&lt;span class="o"&gt;:&lt;/span&gt; &lt;span class="s"&gt;"1"&lt;/span&gt;&lt;span class="p"&gt;}},&lt;/span&gt;
        &lt;span class="n"&gt;bson&lt;/span&gt;&lt;span class="o"&gt;.&lt;/span&gt;&lt;span class="n"&gt;D&lt;/span&gt;&lt;span class="p"&gt;{&lt;/span&gt;
            &lt;span class="p"&gt;{&lt;/span&gt;&lt;span class="n"&gt;Key&lt;/span&gt;&lt;span class="o"&gt;:&lt;/span&gt; &lt;span class="s"&gt;"ok"&lt;/span&gt;&lt;span class="p"&gt;,&lt;/span&gt; &lt;span class="n"&gt;Value&lt;/span&gt;&lt;span class="o"&gt;:&lt;/span&gt; &lt;span class="s"&gt;"1"&lt;/span&gt;&lt;span class="p"&gt;},&lt;/span&gt;
            &lt;span class="p"&gt;{&lt;/span&gt;&lt;span class="n"&gt;Key&lt;/span&gt;&lt;span class="o"&gt;:&lt;/span&gt; &lt;span class="s"&gt;"value"&lt;/span&gt;&lt;span class="p"&gt;,&lt;/span&gt; &lt;span class="n"&gt;Value&lt;/span&gt;&lt;span class="o"&gt;:&lt;/span&gt; &lt;span class="n"&gt;bson&lt;/span&gt;&lt;span class="o"&gt;.&lt;/span&gt;&lt;span class="n"&gt;M&lt;/span&gt;&lt;span class="p"&gt;{&lt;/span&gt;&lt;span class="s"&gt;"_id"&lt;/span&gt;&lt;span class="o"&gt;:&lt;/span&gt; &lt;span class="s"&gt;"custom123"&lt;/span&gt;&lt;span class="p"&gt;,&lt;/span&gt; &lt;span class="s"&gt;"key"&lt;/span&gt;&lt;span class="o"&gt;:&lt;/span&gt; &lt;span class="m"&gt;24&lt;/span&gt;&lt;span class="p"&gt;}},&lt;/span&gt;
        &lt;span class="p"&gt;})&lt;/span&gt;

    &lt;span class="n"&gt;opts&lt;/span&gt; &lt;span class="o"&gt;:=&lt;/span&gt; &lt;span class="n"&gt;options&lt;/span&gt;&lt;span class="o"&gt;.&lt;/span&gt;&lt;span class="n"&gt;Client&lt;/span&gt;&lt;span class="p"&gt;()&lt;/span&gt;
    &lt;span class="n"&gt;opts&lt;/span&gt;&lt;span class="o"&gt;.&lt;/span&gt;&lt;span class="n"&gt;Deployment&lt;/span&gt; &lt;span class="o"&gt;=&lt;/span&gt; &lt;span class="n"&gt;mockDep&lt;/span&gt;

    &lt;span class="n"&gt;client&lt;/span&gt;&lt;span class="p"&gt;,&lt;/span&gt; &lt;span class="n"&gt;err&lt;/span&gt; &lt;span class="o"&gt;:=&lt;/span&gt; &lt;span class="n"&gt;mongo&lt;/span&gt;&lt;span class="o"&gt;.&lt;/span&gt;&lt;span class="n"&gt;Connect&lt;/span&gt;&lt;span class="p"&gt;(&lt;/span&gt;&lt;span class="n"&gt;opts&lt;/span&gt;&lt;span class="p"&gt;)&lt;/span&gt;
    &lt;span class="n"&gt;require&lt;/span&gt;&lt;span class="o"&gt;.&lt;/span&gt;&lt;span class="n"&gt;NoError&lt;/span&gt;&lt;span class="p"&gt;(&lt;/span&gt;&lt;span class="n"&gt;t&lt;/span&gt;&lt;span class="p"&gt;,&lt;/span&gt; &lt;span class="n"&gt;err&lt;/span&gt;&lt;span class="p"&gt;)&lt;/span&gt;
    &lt;span class="k"&gt;defer&lt;/span&gt; &lt;span class="n"&gt;client&lt;/span&gt;&lt;span class="o"&gt;.&lt;/span&gt;&lt;span class="n"&gt;Disconnect&lt;/span&gt;&lt;span class="p"&gt;(&lt;/span&gt;&lt;span class="n"&gt;ctx&lt;/span&gt;&lt;span class="p"&gt;)&lt;/span&gt;

    &lt;span class="n"&gt;err&lt;/span&gt; &lt;span class="o"&gt;:=&lt;/span&gt; &lt;span class="n"&gt;MyRepositoryFunction&lt;/span&gt;&lt;span class="p"&gt;(&lt;/span&gt;&lt;span class="n"&gt;context&lt;/span&gt;&lt;span class="o"&gt;.&lt;/span&gt;&lt;span class="n"&gt;TODO&lt;/span&gt;&lt;span class="p"&gt;(),&lt;/span&gt; &lt;span class="n"&gt;client&lt;/span&gt;&lt;span class="o"&gt;.&lt;/span&gt;&lt;span class="n"&gt;Database&lt;/span&gt;&lt;span class="p"&gt;(&lt;/span&gt;&lt;span class="s"&gt;"mydb"&lt;/span&gt;&lt;span class="p"&gt;)&lt;/span&gt;&lt;span class="o"&gt;.&lt;/span&gt;&lt;span class="n"&gt;Collection&lt;/span&gt;&lt;span class="p"&gt;(&lt;/span&gt;&lt;span class="s"&gt;"mycoll"&lt;/span&gt;&lt;span class="p"&gt;))&lt;/span&gt;

    &lt;span class="n"&gt;assert&lt;/span&gt;&lt;span class="o"&gt;.&lt;/span&gt;&lt;span class="n"&gt;NoError&lt;/span&gt;&lt;span class="p"&gt;(&lt;/span&gt;&lt;span class="n"&gt;t&lt;/span&gt;&lt;span class="p"&gt;,&lt;/span&gt; &lt;span class="n"&gt;err&lt;/span&gt;&lt;span class="p"&gt;,&lt;/span&gt; &lt;span class="s"&gt;"Should have successfully run"&lt;/span&gt;&lt;span class="p"&gt;)&lt;/span&gt;
&lt;span class="p"&gt;}&lt;/span&gt;
&lt;/code&gt;&lt;/pre&gt;

&lt;/div&gt;



</description>
    </item>
    <item>
      <title>Mocking an LLM Embedder Targeting MongoDB Atlas</title>
      <dc:creator>Preston Vasquez</dc:creator>
      <pubDate>Mon, 09 Sep 2024 15:43:50 +0000</pubDate>
      <link>https://forem.com/prestonvasquez/mocking-an-llm-embedder-targeting-mongodb-atlas-1glp</link>
      <guid>https://forem.com/prestonvasquez/mocking-an-llm-embedder-targeting-mongodb-atlas-1glp</guid>
      <description>&lt;p&gt;Implementation library: &lt;a href="https://github.com/prestonvasquez/vectormock" rel="noopener noreferrer"&gt;https://github.com/prestonvasquez/vectormock&lt;/a&gt;&lt;/p&gt;

&lt;p&gt;&lt;strong&gt;Problem&lt;/strong&gt;. Integration testing vector databases with live embedding models can be tricky since results may not always be deterministic. For example, three consecutive similarity searches for "South American Authors" might return:&lt;br&gt;
&lt;/p&gt;

&lt;div class="highlight js-code-highlight"&gt;
&lt;pre class="highlight python"&gt;&lt;code&gt;&lt;span class="p"&gt;{&lt;/span&gt;&lt;span class="n"&gt;text&lt;/span&gt;&lt;span class="p"&gt;:&lt;/span&gt; &lt;span class="sh"&gt;"&lt;/span&gt;&lt;span class="s"&gt;Gabriel García Márquez&lt;/span&gt;&lt;span class="sh"&gt;"&lt;/span&gt;&lt;span class="p"&gt;,&lt;/span&gt; &lt;span class="n"&gt;score&lt;/span&gt;&lt;span class="p"&gt;:&lt;/span&gt; &lt;span class="mf"&gt;0.80&lt;/span&gt;&lt;span class="p"&gt;}&lt;/span&gt; &lt;span class="c1"&gt;# Search 1
&lt;/span&gt;&lt;span class="p"&gt;{&lt;/span&gt;&lt;span class="n"&gt;text&lt;/span&gt;&lt;span class="p"&gt;:&lt;/span&gt; &lt;span class="sh"&gt;"&lt;/span&gt;&lt;span class="s"&gt;Gabriel García Márquez&lt;/span&gt;&lt;span class="sh"&gt;"&lt;/span&gt;&lt;span class="p"&gt;,&lt;/span&gt; &lt;span class="n"&gt;score&lt;/span&gt;&lt;span class="p"&gt;:&lt;/span&gt; &lt;span class="mf"&gt;0.82&lt;/span&gt;&lt;span class="p"&gt;}&lt;/span&gt; &lt;span class="c1"&gt;# Search 2
&lt;/span&gt;&lt;span class="p"&gt;{&lt;/span&gt;&lt;span class="n"&gt;text&lt;/span&gt;&lt;span class="p"&gt;:&lt;/span&gt; &lt;span class="sh"&gt;"&lt;/span&gt;&lt;span class="s"&gt;Gabriel García Márquez&lt;/span&gt;&lt;span class="sh"&gt;"&lt;/span&gt;&lt;span class="p"&gt;,&lt;/span&gt; &lt;span class="n"&gt;score&lt;/span&gt;&lt;span class="p"&gt;:&lt;/span&gt; &lt;span class="mf"&gt;0.78&lt;/span&gt;&lt;span class="p"&gt;}&lt;/span&gt; &lt;span class="c1"&gt;# Search 3
&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;

&lt;/div&gt;


&lt;p&gt;Suppose we expect the above similarity search to result in a score of &lt;code&gt;0.8&lt;/code&gt;. A common instinct to resolve false negatives (i.e. the &lt;code&gt;0.82&lt;/code&gt; and &lt;code&gt;0.78&lt;/code&gt; cases) would be to define an acceptable range for similarity scores:&lt;br&gt;
&lt;/p&gt;
&lt;div class="highlight js-code-highlight"&gt;
&lt;pre class="highlight python"&gt;&lt;code&gt;&lt;span class="k"&gt;def&lt;/span&gt; &lt;span class="nf"&gt;is_score_in_range&lt;/span&gt;&lt;span class="p"&gt;(&lt;/span&gt;&lt;span class="n"&gt;doc&lt;/span&gt;&lt;span class="p"&gt;,&lt;/span&gt; &lt;span class="n"&gt;lower&lt;/span&gt;&lt;span class="p"&gt;,&lt;/span&gt; &lt;span class="n"&gt;upper&lt;/span&gt;&lt;span class="p"&gt;):&lt;/span&gt;
    &lt;span class="k"&gt;return&lt;/span&gt; &lt;span class="n"&gt;lower&lt;/span&gt; &lt;span class="o"&gt;&amp;lt;=&lt;/span&gt; &lt;span class="n"&gt;doc&lt;/span&gt;&lt;span class="p"&gt;.&lt;/span&gt;&lt;span class="n"&gt;score&lt;/span&gt; &lt;span class="o"&gt;&amp;lt;=&lt;/span&gt; &lt;span class="n"&gt;upper&lt;/span&gt;

&lt;span class="k"&gt;class&lt;/span&gt; &lt;span class="nc"&gt;TestSimilarityScore&lt;/span&gt;&lt;span class="p"&gt;(&lt;/span&gt;&lt;span class="n"&gt;unittest&lt;/span&gt;&lt;span class="p"&gt;.&lt;/span&gt;&lt;span class="n"&gt;TestCase&lt;/span&gt;&lt;span class="p"&gt;):&lt;/span&gt;
    &lt;span class="k"&gt;def&lt;/span&gt; &lt;span class="nf"&gt;test_doc_scores&lt;/span&gt;&lt;span class="p"&gt;(&lt;/span&gt;&lt;span class="n"&gt;self&lt;/span&gt;&lt;span class="p"&gt;):&lt;/span&gt;
        &lt;span class="n"&gt;docs&lt;/span&gt; &lt;span class="o"&gt;=&lt;/span&gt; &lt;span class="p"&gt;[{&lt;/span&gt;&lt;span class="sh"&gt;"&lt;/span&gt;&lt;span class="s"&gt;score&lt;/span&gt;&lt;span class="sh"&gt;"&lt;/span&gt;&lt;span class="p"&gt;:&lt;/span&gt; &lt;span class="mf"&gt;0.80&lt;/span&gt;&lt;span class="p"&gt;},&lt;/span&gt; &lt;span class="p"&gt;{&lt;/span&gt;&lt;span class="sh"&gt;"&lt;/span&gt;&lt;span class="s"&gt;score&lt;/span&gt;&lt;span class="sh"&gt;"&lt;/span&gt;&lt;span class="p"&gt;:&lt;/span&gt; &lt;span class="mf"&gt;0.82&lt;/span&gt;&lt;span class="p"&gt;},&lt;/span&gt; &lt;span class="p"&gt;{&lt;/span&gt;&lt;span class="sh"&gt;"&lt;/span&gt;&lt;span class="s"&gt;score&lt;/span&gt;&lt;span class="sh"&gt;"&lt;/span&gt;&lt;span class="p"&gt;:&lt;/span&gt; &lt;span class="mf"&gt;0.78&lt;/span&gt;&lt;span class="p"&gt;}]&lt;/span&gt;
        &lt;span class="k"&gt;for&lt;/span&gt; &lt;span class="n"&gt;doc&lt;/span&gt; &lt;span class="ow"&gt;in&lt;/span&gt; &lt;span class="n"&gt;docs&lt;/span&gt;&lt;span class="p"&gt;:&lt;/span&gt;
            &lt;span class="c1"&gt;# If we expect 0.80, then the delta = 2.
&lt;/span&gt;            &lt;span class="n"&gt;self&lt;/span&gt;&lt;span class="p"&gt;.&lt;/span&gt;&lt;span class="nf"&gt;assertTrue&lt;/span&gt;&lt;span class="p"&gt;(&lt;/span&gt;&lt;span class="nf"&gt;is_score_in_range&lt;/span&gt;&lt;span class="p"&gt;(&lt;/span&gt;&lt;span class="n"&gt;doc&lt;/span&gt;&lt;span class="p"&gt;,&lt;/span&gt; &lt;span class="mf"&gt;0.78&lt;/span&gt;&lt;span class="p"&gt;,&lt;/span&gt; &lt;span class="mf"&gt;0.82&lt;/span&gt;&lt;span class="p"&gt;))&lt;/span&gt;
&lt;/code&gt;&lt;/pre&gt;

&lt;/div&gt;


&lt;p&gt;Of course, we are met with a difficult question: how do you set the delta? Large deltas increase false positives; small deltas increases false negatives:&lt;br&gt;
&lt;/p&gt;
&lt;div class="highlight js-code-highlight"&gt;
&lt;pre class="highlight python"&gt;&lt;code&gt;&lt;span class="c1"&gt;# Wouldn't pass with delta=2
&lt;/span&gt;&lt;span class="p"&gt;{&lt;/span&gt;&lt;span class="n"&gt;text&lt;/span&gt;&lt;span class="p"&gt;:&lt;/span&gt; &lt;span class="sh"&gt;"&lt;/span&gt;&lt;span class="s"&gt;Gabriel García Márquez&lt;/span&gt;&lt;span class="sh"&gt;"&lt;/span&gt;&lt;span class="p"&gt;,&lt;/span&gt; &lt;span class="n"&gt;score&lt;/span&gt;&lt;span class="p"&gt;:&lt;/span&gt; &lt;span class="mf"&gt;0.779&lt;/span&gt;&lt;span class="p"&gt;}&lt;/span&gt;

&lt;span class="c1"&gt;# Would pass with delta=40
&lt;/span&gt;&lt;span class="p"&gt;{&lt;/span&gt;&lt;span class="n"&gt;text&lt;/span&gt;&lt;span class="p"&gt;:&lt;/span&gt; &lt;span class="sh"&gt;"&lt;/span&gt;&lt;span class="s"&gt;Gabriel García Márquez&lt;/span&gt;&lt;span class="sh"&gt;"&lt;/span&gt;&lt;span class="p"&gt;,&lt;/span&gt; &lt;span class="n"&gt;score&lt;/span&gt;&lt;span class="p"&gt;:&lt;/span&gt; &lt;span class="mf"&gt;0.41&lt;/span&gt;&lt;span class="p"&gt;}&lt;/span&gt;
&lt;/code&gt;&lt;/pre&gt;

&lt;/div&gt;


&lt;p&gt;Additionally, embedding models may have rate limits, quotas, or per-usage fees. All of which add to the difficulty of maintaining robust tests.&lt;/p&gt;

&lt;p&gt;&lt;strong&gt;Goal&lt;/strong&gt;. This article shows how to overcome these obstacles by mocking an embedding model for MongoDB Atlas, making integration tests more predictable and reliable. For example, say we want the query "South American Authors" to always return the following three documents (without variation) for a vector database integration test:&lt;br&gt;
&lt;/p&gt;
&lt;div class="highlight js-code-highlight"&gt;
&lt;pre class="highlight python"&gt;&lt;code&gt;&lt;span class="p"&gt;{&lt;/span&gt;&lt;span class="n"&gt;text&lt;/span&gt;&lt;span class="p"&gt;:&lt;/span&gt; &lt;span class="sh"&gt;"&lt;/span&gt;&lt;span class="s"&gt;Gabriel García Márquez&lt;/span&gt;&lt;span class="sh"&gt;"&lt;/span&gt;&lt;span class="p"&gt;,&lt;/span&gt; &lt;span class="n"&gt;score&lt;/span&gt;&lt;span class="p"&gt;:&lt;/span&gt; &lt;span class="mf"&gt;0.80&lt;/span&gt;&lt;span class="p"&gt;}&lt;/span&gt;
&lt;span class="p"&gt;{&lt;/span&gt;&lt;span class="n"&gt;text&lt;/span&gt;&lt;span class="p"&gt;:&lt;/span&gt; &lt;span class="sh"&gt;"&lt;/span&gt;&lt;span class="s"&gt;Gabriela Mistral&lt;/span&gt;&lt;span class="sh"&gt;"&lt;/span&gt;&lt;span class="p"&gt;,&lt;/span&gt; &lt;span class="n"&gt;score&lt;/span&gt;&lt;span class="p"&gt;:&lt;/span&gt; &lt;span class="mf"&gt;0.94&lt;/span&gt;&lt;span class="p"&gt;}&lt;/span&gt;       
&lt;span class="p"&gt;{&lt;/span&gt;&lt;span class="n"&gt;text&lt;/span&gt;&lt;span class="p"&gt;:&lt;/span&gt; &lt;span class="sh"&gt;"&lt;/span&gt;&lt;span class="s"&gt;Miguel de Cervantes&lt;/span&gt;&lt;span class="sh"&gt;"&lt;/span&gt;&lt;span class="p"&gt;,&lt;/span&gt; &lt;span class="n"&gt;score&lt;/span&gt;&lt;span class="p"&gt;:&lt;/span&gt; &lt;span class="mf"&gt;0.07&lt;/span&gt;&lt;span class="p"&gt;}&lt;/span&gt;   
&lt;/code&gt;&lt;/pre&gt;

&lt;/div&gt;


&lt;p&gt;This guide is specifically focused on similarity scores derived through the &lt;a href="https://en.wikipedia.org/wiki/Dot_product" rel="noopener noreferrer"&gt;dot product&lt;/a&gt;.&lt;/p&gt;

&lt;p&gt;To simplify things, we'll focus on &lt;a href="https://python.langchain.com/v0.2/docs/introduction/" rel="noopener noreferrer"&gt;LangChain&lt;/a&gt;'s definition of an embedding model, which requires the implementation of two key functions: &lt;a href="https://python.langchain.com/v0.2/docs/how_to/embed_text/#embed_query" rel="noopener noreferrer"&gt;&lt;code&gt;embed_query&lt;/code&gt;&lt;/a&gt; and &lt;a href="https://python.langchain.com/v0.2/docs/how_to/embed_text/#embed_documents" rel="noopener noreferrer"&gt;&lt;code&gt;embed_documents&lt;/code&gt;&lt;/a&gt;.&lt;/p&gt;

&lt;p&gt;&lt;strong&gt;Deriving &lt;code&gt;embed_query&lt;/code&gt;&lt;/strong&gt;. The goal of an &lt;a href="https://en.wikipedia.org/wiki/Word_embedding" rel="noopener noreferrer"&gt;embedding model&lt;/a&gt; (embedder) is to transform 

&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;
-queries to  
&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;
 
&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;
-dimensional vectors (an embedding) such that the relationships between different inputs are preserved in the embedding space.&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%2F5fy652hafgl7ifxv3an7.png" 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%2F5fy652hafgl7ifxv3an7.png" alt="Image description" width="800" height="181"&gt;&lt;/a&gt;&lt;/p&gt;

&lt;p&gt;So let 
&lt;span class="katex-element"&gt;
  &lt;span class="katex"&gt;&lt;span class="katex-mathml"&gt;qq&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;q&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;
&lt;/span&gt;
 be a mapping from a query vector 
&lt;span class="katex-element"&gt;
  &lt;span class="katex"&gt;&lt;span class="katex-mathml"&gt;v∗\mathbf{v}^{*} &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 mathbf"&gt;v&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;∗&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&gt;
 to a set of similar vectors defined as the column space of 
&lt;span class="katex-element"&gt;
  &lt;span class="katex"&gt;&lt;span class="katex-mathml"&gt;E∈Rn×N\mathbf{E} \in \mathbb{R}^{n\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 mathbf"&gt;E&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 mathbb"&gt;R&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 mathnormal mtight"&gt;n&lt;/span&gt;&lt;span class="mbin mtight"&gt;×&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&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;
&lt;/span&gt;
, up-to the similarity score algorithm (e.g. dot product). &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;q:v∗→C(E)
\begin{equation}
q:\mathbf{v}^* \rightarrow \text{C}(\mathbf{E})
\end{equation}
&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="mtable"&gt;&lt;span class="col-align-c"&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 mathnormal"&gt;q&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 class="mord"&gt;&lt;span class="mord mathbf"&gt;v&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="mbin mtight"&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 class="mord text"&gt;&lt;span class="mord"&gt;C&lt;/span&gt;&lt;/span&gt;&lt;span class="mopen"&gt;(&lt;/span&gt;&lt;span class="mord mathbf"&gt;E&lt;/span&gt;&lt;span class="mclose"&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="tag"&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="eqn-num"&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;/div&gt;



&lt;p&gt;We want to define 
&lt;span class="katex-element"&gt;
  &lt;span class="katex"&gt;&lt;span class="katex-mathml"&gt;v∗\mathbf{v}^*&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 mathbf"&gt;v&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="mbin mtight"&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;
 such that the following holds for all 
&lt;span class="katex-element"&gt;
  &lt;span class="katex"&gt;&lt;span class="katex-mathml"&gt;v∈C(E)\mathbf{v} \in \text{C}(\mathbf{E})&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 mathbf"&gt;v&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 text"&gt;&lt;span class="mord"&gt;C&lt;/span&gt;&lt;/span&gt;&lt;span class="mopen"&gt;(&lt;/span&gt;&lt;span class="mord mathbf"&gt;E&lt;/span&gt;&lt;span class="mclose"&gt;)&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;v∗⋅v=S^
\begin{equation}
\mathbf{v}^{*} \cdot \mathbf{v} = \hat{S}
\end{equation}
&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="mtable"&gt;&lt;span class="col-align-c"&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 mathbf"&gt;v&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;∗&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="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 mathbf"&gt;v&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 class="mord accent"&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="mord mathnormal"&gt;S&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="accent-body"&gt;&lt;span class="mord"&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&gt;&lt;/span&gt;&lt;span class="tag"&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="eqn-num"&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;/div&gt;


&lt;p&gt;Where 
&lt;span class="katex-element"&gt;
  &lt;span class="katex"&gt;&lt;span class="katex-mathml"&gt;S^\hat{S}&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 accent"&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="mord mathnormal"&gt;S&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="accent-body"&gt;&lt;span class="mord"&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;
 is derived from normalizing the similarity score. For example, Atlas Vector Search uses the following &lt;a href="https://www.mongodb.com/docs/atlas/atlas-vector-search/vector-search-stage/#atlas-vector-search-score" rel="noopener noreferrer"&gt;algorithm&lt;/a&gt; to normalize the score using dot product:&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;S=12(1+v∗⋅v)
\begin{equation*}
S = \frac{1}{2}\left(1+ \mathbf{v}^{*} \cdot \mathbf{v}\right)
\end{equation*}
&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="mtable"&gt;&lt;span class="col-align-c"&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 mathnormal"&gt;S&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 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;2&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;1&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="minner"&gt;&lt;span class="mopen delimcenter"&gt;(&lt;/span&gt;&lt;span class="mord"&gt;1&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;&lt;span class="mord mathbf"&gt;v&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;∗&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="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 mathbf"&gt;v&lt;/span&gt;&lt;span class="mclose delimcenter"&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;Therefore, 
&lt;span class="katex-element"&gt;
  &lt;span class="katex"&gt;&lt;span class="katex-mathml"&gt;S^=2S−1\hat{S}=2S - 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 accent"&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="mord mathnormal"&gt;S&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="accent-body"&gt;&lt;span class="mord"&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 mathnormal"&gt;S&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;
. &lt;/p&gt;

&lt;p&gt;Using our example, let 
&lt;span class="katex-element"&gt;
  &lt;span class="katex"&gt;&lt;span class="katex-mathml"&gt;vA∗\mathbf{v_A}^*&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"&gt;&lt;span class="mord mathbf"&gt;v&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 mathbf mtight"&gt;A&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="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="mbin mtight"&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;
 be the query vector for "South American Authors". For simplicity, embeddable documents are just a tuple 
&lt;span class="katex-element"&gt;
  &lt;span class="katex"&gt;&lt;span class="katex-mathml"&gt;(P,S)\left(P, S\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 delimcenter"&gt;(&lt;/span&gt;&lt;span class="mord mathnormal"&gt;P&lt;/span&gt;&lt;span class="mpunct"&gt;,&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;span class="mord mathnormal"&gt;S&lt;/span&gt;&lt;span class="mclose delimcenter"&gt;)&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;
&lt;/span&gt;
 where 
&lt;span class="katex-element"&gt;
  &lt;span class="katex"&gt;&lt;span class="katex-mathml"&gt;PP&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;P&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;
&lt;/span&gt;
 is the textual representation of a vector in 
&lt;span class="katex-element"&gt;
  &lt;span class="katex"&gt;&lt;span class="katex-mathml"&gt;C(E)\text{C}(\mathbf{E})&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;C&lt;/span&gt;&lt;/span&gt;&lt;span class="mopen"&gt;(&lt;/span&gt;&lt;span class="mord mathbf"&gt;E&lt;/span&gt;&lt;span class="mclose"&gt;)&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;SS&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;S&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;
&lt;/span&gt;
 is the desired score as it relates to the query vector. If we call 
&lt;span class="katex-element"&gt;
  &lt;span class="katex"&gt;&lt;span class="katex-mathml"&gt;ϕ\phi&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;ϕ&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;
&lt;/span&gt;
 the mock &lt;a href="https://en.wikipedia.org/wiki/Transformer_(deep_learning_architecture)" rel="noopener noreferrer"&gt;transformer&lt;/a&gt;, then let 
&lt;span class="katex-element"&gt;
  &lt;span class="katex"&gt;&lt;span class="katex-mathml"&gt;v1,v2,v3\mathbf{v}_1, \mathbf{v}_2, \mathbf{v}_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"&gt;&lt;span class="mord mathbf"&gt;v&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;1&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="mpunct"&gt;,&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mord mathbf"&gt;v&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;2&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="mpunct"&gt;,&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mord mathbf"&gt;v&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;3&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;
 be the embeddings for Márquez, Mistral, and Cervantes respectively. Without loss of generality:&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;ϕ(Maˊrquez,0.80)→v1such thatvA∗⋅v1=0.6
\begin{equation*}
\phi\left(\text{M\'{a}rquez}, 0.80\right) \rightarrow \mathbf{v}_1 \quad \text{such that} \quad \mathbf{v_A}^* \cdot \mathbf{v}_1 = 0.6
\end{equation*}
&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="mtable"&gt;&lt;span class="col-align-c"&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 mathnormal"&gt;ϕ&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;span class="minner"&gt;&lt;span class="mopen delimcenter"&gt;(&lt;/span&gt;&lt;span class="mord text"&gt;&lt;span class="mord"&gt;M&lt;/span&gt;&lt;span class="mord accent"&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="mord"&gt;&lt;span class="mord"&gt;a&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="accent-body"&gt;&lt;span class="mord"&gt;ˊ&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;rquez&lt;/span&gt;&lt;/span&gt;&lt;span class="mpunct"&gt;,&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;span class="mord"&gt;0.80&lt;/span&gt;&lt;span class="mclose delimcenter"&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 class="mord"&gt;&lt;span class="mord mathbf"&gt;v&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;1&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="mord text"&gt;&lt;span class="mord"&gt;such that&lt;/span&gt;&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mord"&gt;&lt;span class="mord mathbf"&gt;v&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 mathbf mtight"&gt;A&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="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="mbin mtight"&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 class="mord"&gt;&lt;span class="mord mathbf"&gt;v&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;1&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 class="mord"&gt;0.6&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;And to complete the mock embedding, we want:&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;q(vA∗)→[v1,v2,v3]
\begin{equation*}
q\left(\mathbf{v_A}^{*}\right) \to \left[\mathbf{v}_1, \mathbf{v}_2,  \mathbf{v}_3 \right]
\end{equation*}
&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="mtable"&gt;&lt;span class="col-align-c"&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 mathnormal"&gt;q&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;span class="minner"&gt;&lt;span class="mopen delimcenter"&gt;(&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mord"&gt;&lt;span class="mord mathbf"&gt;v&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 mathbf mtight"&gt;A&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="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;∗&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="mclose delimcenter"&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 class="minner"&gt;&lt;span class="mopen delimcenter"&gt;[&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mord mathbf"&gt;v&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;1&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="mpunct"&gt;,&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mord mathbf"&gt;v&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;2&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="mpunct"&gt;,&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mord mathbf"&gt;v&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;3&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="mclose delimcenter"&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;Looking at libraries like &lt;a href="https://github.com/langchain-ai/langchain" rel="noopener noreferrer"&gt;LangChain&lt;/a&gt;, 
&lt;span class="katex-element"&gt;
  &lt;span class="katex"&gt;&lt;span class="katex-mathml"&gt;qq&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;q&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;
&lt;/span&gt;
 is analogous to &lt;code&gt;similarity_search&lt;/code&gt; and 
&lt;span class="katex-element"&gt;
  &lt;span class="katex"&gt;&lt;span class="katex-mathml"&gt;ϕ\phi&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;ϕ&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;
&lt;/span&gt;
 is &lt;code&gt;embed_documents&lt;/code&gt;. 
&lt;span class="katex-element"&gt;
  &lt;span class="katex"&gt;&lt;span class="katex-mathml"&gt;C(E)C(\mathbf{E})&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="mopen"&gt;(&lt;/span&gt;&lt;span class="mord mathbf"&gt;E&lt;/span&gt;&lt;span class="mclose"&gt;)&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;
&lt;/span&gt;
 is a subset of the vector store (e.g. a MongoDB collection) and if it was embedded by &lt;code&gt;text-embedding-3-small&lt;/code&gt;, for example, 
&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;
 would be 
&lt;span class="katex-element"&gt;
  &lt;span class="katex"&gt;&lt;span class="katex-mathml"&gt;15361536&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;1536&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;
&lt;/span&gt;
. Additionally, since 
&lt;span class="katex-element"&gt;
  &lt;span class="katex"&gt;&lt;span class="katex-mathml"&gt;vA∗\mathbf{v_A}^*&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"&gt;&lt;span class="mord mathbf"&gt;v&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 mathbf mtight"&gt;A&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="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="mbin mtight"&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;
 is fixed, then &lt;code&gt;embed_query&lt;/code&gt; is simply an accessor:&lt;br&gt;
&lt;/p&gt;

&lt;div class="highlight js-code-highlight"&gt;
&lt;pre class="highlight python"&gt;&lt;code&gt;&lt;span class="k"&gt;def&lt;/span&gt; &lt;span class="nf"&gt;embed_query&lt;/span&gt;&lt;span class="p"&gt;(&lt;/span&gt;&lt;span class="n"&gt;self&lt;/span&gt;&lt;span class="p"&gt;,&lt;/span&gt; &lt;span class="n"&gt;context&lt;/span&gt;&lt;span class="p"&gt;,&lt;/span&gt; &lt;span class="n"&gt;text&lt;/span&gt;&lt;span class="p"&gt;):&lt;/span&gt;
    &lt;span class="c1"&gt;# Return the query vector stored in the object (emb.query_vector)
&lt;/span&gt;    &lt;span class="k"&gt;return&lt;/span&gt; &lt;span class="n"&gt;self&lt;/span&gt;&lt;span class="p"&gt;.&lt;/span&gt;&lt;span class="n"&gt;query_vector&lt;/span&gt;&lt;span class="p"&gt;,&lt;/span&gt; &lt;span class="bp"&gt;None&lt;/span&gt;
&lt;/code&gt;&lt;/pre&gt;

&lt;/div&gt;


&lt;p&gt;The next step is to define an algorithm for deriving 
&lt;span class="katex-element"&gt;
  &lt;span class="katex"&gt;&lt;span class="katex-mathml"&gt;ϕ\phi&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;ϕ&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;
&lt;/span&gt;
. But before we get into that, enjoy this (&lt;a href="https://xkcd.com/1838/):" rel="noopener noreferrer"&gt;https://xkcd.com/1838/):&lt;/a&gt;&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%2F1qqbke9s0r69x2ie86ug.png" 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%2F1qqbke9s0r69x2ie86ug.png" alt="Image description" width="742" height="877"&gt;&lt;/a&gt;&lt;/p&gt;

&lt;p&gt;&lt;strong&gt;Deriving &lt;code&gt;embed_documents&lt;/code&gt;&lt;/strong&gt;. Generally, 
&lt;span class="katex-element"&gt;
  &lt;span class="katex"&gt;&lt;span class="katex-mathml"&gt;ϕv∗\phi^{\mathbf{v}^{*}}&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;ϕ&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;&lt;span class="mord mathbf mtight"&gt;v&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-size3 size1 mtight"&gt;&lt;span class="mord mtight"&gt;&lt;span class="mord mtight"&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;/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;
 maps a document 
&lt;span class="katex-element"&gt;
  &lt;span class="katex"&gt;&lt;span class="katex-mathml"&gt;(P,S)\left(P, S\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 delimcenter"&gt;(&lt;/span&gt;&lt;span class="mord mathnormal"&gt;P&lt;/span&gt;&lt;span class="mpunct"&gt;,&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;span class="mord mathnormal"&gt;S&lt;/span&gt;&lt;span class="mclose delimcenter"&gt;)&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;
&lt;/span&gt;
 to a vector 
&lt;span class="katex-element"&gt;
  &lt;span class="katex"&gt;&lt;span class="katex-mathml"&gt;v\mathbf{v}&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 mathbf"&gt;v&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;
&lt;/span&gt;
 such that 
&lt;span class="katex-element"&gt;
  &lt;span class="katex"&gt;&lt;span class="katex-mathml"&gt;v∗⋅v=2S−1\mathbf{v}^{*} \cdot \mathbf{v} = 2S - 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;&lt;span class="mord mathbf"&gt;v&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;∗&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="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 mathbf"&gt;v&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 mathnormal"&gt;S&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;
:&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;ϕv∗:(P,S)→vsuch thatv∗⋅v=2S−1
\begin{equation}
\phi^{\mathbf{v}^{*}}: \left(P, S\right) \rightarrow \mathbf{v} \quad \text{such that} \quad \mathbf{v}^* \cdot \mathbf{v} = 2S - 1
\end{equation}
&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="mtable"&gt;&lt;span class="col-align-c"&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;ϕ&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;&lt;span class="mord mathbf mtight"&gt;v&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-size3 size1 mtight"&gt;&lt;span class="mord mtight"&gt;&lt;span class="mord mtight"&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;/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 class="minner"&gt;&lt;span class="mopen delimcenter"&gt;(&lt;/span&gt;&lt;span class="mord mathnormal"&gt;P&lt;/span&gt;&lt;span class="mpunct"&gt;,&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;span class="mord mathnormal"&gt;S&lt;/span&gt;&lt;span class="mclose delimcenter"&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 class="mord mathbf"&gt;v&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;span class="mord text"&gt;&lt;span class="mord"&gt;such that&lt;/span&gt;&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mord mathbf"&gt;v&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="mbin mtight"&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 class="mord mathbf"&gt;v&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 class="mord"&gt;2&lt;/span&gt;&lt;span class="mord mathnormal"&gt;S&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&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="tag"&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="eqn-num"&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;/div&gt;



&lt;p&gt;In other words, 
&lt;span class="katex-element"&gt;
  &lt;span class="katex"&gt;&lt;span class="katex-mathml"&gt;ϕv∗\phi^{\mathbf{v}^{*}}&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;ϕ&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;&lt;span class="mord mathbf mtight"&gt;v&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-size3 size1 mtight"&gt;&lt;span class="mord mtight"&gt;&lt;span class="mord mtight"&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;/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;
 is used to generate the vectors that compose 
&lt;span class="katex-element"&gt;
  &lt;span class="katex"&gt;&lt;span class="katex-mathml"&gt;E\mathbf{E} &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 mathbf"&gt;E&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;
&lt;/span&gt;
 from (eq. 1). By definition (eq. 2), since 
&lt;span class="katex-element"&gt;
  &lt;span class="katex"&gt;&lt;span class="katex-mathml"&gt;Rank(E)=N\text{Rank}(\mathbf{E}) = 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 text"&gt;&lt;span class="mord"&gt;Rank&lt;/span&gt;&lt;/span&gt;&lt;span class="mopen"&gt;(&lt;/span&gt;&lt;span class="mord mathbf"&gt;E&lt;/span&gt;&lt;span class="mclose"&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 mathnormal"&gt;N&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;
&lt;/span&gt;
 and the dimension of 
&lt;span class="katex-element"&gt;
  &lt;span class="katex"&gt;&lt;span class="katex-mathml"&gt;C(E)\text{C}(\mathbf{E})&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;C&lt;/span&gt;&lt;/span&gt;&lt;span class="mopen"&gt;(&lt;/span&gt;&lt;span class="mord mathbf"&gt;E&lt;/span&gt;&lt;span class="mclose"&gt;)&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;
&lt;/span&gt;
 is also 
&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;
, the vectors in 
&lt;span class="katex-element"&gt;
  &lt;span class="katex"&gt;&lt;span class="katex-mathml"&gt;C(E)\text{C}(\mathbf{E})&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;C&lt;/span&gt;&lt;/span&gt;&lt;span class="mopen"&gt;(&lt;/span&gt;&lt;span class="mord mathbf"&gt;E&lt;/span&gt;&lt;span class="mclose"&gt;)&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;
&lt;/span&gt;
 must be linearly independent. Therefore defining 
&lt;span class="katex-element"&gt;
  &lt;span class="katex"&gt;&lt;span class="katex-mathml"&gt;ϕv∗\phi^{\mathbf{v}^{*}}&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;ϕ&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;&lt;span class="mord mathbf mtight"&gt;v&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-size3 size1 mtight"&gt;&lt;span class="mord mtight"&gt;&lt;span class="mord mtight"&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;/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;
 can be simplified into two conditions that must both hold:&lt;/p&gt;

&lt;ol&gt;
&lt;li&gt;All vectors in the set 
&lt;span class="katex-element"&gt;
  &lt;span class="katex"&gt;&lt;span class="katex-mathml"&gt;v1,v2,…,vN,v∗{\mathbf{v_1}, \mathbf{v_2}, \ldots, \mathbf{v_N}, \mathbf{v^*}}&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"&gt;&lt;span class="mord mathbf"&gt;v&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 mathbf mtight"&gt;1&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="mpunct"&gt;,&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mord mathbf"&gt;v&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 mathbf mtight"&gt;2&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="mpunct"&gt;,&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;span class="minner"&gt;…&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;span class="mpunct"&gt;,&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mord mathbf"&gt;v&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 mathbf mtight"&gt;N&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="mpunct"&gt;,&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mord mathbf"&gt;v&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="mbin mtight"&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;/span&gt;
 are linearly independent (Algorithm A2).&lt;/li&gt;
&lt;li&gt;
&lt;span class="katex-element"&gt;
  &lt;span class="katex"&gt;&lt;span class="katex-mathml"&gt;v⋅v∗=2Sv−1\mathbf{v} \cdot \mathbf{v^*} = 2S_{\mathbf{v}} - 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 mathbf"&gt;v&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 mathbf"&gt;v&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="mbin mtight"&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="mord"&gt;2&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mord mathnormal"&gt;S&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 mathbf mtight"&gt;v&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;1&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;
&lt;/span&gt;
, where 
&lt;span class="katex-element"&gt;
  &lt;span class="katex"&gt;&lt;span class="katex-mathml"&gt;v∈C(E)\mathbf{v} \in \text{C}(\mathbf{E})&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 mathbf"&gt;v&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 text"&gt;&lt;span class="mord"&gt;C&lt;/span&gt;&lt;/span&gt;&lt;span class="mopen"&gt;(&lt;/span&gt;&lt;span class="mord mathbf"&gt;E&lt;/span&gt;&lt;span class="mclose"&gt;)&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;
&lt;/span&gt;
 (Algorithm A3).&lt;/li&gt;
&lt;/ol&gt;

&lt;p&gt;Without loss of generality, to satisfy condition 1, we need at least one 
&lt;span class="katex-element"&gt;
  &lt;span class="katex"&gt;&lt;span class="katex-mathml"&gt;viv_i&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;v&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 mathnormal mtight"&gt;i&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;
 such that the following does not hold:&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;v1v1∗=v2v2∗=…=vn−1vn−1∗
\begin{equation*}

\frac{v_1}{v_1^*} = \frac{v_2}{v_2^*} = \ldots = \frac{v_{n-1}}{v_{n-1}^*}

\end{equation*}
&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="mtable"&gt;&lt;span class="col-align-c"&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="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;&lt;span class="mord mathnormal"&gt;v&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;1&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="mbin mtight"&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="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;&lt;span class="mord mathnormal"&gt;v&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;1&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="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 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;&lt;span class="mord mathnormal"&gt;v&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;2&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="mbin mtight"&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="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;&lt;span class="mord mathnormal"&gt;v&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;2&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="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 class="minner"&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 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;&lt;span class="mord mathnormal"&gt;v&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="mbin 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 class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mbin mtight"&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="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;&lt;span class="mord mathnormal"&gt;v&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="mbin 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="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="mclose nulldelimiter"&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;This ensures that 
&lt;span class="katex-element"&gt;
  &lt;span class="katex"&gt;&lt;span class="katex-mathml"&gt;v\mathbf{v}&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 mathbf"&gt;v&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;
&lt;/span&gt;
 is not a scalar multiple of 
&lt;span class="katex-element"&gt;
  &lt;span class="katex"&gt;&lt;span class="katex-mathml"&gt;v∗\mathbf{v}^* &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 mathbf"&gt;v&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="mbin mtight"&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;
. To guarantee condition 2 is met, we update 
&lt;span class="katex-element"&gt;
  &lt;span class="katex"&gt;&lt;span class="katex-mathml"&gt;vnv_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"&gt;&lt;span class="mord mathnormal"&gt;v&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 mathnormal mtight"&gt;n&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;/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;vn=1vn∗(2S−1−∑i=1n−1vi∗vi)
\begin{equation}
v_n =  \frac{1}{v_n^*} \left(2S - 1 - \sum_{i=1}^{n-1} v_i^*v_i \right)
\end{equation}
&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="mtable"&gt;&lt;span class="col-align-c"&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;v&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 mathnormal mtight"&gt;n&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 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;&lt;span class="mord mathnormal"&gt;v&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 mathnormal mtight"&gt;n&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="mbin mtight"&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="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;1&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="minner"&gt;&lt;span class="mopen delimcenter"&gt;&lt;span class="delimsizing size4"&gt;(&lt;/span&gt;&lt;/span&gt;&lt;span class="mord"&gt;2&lt;/span&gt;&lt;span class="mord mathnormal"&gt;S&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="mspace"&gt;&lt;/span&gt;&lt;span class="mbin"&gt;−&lt;/span&gt;&lt;span class="mspace"&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;i&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 class="mbin 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="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;v&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 mathnormal mtight"&gt;i&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="mbin mtight"&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;v&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 mathnormal mtight"&gt;i&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="mclose delimcenter"&gt;&lt;span class="delimsizing size4"&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&gt;&lt;/span&gt;&lt;span class="tag"&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="eqn-num"&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;/div&gt;


&lt;p&gt;One &lt;em&gt;very&lt;/em&gt; important note: this method will not guarantee the final element falls in the range of 
&lt;span class="katex-element"&gt;
  &lt;span class="katex"&gt;&lt;span class="katex-mathml"&gt;[−1,1][-1, 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="mopen"&gt;[&lt;/span&gt;&lt;span class="mord"&gt;−&lt;/span&gt;&lt;span class="mord"&gt;1&lt;/span&gt;&lt;span class="mpunct"&gt;,&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;span class="mord"&gt;1&lt;/span&gt;&lt;span class="mclose"&gt;]&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;
&lt;/span&gt;
. This is not a requirement in MongoDB Atlas, which is why we don't worry about normalizing it in this guide.&lt;/p&gt;

&lt;p&gt;Finally, it follows that algorithm for mocking &lt;code&gt;embed_documents&lt;/code&gt; becomes this:&lt;br&gt;
&lt;/p&gt;

&lt;div class="highlight js-code-highlight"&gt;
&lt;pre class="highlight python"&gt;&lt;code&gt;&lt;span class="k"&gt;def&lt;/span&gt; &lt;span class="nf"&gt;embed_documents&lt;/span&gt;&lt;span class="p"&gt;(&lt;/span&gt;&lt;span class="n"&gt;ctx&lt;/span&gt;&lt;span class="p"&gt;,&lt;/span&gt; &lt;span class="n"&gt;emb&lt;/span&gt;&lt;span class="p"&gt;,&lt;/span&gt; &lt;span class="n"&gt;texts&lt;/span&gt;&lt;span class="p"&gt;):&lt;/span&gt;
    &lt;span class="n"&gt;vectors&lt;/span&gt; &lt;span class="o"&gt;=&lt;/span&gt; &lt;span class="p"&gt;[]&lt;/span&gt;

    &lt;span class="k"&gt;for&lt;/span&gt; &lt;span class="n"&gt;text&lt;/span&gt; &lt;span class="ow"&gt;in&lt;/span&gt; &lt;span class="n"&gt;texts&lt;/span&gt;&lt;span class="p"&gt;:&lt;/span&gt;
        &lt;span class="c1"&gt;# If the text doesn't exist in the document set, return a zero vector
&lt;/span&gt;        &lt;span class="k"&gt;if&lt;/span&gt; &lt;span class="n"&gt;text&lt;/span&gt; &lt;span class="ow"&gt;not&lt;/span&gt; &lt;span class="ow"&gt;in&lt;/span&gt; &lt;span class="n"&gt;emb&lt;/span&gt;&lt;span class="p"&gt;[&lt;/span&gt;&lt;span class="sh"&gt;"&lt;/span&gt;&lt;span class="s"&gt;docs&lt;/span&gt;&lt;span class="sh"&gt;"&lt;/span&gt;&lt;span class="p"&gt;]:&lt;/span&gt;
            &lt;span class="n"&gt;vectors&lt;/span&gt;&lt;span class="p"&gt;.&lt;/span&gt;&lt;span class="nf"&gt;append&lt;/span&gt;&lt;span class="p"&gt;([&lt;/span&gt;&lt;span class="mf"&gt;0.0&lt;/span&gt;&lt;span class="p"&gt;]&lt;/span&gt; &lt;span class="o"&gt;*&lt;/span&gt; &lt;span class="nf"&gt;len&lt;/span&gt;&lt;span class="p"&gt;(&lt;/span&gt;&lt;span class="n"&gt;emb&lt;/span&gt;&lt;span class="p"&gt;[&lt;/span&gt;&lt;span class="sh"&gt;"&lt;/span&gt;&lt;span class="s"&gt;query_vector&lt;/span&gt;&lt;span class="sh"&gt;"&lt;/span&gt;&lt;span class="p"&gt;]))&lt;/span&gt;
            &lt;span class="k"&gt;continue&lt;/span&gt;

        &lt;span class="c1"&gt;# If the vector already exists, return the existing one
&lt;/span&gt;        &lt;span class="k"&gt;if&lt;/span&gt; &lt;span class="n"&gt;text&lt;/span&gt; &lt;span class="ow"&gt;in&lt;/span&gt; &lt;span class="n"&gt;emb&lt;/span&gt;&lt;span class="p"&gt;[&lt;/span&gt;&lt;span class="sh"&gt;"&lt;/span&gt;&lt;span class="s"&gt;doc_vectors&lt;/span&gt;&lt;span class="sh"&gt;"&lt;/span&gt;&lt;span class="p"&gt;]:&lt;/span&gt;
            &lt;span class="n"&gt;vectors&lt;/span&gt;&lt;span class="p"&gt;.&lt;/span&gt;&lt;span class="nf"&gt;append&lt;/span&gt;&lt;span class="p"&gt;(&lt;/span&gt;&lt;span class="n"&gt;emb&lt;/span&gt;&lt;span class="p"&gt;[&lt;/span&gt;&lt;span class="sh"&gt;"&lt;/span&gt;&lt;span class="s"&gt;doc_vectors&lt;/span&gt;&lt;span class="sh"&gt;"&lt;/span&gt;&lt;span class="p"&gt;][&lt;/span&gt;&lt;span class="n"&gt;text&lt;/span&gt;&lt;span class="p"&gt;])&lt;/span&gt;
            &lt;span class="k"&gt;continue&lt;/span&gt;

        &lt;span class="c1"&gt;# Otherwise, generate a new orthogonal vector
&lt;/span&gt;        &lt;span class="n"&gt;new_vector_basis&lt;/span&gt; &lt;span class="o"&gt;=&lt;/span&gt; &lt;span class="nf"&gt;new_orthogonal_vector&lt;/span&gt;&lt;span class="p"&gt;(&lt;/span&gt;&lt;span class="nf"&gt;len&lt;/span&gt;&lt;span class="p"&gt;(&lt;/span&gt;&lt;span class="n"&gt;emb&lt;/span&gt;&lt;span class="p"&gt;[&lt;/span&gt;&lt;span class="sh"&gt;"&lt;/span&gt;&lt;span class="s"&gt;query_vector&lt;/span&gt;&lt;span class="sh"&gt;"&lt;/span&gt;&lt;span class="p"&gt;]),&lt;/span&gt; &lt;span class="o"&gt;*&lt;/span&gt;&lt;span class="nf"&gt;existing_vectors&lt;/span&gt;&lt;span class="p"&gt;(&lt;/span&gt;&lt;span class="n"&gt;emb&lt;/span&gt;&lt;span class="p"&gt;))&lt;/span&gt;

        &lt;span class="c1"&gt;# Update the last element of the vector so that 
&lt;/span&gt;        &lt;span class="c1"&gt;# v^* * v = 2S - 1.
&lt;/span&gt;        &lt;span class="n"&gt;doc&lt;/span&gt; &lt;span class="o"&gt;=&lt;/span&gt; &lt;span class="n"&gt;emb&lt;/span&gt;&lt;span class="p"&gt;[&lt;/span&gt;&lt;span class="sh"&gt;"&lt;/span&gt;&lt;span class="s"&gt;docs&lt;/span&gt;&lt;span class="sh"&gt;"&lt;/span&gt;&lt;span class="p"&gt;][&lt;/span&gt;&lt;span class="n"&gt;text&lt;/span&gt;&lt;span class="p"&gt;]&lt;/span&gt;
        &lt;span class="n"&gt;new_vector&lt;/span&gt; &lt;span class="o"&gt;=&lt;/span&gt; &lt;span class="nf"&gt;new_score_vector&lt;/span&gt;&lt;span class="p"&gt;(&lt;/span&gt;&lt;span class="n"&gt;doc&lt;/span&gt;&lt;span class="p"&gt;[&lt;/span&gt;&lt;span class="sh"&gt;"&lt;/span&gt;&lt;span class="s"&gt;Score&lt;/span&gt;&lt;span class="sh"&gt;"&lt;/span&gt;&lt;span class="p"&gt;],&lt;/span&gt; &lt;span class="n"&gt;emb&lt;/span&gt;&lt;span class="p"&gt;[&lt;/span&gt;&lt;span class="sh"&gt;"&lt;/span&gt;&lt;span class="s"&gt;query_vector&lt;/span&gt;&lt;span class="sh"&gt;"&lt;/span&gt;&lt;span class="p"&gt;],&lt;/span&gt; &lt;span class="n"&gt;new_vector_basis&lt;/span&gt;&lt;span class="p"&gt;)&lt;/span&gt;

        &lt;span class="c1"&gt;# Store and return the new vector
&lt;/span&gt;        &lt;span class="n"&gt;vectors&lt;/span&gt;&lt;span class="p"&gt;.&lt;/span&gt;&lt;span class="nf"&gt;append&lt;/span&gt;&lt;span class="p"&gt;(&lt;/span&gt;&lt;span class="n"&gt;new_vector&lt;/span&gt;&lt;span class="p"&gt;)&lt;/span&gt;
        &lt;span class="n"&gt;emb&lt;/span&gt;&lt;span class="p"&gt;[&lt;/span&gt;&lt;span class="sh"&gt;"&lt;/span&gt;&lt;span class="s"&gt;doc_vectors&lt;/span&gt;&lt;span class="sh"&gt;"&lt;/span&gt;&lt;span class="p"&gt;][&lt;/span&gt;&lt;span class="n"&gt;text&lt;/span&gt;&lt;span class="p"&gt;]&lt;/span&gt; &lt;span class="o"&gt;=&lt;/span&gt; &lt;span class="n"&gt;new_vector&lt;/span&gt;

    &lt;span class="k"&gt;return&lt;/span&gt; &lt;span class="n"&gt;vectors&lt;/span&gt;
&lt;/code&gt;&lt;/pre&gt;

&lt;/div&gt;


&lt;p&gt;&lt;strong&gt;Example&lt;/strong&gt;. Lets generate 
&lt;span class="katex-element"&gt;
  &lt;span class="katex"&gt;&lt;span class="katex-mathml"&gt;E\mathbf{E}&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 mathbf"&gt;E&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;
&lt;/span&gt;
 for our "South American Author" example. For simplicity, say 
&lt;span class="katex-element"&gt;
  &lt;span class="katex"&gt;&lt;span class="katex-mathml"&gt;n=3n = 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 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;3&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;
&lt;/span&gt;
. First we generate the query vector using Algorithm A1:&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;vA∗=(−0.5,0.25,0.71)
\begin{equation*}
\mathbf{v}_A^{*} = (-0.5, 0.25, 0.71)
\end{equation*}
&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="mtable"&gt;&lt;span class="col-align-c"&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 mathbf"&gt;v&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 mathnormal mtight"&gt;A&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 mtight"&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="mspace"&gt;&lt;/span&gt;&lt;span class="mrel"&gt;=&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;span class="mopen"&gt;(&lt;/span&gt;&lt;span class="mord"&gt;−&lt;/span&gt;&lt;span class="mord"&gt;0.5&lt;/span&gt;&lt;span class="mpunct"&gt;,&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;span class="mord"&gt;0.25&lt;/span&gt;&lt;span class="mpunct"&gt;,&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;span class="mord"&gt;0.71&lt;/span&gt;&lt;span class="mclose"&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;Using &lt;code&gt;new_orthogonal_vector&lt;/code&gt; from Algorithm A2, create a vector orthogonal to 
&lt;span class="katex-element"&gt;
  &lt;span class="katex"&gt;&lt;span class="katex-mathml"&gt;vA∗\mathbf{v}_A^{*}&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 mathbf"&gt;v&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 mathnormal mtight"&gt;A&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 mtight"&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;
 for Márquez:&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;v1=(0.6,−0.8,v13)
\begin{equation*}
\mathbf{v}1 = (0.6, -0.8, v{13})
\end{equation*}
&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="mtable"&gt;&lt;span class="col-align-c"&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 mathbf"&gt;v&lt;/span&gt;&lt;span class="mord"&gt;1&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 class="mopen"&gt;(&lt;/span&gt;&lt;span class="mord"&gt;0.6&lt;/span&gt;&lt;span class="mpunct"&gt;,&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;span class="mord"&gt;−&lt;/span&gt;&lt;span class="mord"&gt;0.8&lt;/span&gt;&lt;span class="mpunct"&gt;,&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;span class="mord mathnormal"&gt;v&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mord"&gt;13&lt;/span&gt;&lt;/span&gt;&lt;span class="mclose"&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;From eq. 3 it follows that we need to update 
&lt;span class="katex-element"&gt;
  &lt;span class="katex"&gt;&lt;span class="katex-mathml"&gt;v13v_{13}&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;v&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;13&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;
 such that 
&lt;span class="katex-element"&gt;
  &lt;span class="katex"&gt;&lt;span class="katex-mathml"&gt;vA∗⋅v1=0.6\mathbf{v}_A^{*} \cdot \mathbf{v}_1 = 0.6&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 mathbf"&gt;v&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 mathnormal mtight"&gt;A&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 mtight"&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="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 mathbf"&gt;v&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;1&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;0.6&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;
&lt;/span&gt;
. Using eq. 4 we get that &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;vA∗⋅v1=(−0.5)(0.6)+(0.25)(−0.8)+(0.71)(v13)
\begin{equation*}
\mathbf{v}{A}^{*} \cdot \mathbf{v}_1 = (-0.5)(0.6) + (0.25)(-0.8) + (0.71)(v{13})
\end{equation*}
&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="mtable"&gt;&lt;span class="col-align-c"&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 mathbf"&gt;v&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&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;∗&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="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;&lt;span class="mord mathbf"&gt;v&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;1&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 class="mopen"&gt;(&lt;/span&gt;&lt;span class="mord"&gt;−&lt;/span&gt;&lt;span class="mord"&gt;0.5&lt;/span&gt;&lt;span class="mclose"&gt;)&lt;/span&gt;&lt;span class="mopen"&gt;(&lt;/span&gt;&lt;span class="mord"&gt;0.6&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 class="mopen"&gt;(&lt;/span&gt;&lt;span class="mord"&gt;0.25&lt;/span&gt;&lt;span class="mclose"&gt;)&lt;/span&gt;&lt;span class="mopen"&gt;(&lt;/span&gt;&lt;span class="mord"&gt;−&lt;/span&gt;&lt;span class="mord"&gt;0.8&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 class="mopen"&gt;(&lt;/span&gt;&lt;span class="mord"&gt;0.71&lt;/span&gt;&lt;span class="mclose"&gt;)&lt;/span&gt;&lt;span class="mopen"&gt;(&lt;/span&gt;&lt;span class="mord mathnormal"&gt;v&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mord"&gt;13&lt;/span&gt;&lt;/span&gt;&lt;span class="mclose"&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;Substituting the known values:&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;−0.3−0.2+0.71v13=0.6
\begin{equation*}
-0.3 - 0.2 + 0.71 v_{13} = 0.6
\end{equation*}
&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="mtable"&gt;&lt;span class="col-align-c"&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&gt;&lt;span class="mord"&gt;0.3&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;0.2&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 class="mord"&gt;0.71&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mord mathnormal"&gt;v&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;13&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 class="mord"&gt;0.6&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;Solving for 
&lt;span class="katex-element"&gt;
  &lt;span class="katex"&gt;&lt;span class="katex-mathml"&gt;v13v_{13}&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;v&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;13&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;/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;0.71v13=1.1  ⟹  v13=1.10.71≈1.549
\begin{equation*}
0.71 v_{13} = 1.1 \implies v_{13} = \frac{1.1}{0.71} \approx 1.549
\end{equation*}
&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="mtable"&gt;&lt;span class="col-align-c"&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;0.71&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mord mathnormal"&gt;v&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;13&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 class="mord"&gt;1.1&lt;/span&gt;&lt;span class="mspace"&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 class="mspace"&gt;&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mord mathnormal"&gt;v&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;13&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 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;0.71&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;1.1&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 class="mord"&gt;1.549&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;Thus, the updated vector 
&lt;span class="katex-element"&gt;
  &lt;span class="katex"&gt;&lt;span class="katex-mathml"&gt;v1\mathbf{v}_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;&lt;span class="mord mathbf"&gt;v&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;1&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;
 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;v1=(0.6,−0.8,1.549)
\begin{equation*}
\mathbf{v}_1 = (0.6, -0.8, 1.549)
\end{equation*}
&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="mtable"&gt;&lt;span class="col-align-c"&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 mathbf"&gt;v&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;1&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 class="mopen"&gt;(&lt;/span&gt;&lt;span class="mord"&gt;0.6&lt;/span&gt;&lt;span class="mpunct"&gt;,&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;span class="mord"&gt;−&lt;/span&gt;&lt;span class="mord"&gt;0.8&lt;/span&gt;&lt;span class="mpunct"&gt;,&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;span class="mord"&gt;1.549&lt;/span&gt;&lt;span class="mclose"&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;See &lt;a href="https://github.com/prestonvasquez/vectormock/blob/main/examples/mongodb/main.go" rel="noopener noreferrer"&gt;here&lt;/a&gt; for a working example using &lt;a href="https://github.com/tmc/langchaingo" rel="noopener noreferrer"&gt;LangChainGo&lt;/a&gt;. Note, this will require a MongoDB Atlas Cluster with a vector search index. Check out the &lt;a href="https://www.mongodb.com/docs/atlas/atlas-vector-search/tutorials" rel="noopener noreferrer"&gt;tutorial&lt;/a&gt;! &lt;/p&gt;

&lt;p&gt;&lt;strong&gt;Algorithms&lt;/strong&gt;. Here is a list of algorithms used to mock an embedding model. &lt;/p&gt;

&lt;p&gt;A1. Create a vector of normalized 32-bit floats&lt;br&gt;
&lt;/p&gt;

&lt;div class="highlight js-code-highlight"&gt;
&lt;pre class="highlight python"&gt;&lt;code&gt;&lt;span class="k"&gt;def&lt;/span&gt; &lt;span class="nf"&gt;new_normalized_float32&lt;/span&gt;&lt;span class="p"&gt;():&lt;/span&gt;
    &lt;span class="k"&gt;try&lt;/span&gt;&lt;span class="p"&gt;:&lt;/span&gt;
        &lt;span class="c1"&gt;# Generate a random integer in the range [0, 2^24)
&lt;/span&gt;        &lt;span class="n"&gt;max_value&lt;/span&gt; &lt;span class="o"&gt;=&lt;/span&gt; &lt;span class="mi"&gt;1&lt;/span&gt; &lt;span class="o"&gt;&amp;lt;&amp;lt;&lt;/span&gt; &lt;span class="mi"&gt;24&lt;/span&gt;
        &lt;span class="n"&gt;n&lt;/span&gt; &lt;span class="o"&gt;=&lt;/span&gt; &lt;span class="n"&gt;random&lt;/span&gt;&lt;span class="p"&gt;.&lt;/span&gt;&lt;span class="nf"&gt;randint&lt;/span&gt;&lt;span class="p"&gt;(&lt;/span&gt;&lt;span class="mi"&gt;0&lt;/span&gt;&lt;span class="p"&gt;,&lt;/span&gt; &lt;span class="n"&gt;max_value&lt;/span&gt; &lt;span class="o"&gt;-&lt;/span&gt; &lt;span class="mi"&gt;1&lt;/span&gt;&lt;span class="p"&gt;)&lt;/span&gt;

        &lt;span class="c1"&gt;# Normalize it to the range [-1, 1]
&lt;/span&gt;        &lt;span class="n"&gt;normalized_float&lt;/span&gt; &lt;span class="o"&gt;=&lt;/span&gt; &lt;span class="mf"&gt;2.0&lt;/span&gt; &lt;span class="o"&gt;*&lt;/span&gt; &lt;span class="p"&gt;(&lt;/span&gt;&lt;span class="nf"&gt;float&lt;/span&gt;&lt;span class="p"&gt;(&lt;/span&gt;&lt;span class="n"&gt;n&lt;/span&gt;&lt;span class="p"&gt;)&lt;/span&gt; &lt;span class="o"&gt;/&lt;/span&gt; &lt;span class="n"&gt;max_value&lt;/span&gt;&lt;span class="p"&gt;)&lt;/span&gt; &lt;span class="o"&gt;-&lt;/span&gt; &lt;span class="mf"&gt;1.0&lt;/span&gt;

        &lt;span class="k"&gt;return&lt;/span&gt; &lt;span class="n"&gt;normalized_float&lt;/span&gt;
    &lt;span class="k"&gt;except&lt;/span&gt; &lt;span class="nb"&gt;Exception&lt;/span&gt; &lt;span class="k"&gt;as&lt;/span&gt; &lt;span class="n"&gt;e&lt;/span&gt;&lt;span class="p"&gt;:&lt;/span&gt;
        &lt;span class="k"&gt;raise&lt;/span&gt; &lt;span class="nc"&gt;Exception&lt;/span&gt;&lt;span class="p"&gt;(&lt;/span&gt;&lt;span class="sh"&gt;"&lt;/span&gt;&lt;span class="s"&gt;Failed to normalize float32&lt;/span&gt;&lt;span class="sh"&gt;"&lt;/span&gt;&lt;span class="p"&gt;)&lt;/span&gt; &lt;span class="k"&gt;from&lt;/span&gt; &lt;span class="n"&gt;e&lt;/span&gt;

&lt;span class="k"&gt;def&lt;/span&gt; &lt;span class="nf"&gt;new_normalized_vector&lt;/span&gt;&lt;span class="p"&gt;(&lt;/span&gt;&lt;span class="n"&gt;n&lt;/span&gt;&lt;span class="p"&gt;):&lt;/span&gt;
    &lt;span class="n"&gt;vector&lt;/span&gt; &lt;span class="o"&gt;=&lt;/span&gt; &lt;span class="p"&gt;[&lt;/span&gt;&lt;span class="mf"&gt;0.0&lt;/span&gt;&lt;span class="p"&gt;]&lt;/span&gt; &lt;span class="o"&gt;*&lt;/span&gt; &lt;span class="n"&gt;n&lt;/span&gt;
    &lt;span class="k"&gt;for&lt;/span&gt; &lt;span class="n"&gt;i&lt;/span&gt; &lt;span class="ow"&gt;in&lt;/span&gt; &lt;span class="nf"&gt;range&lt;/span&gt;&lt;span class="p"&gt;(&lt;/span&gt;&lt;span class="n"&gt;n&lt;/span&gt;&lt;span class="p"&gt;):&lt;/span&gt;
        &lt;span class="n"&gt;vector&lt;/span&gt;&lt;span class="p"&gt;[&lt;/span&gt;&lt;span class="n"&gt;i&lt;/span&gt;&lt;span class="p"&gt;]&lt;/span&gt; &lt;span class="o"&gt;=&lt;/span&gt; &lt;span class="nf"&gt;new_normalized_float32&lt;/span&gt;&lt;span class="p"&gt;()&lt;/span&gt;

    &lt;span class="k"&gt;return&lt;/span&gt; &lt;span class="n"&gt;vector&lt;/span&gt;
&lt;/code&gt;&lt;/pre&gt;

&lt;/div&gt;



&lt;p&gt;A2. Create 
&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;
 
&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;
-dimensional normalized vectors&lt;br&gt;
&lt;/p&gt;

&lt;div class="highlight js-code-highlight"&gt;
&lt;pre class="highlight python"&gt;&lt;code&gt;&lt;span class="c1"&gt;# Compute the dot product between two lists (vectors) of floats.
&lt;/span&gt;&lt;span class="k"&gt;def&lt;/span&gt; &lt;span class="nf"&gt;dot_product&lt;/span&gt;&lt;span class="p"&gt;(&lt;/span&gt;&lt;span class="n"&gt;v1&lt;/span&gt;&lt;span class="p"&gt;,&lt;/span&gt; &lt;span class="n"&gt;v2&lt;/span&gt;&lt;span class="p"&gt;):&lt;/span&gt;
    &lt;span class="nb"&gt;sum&lt;/span&gt; &lt;span class="o"&gt;=&lt;/span&gt; &lt;span class="mf"&gt;0.0&lt;/span&gt;
    &lt;span class="k"&gt;for&lt;/span&gt; &lt;span class="n"&gt;i&lt;/span&gt; &lt;span class="ow"&gt;in&lt;/span&gt; &lt;span class="nf"&gt;range&lt;/span&gt;&lt;span class="p"&gt;(&lt;/span&gt;&lt;span class="nf"&gt;len&lt;/span&gt;&lt;span class="p"&gt;(&lt;/span&gt;&lt;span class="n"&gt;v1&lt;/span&gt;&lt;span class="p"&gt;)):&lt;/span&gt;
        &lt;span class="nb"&gt;sum&lt;/span&gt; &lt;span class="o"&gt;+=&lt;/span&gt; &lt;span class="n"&gt;v1&lt;/span&gt;&lt;span class="p"&gt;[&lt;/span&gt;&lt;span class="n"&gt;i&lt;/span&gt;&lt;span class="p"&gt;]&lt;/span&gt; &lt;span class="o"&gt;*&lt;/span&gt; &lt;span class="n"&gt;v2&lt;/span&gt;&lt;span class="p"&gt;[&lt;/span&gt;&lt;span class="n"&gt;i&lt;/span&gt;&lt;span class="p"&gt;]&lt;/span&gt;
    &lt;span class="k"&gt;return&lt;/span&gt; &lt;span class="nb"&gt;sum&lt;/span&gt;

&lt;span class="c1"&gt;# Use Gram-Schmidt to return a vector orthogonal to the basis,
# assuming the vectors in the basis are linearly independent.
&lt;/span&gt;&lt;span class="k"&gt;def&lt;/span&gt; &lt;span class="nf"&gt;new_orthogonal_vector&lt;/span&gt;&lt;span class="p"&gt;(&lt;/span&gt;&lt;span class="n"&gt;dim&lt;/span&gt;&lt;span class="p"&gt;,&lt;/span&gt; &lt;span class="o"&gt;*&lt;/span&gt;&lt;span class="n"&gt;basis&lt;/span&gt;&lt;span class="p"&gt;):&lt;/span&gt;
    &lt;span class="n"&gt;candidate&lt;/span&gt; &lt;span class="o"&gt;=&lt;/span&gt; &lt;span class="nf"&gt;new_normalized_vector&lt;/span&gt;&lt;span class="p"&gt;(&lt;/span&gt;&lt;span class="n"&gt;dim&lt;/span&gt;&lt;span class="p"&gt;)&lt;/span&gt;

    &lt;span class="k"&gt;for&lt;/span&gt; &lt;span class="n"&gt;b&lt;/span&gt; &lt;span class="ow"&gt;in&lt;/span&gt; &lt;span class="n"&gt;basis&lt;/span&gt;&lt;span class="p"&gt;:&lt;/span&gt;
        &lt;span class="n"&gt;dp&lt;/span&gt; &lt;span class="o"&gt;=&lt;/span&gt; &lt;span class="nf"&gt;dot_product&lt;/span&gt;&lt;span class="p"&gt;(&lt;/span&gt;&lt;span class="n"&gt;candidate&lt;/span&gt;&lt;span class="p"&gt;,&lt;/span&gt; &lt;span class="n"&gt;b&lt;/span&gt;&lt;span class="p"&gt;)&lt;/span&gt;
        &lt;span class="n"&gt;basis_norm&lt;/span&gt; &lt;span class="o"&gt;=&lt;/span&gt; &lt;span class="nf"&gt;dot_product&lt;/span&gt;&lt;span class="p"&gt;(&lt;/span&gt;&lt;span class="n"&gt;b&lt;/span&gt;&lt;span class="p"&gt;,&lt;/span&gt; &lt;span class="n"&gt;b&lt;/span&gt;&lt;span class="p"&gt;)&lt;/span&gt;

        &lt;span class="k"&gt;for&lt;/span&gt; &lt;span class="n"&gt;i&lt;/span&gt; &lt;span class="ow"&gt;in&lt;/span&gt; &lt;span class="nf"&gt;range&lt;/span&gt;&lt;span class="p"&gt;(&lt;/span&gt;&lt;span class="nf"&gt;len&lt;/span&gt;&lt;span class="p"&gt;(&lt;/span&gt;&lt;span class="n"&gt;candidate&lt;/span&gt;&lt;span class="p"&gt;)):&lt;/span&gt;
            &lt;span class="n"&gt;candidate&lt;/span&gt;&lt;span class="p"&gt;[&lt;/span&gt;&lt;span class="n"&gt;i&lt;/span&gt;&lt;span class="p"&gt;]&lt;/span&gt; &lt;span class="o"&gt;-=&lt;/span&gt; &lt;span class="p"&gt;(&lt;/span&gt;&lt;span class="n"&gt;dp&lt;/span&gt; &lt;span class="o"&gt;/&lt;/span&gt; &lt;span class="n"&gt;basis_norm&lt;/span&gt;&lt;span class="p"&gt;)&lt;/span&gt; &lt;span class="o"&gt;*&lt;/span&gt; &lt;span class="n"&gt;b&lt;/span&gt;&lt;span class="p"&gt;[&lt;/span&gt;&lt;span class="n"&gt;i&lt;/span&gt;&lt;span class="p"&gt;]&lt;/span&gt;

    &lt;span class="k"&gt;return&lt;/span&gt; &lt;span class="n"&gt;candidate&lt;/span&gt;

&lt;span class="c1"&gt;# Make n linearly independent vectors of size dim.
&lt;/span&gt;&lt;span class="k"&gt;def&lt;/span&gt; &lt;span class="nf"&gt;new_linearly_independent_vectors&lt;/span&gt;&lt;span class="p"&gt;(&lt;/span&gt;&lt;span class="n"&gt;n&lt;/span&gt;&lt;span class="p"&gt;,&lt;/span&gt; &lt;span class="n"&gt;dim&lt;/span&gt;&lt;span class="p"&gt;):&lt;/span&gt;
    &lt;span class="n"&gt;vectors&lt;/span&gt; &lt;span class="o"&gt;=&lt;/span&gt; &lt;span class="p"&gt;[]&lt;/span&gt;

    &lt;span class="k"&gt;for&lt;/span&gt; &lt;span class="n"&gt;_&lt;/span&gt; &lt;span class="ow"&gt;in&lt;/span&gt; &lt;span class="nf"&gt;range&lt;/span&gt;&lt;span class="p"&gt;(&lt;/span&gt;&lt;span class="n"&gt;n&lt;/span&gt;&lt;span class="p"&gt;):&lt;/span&gt;
        &lt;span class="n"&gt;v&lt;/span&gt; &lt;span class="o"&gt;=&lt;/span&gt; &lt;span class="nf"&gt;new_orthogonal_vector&lt;/span&gt;&lt;span class="p"&gt;(&lt;/span&gt;&lt;span class="n"&gt;dim&lt;/span&gt;&lt;span class="p"&gt;,&lt;/span&gt; &lt;span class="o"&gt;*&lt;/span&gt;&lt;span class="n"&gt;vectors&lt;/span&gt;&lt;span class="p"&gt;)&lt;/span&gt;
        &lt;span class="n"&gt;vectors&lt;/span&gt;&lt;span class="p"&gt;.&lt;/span&gt;&lt;span class="nf"&gt;append&lt;/span&gt;&lt;span class="p"&gt;(&lt;/span&gt;&lt;span class="n"&gt;v&lt;/span&gt;&lt;span class="p"&gt;)&lt;/span&gt;

    &lt;span class="k"&gt;return&lt;/span&gt; &lt;span class="n"&gt;vectors&lt;/span&gt;
&lt;/code&gt;&lt;/pre&gt;

&lt;/div&gt;



&lt;p&gt;A3. Update two linearly independent vectors such that their dot product is a desired value.&lt;br&gt;
&lt;/p&gt;

&lt;div class="highlight js-code-highlight"&gt;
&lt;pre class="highlight python"&gt;&lt;code&gt;&lt;span class="c1"&gt;# Update the basis vector such that qvector * basis = 2S - 1.
&lt;/span&gt;&lt;span class="k"&gt;def&lt;/span&gt; &lt;span class="nf"&gt;new_score_vector&lt;/span&gt;&lt;span class="p"&gt;(&lt;/span&gt;&lt;span class="n"&gt;S&lt;/span&gt;&lt;span class="p"&gt;,&lt;/span&gt; &lt;span class="n"&gt;qvector&lt;/span&gt;&lt;span class="p"&gt;,&lt;/span&gt; &lt;span class="n"&gt;basis&lt;/span&gt;&lt;span class="p"&gt;):&lt;/span&gt;
    &lt;span class="n"&gt;sum_value&lt;/span&gt; &lt;span class="o"&gt;=&lt;/span&gt; &lt;span class="mf"&gt;0.0&lt;/span&gt;

    &lt;span class="c1"&gt;# Populate v2 up to dim-1.
&lt;/span&gt;    &lt;span class="k"&gt;for&lt;/span&gt; &lt;span class="n"&gt;i&lt;/span&gt; &lt;span class="ow"&gt;in&lt;/span&gt; &lt;span class="nf"&gt;range&lt;/span&gt;&lt;span class="p"&gt;(&lt;/span&gt;&lt;span class="nf"&gt;len&lt;/span&gt;&lt;span class="p"&gt;(&lt;/span&gt;&lt;span class="n"&gt;qvector&lt;/span&gt;&lt;span class="p"&gt;)&lt;/span&gt; &lt;span class="o"&gt;-&lt;/span&gt; &lt;span class="mi"&gt;1&lt;/span&gt;&lt;span class="p"&gt;):&lt;/span&gt;
        &lt;span class="n"&gt;sum_value&lt;/span&gt; &lt;span class="o"&gt;+=&lt;/span&gt; &lt;span class="n"&gt;qvector&lt;/span&gt;&lt;span class="p"&gt;[&lt;/span&gt;&lt;span class="n"&gt;i&lt;/span&gt;&lt;span class="p"&gt;]&lt;/span&gt; &lt;span class="o"&gt;*&lt;/span&gt; &lt;span class="n"&gt;basis&lt;/span&gt;&lt;span class="p"&gt;[&lt;/span&gt;&lt;span class="n"&gt;i&lt;/span&gt;&lt;span class="p"&gt;]&lt;/span&gt;

    &lt;span class="c1"&gt;# Calculate v_{2, dim} such that v1 * v2 = 2S - 1:
&lt;/span&gt;    &lt;span class="n"&gt;basis&lt;/span&gt;&lt;span class="p"&gt;[&lt;/span&gt;&lt;span class="o"&gt;-&lt;/span&gt;&lt;span class="mi"&gt;1&lt;/span&gt;&lt;span class="p"&gt;]&lt;/span&gt; &lt;span class="o"&gt;=&lt;/span&gt; &lt;span class="p"&gt;(&lt;/span&gt;&lt;span class="mi"&gt;2&lt;/span&gt; &lt;span class="o"&gt;*&lt;/span&gt; &lt;span class="n"&gt;S&lt;/span&gt; &lt;span class="o"&gt;-&lt;/span&gt; &lt;span class="mi"&gt;1&lt;/span&gt; &lt;span class="o"&gt;-&lt;/span&gt; &lt;span class="n"&gt;sum_value&lt;/span&gt;&lt;span class="p"&gt;)&lt;/span&gt; &lt;span class="o"&gt;/&lt;/span&gt; &lt;span class="n"&gt;qvector&lt;/span&gt;&lt;span class="p"&gt;[&lt;/span&gt;&lt;span class="o"&gt;-&lt;/span&gt;&lt;span class="mi"&gt;1&lt;/span&gt;&lt;span class="p"&gt;]&lt;/span&gt;

    &lt;span class="c1"&gt;# If the vectors are not linearly independent, regenerate the dim-1 elements of v2.
&lt;/span&gt;    &lt;span class="k"&gt;if&lt;/span&gt; &lt;span class="ow"&gt;not&lt;/span&gt; &lt;span class="nf"&gt;linearly_independent&lt;/span&gt;&lt;span class="p"&gt;(&lt;/span&gt;&lt;span class="n"&gt;qvector&lt;/span&gt;&lt;span class="p"&gt;,&lt;/span&gt; &lt;span class="n"&gt;basis&lt;/span&gt;&lt;span class="p"&gt;):&lt;/span&gt;
        &lt;span class="k"&gt;return&lt;/span&gt; &lt;span class="nf"&gt;new_score_vector&lt;/span&gt;&lt;span class="p"&gt;(&lt;/span&gt;&lt;span class="n"&gt;S&lt;/span&gt;&lt;span class="p"&gt;,&lt;/span&gt; &lt;span class="n"&gt;qvector&lt;/span&gt;&lt;span class="p"&gt;,&lt;/span&gt; &lt;span class="n"&gt;basis&lt;/span&gt;&lt;span class="p"&gt;)&lt;/span&gt;

    &lt;span class="k"&gt;return&lt;/span&gt; &lt;span class="n"&gt;basis&lt;/span&gt;
&lt;/code&gt;&lt;/pre&gt;

&lt;/div&gt;



</description>
      <category>llm</category>
      <category>vectordatabase</category>
      <category>mongodb</category>
      <category>testing</category>
    </item>
    <item>
      <title>Dockerizing a MongoDB Replica Set With TLS/SSL</title>
      <dc:creator>Preston Vasquez</dc:creator>
      <pubDate>Mon, 07 Nov 2022 16:50:06 +0000</pubDate>
      <link>https://forem.com/prestonvasquez/dockerizing-a-mongodb-replica-set-with-tlsssl-20e2</link>
      <guid>https://forem.com/prestonvasquez/dockerizing-a-mongodb-replica-set-with-tlsssl-20e2</guid>
      <description>&lt;h2&gt;
  
  
  Introduction
&lt;/h2&gt;

&lt;p&gt;The goal of this guide is to create a local SSL-enabled &lt;a href="https://www.mongodb.com/docs/manual/replication/" rel="noopener noreferrer"&gt;MongoDB Replica Set&lt;/a&gt; using &lt;a href="https://docs.docker.com/compose/" rel="noopener noreferrer"&gt;docker-compose&lt;/a&gt;.&lt;/p&gt;

&lt;p&gt;TL;DR: &lt;a href="https://github.com/prestonvasquez/mongodb-rs-demo" rel="noopener noreferrer"&gt;Here&lt;/a&gt; is a demo repository.&lt;/p&gt;

&lt;h3&gt;
  
  
  Prerequisites
&lt;/h3&gt;

&lt;p&gt;&lt;a href="https://docs.docker.com/get-docker/" rel="noopener noreferrer"&gt;Docker&lt;/a&gt; is the only pre-requisite, but if you want to connect to the mongo shell locally (recommended) you will need &lt;a href="https://www.mongodb.com/docs/manual/installation/" rel="noopener noreferrer"&gt;MongoDB&lt;/a&gt;.&lt;/p&gt;

&lt;h2&gt;
  
  
  File Structure
&lt;/h2&gt;

&lt;p&gt;Here is the file structure used in this tutorial:&lt;br&gt;
&lt;/p&gt;

&lt;div class="highlight js-code-highlight"&gt;
&lt;pre class="highlight plaintext"&gt;&lt;code&gt;.db/
config/
├─ mongodb.conf
ssl/
├─ ca.pem
├─ server.pem
docker-compose.yml
init.sh
run.sh
wait-for-mongo.sh
&lt;/code&gt;&lt;/pre&gt;

&lt;/div&gt;



&lt;h3&gt;
  
  
  .db/*
&lt;/h3&gt;

&lt;p&gt;This is the &lt;a href="https://www.mongodb.com/docs/manual/reference/configuration-options/#mongodb-setting-storage.dbPath" rel="noopener noreferrer"&gt;database path&lt;/a&gt; for the mongod configuration and should be generated automatically by docker-compose.&lt;/p&gt;

&lt;h3&gt;
  
  
  ssl/*
&lt;/h3&gt;

&lt;p&gt;The files &lt;code&gt;ssl/ca.pem&lt;/code&gt; and &lt;code&gt;ssl/server.pem&lt;/code&gt; are the &lt;a href="https://www.mongodb.com/docs/manual/reference/program/mongos/#std-option-mongos.--tlsCAFile" rel="noopener noreferrer"&gt;TLS CA file&lt;/a&gt; and &lt;a href="https://www.mongodb.com/docs/manual/reference/program/mongos/#std-option-mongos.--tlsCertificateKeyFile" rel="noopener noreferrer"&gt;TLS certificate key file&lt;/a&gt;, respectively. The data in the ssl/ directory is used by the configuration file to initialize a MongoDB server with SSL enabled. This directory will be volumized on the Docker container for the primary replicant.&lt;/p&gt;

&lt;p&gt;This tutorial assumes prior knowledge of TLS/SSL as well as access to valid certificates. For more information on generating certificates using &lt;code&gt;openssl&lt;/code&gt;, see &lt;a href="https://rajanmaharjan.medium.com/secure-your-mongodb-connections-ssl-tls-92e2addb3c89" rel="noopener noreferrer"&gt;this&lt;/a&gt; guide.&lt;/p&gt;

&lt;h3&gt;
  
  
  config/mongodb.conf
&lt;/h3&gt;

&lt;p&gt;This is the &lt;a href="https://www.mongodb.com/docs/manual/reference/configuration-options/" rel="noopener noreferrer"&gt;configuration file&lt;/a&gt; used to configure the mongod instance at startup and will be be volumized on the Docker container for the primary replicant:&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="na"&gt;net&lt;/span&gt;&lt;span class="pi"&gt;:&lt;/span&gt;
  &lt;span class="na"&gt;ssl&lt;/span&gt;&lt;span class="pi"&gt;:&lt;/span&gt;
    &lt;span class="na"&gt;mode&lt;/span&gt;&lt;span class="pi"&gt;:&lt;/span&gt; &lt;span class="s"&gt;requireSSL&lt;/span&gt;
    &lt;span class="na"&gt;PEMKeyFile&lt;/span&gt;&lt;span class="pi"&gt;:&lt;/span&gt; &lt;span class="s2"&gt;"&lt;/span&gt;&lt;span class="s"&gt;/data/ssl/server.pem"&lt;/span&gt;
    &lt;span class="na"&gt;CAFile&lt;/span&gt;&lt;span class="pi"&gt;:&lt;/span&gt; &lt;span class="s2"&gt;"&lt;/span&gt;&lt;span class="s"&gt;/data/ssl/ca.pem"&lt;/span&gt;
&lt;span class="na"&gt;storage&lt;/span&gt;&lt;span class="pi"&gt;:&lt;/span&gt;
  &lt;span class="na"&gt;dbPath&lt;/span&gt;&lt;span class="pi"&gt;:&lt;/span&gt; &lt;span class="s2"&gt;"&lt;/span&gt;&lt;span class="s"&gt;/data/db"&lt;/span&gt;
&lt;span class="na"&gt;net&lt;/span&gt;&lt;span class="pi"&gt;:&lt;/span&gt;
  &lt;span class="na"&gt;bindIp&lt;/span&gt;&lt;span class="pi"&gt;:&lt;/span&gt; &lt;span class="s"&gt;127.0.0.1&lt;/span&gt;
  &lt;span class="na"&gt;port&lt;/span&gt;&lt;span class="pi"&gt;:&lt;/span&gt; &lt;span class="m"&gt;27017&lt;/span&gt;
&lt;/code&gt;&lt;/pre&gt;

&lt;/div&gt;



&lt;h3&gt;
  
  
  docker-compose.yml
&lt;/h3&gt;

&lt;p&gt;In our docker-compose file, we create three nodes: &lt;code&gt;mongo1&lt;/code&gt; (primary), &lt;code&gt;mongo2&lt;/code&gt;, and &lt;code&gt;mongo3&lt;/code&gt;. These replicant containers use the &lt;a href="https://docs.docker.com/network/bridge/" rel="noopener noreferrer"&gt;bridge network&lt;/a&gt; &lt;code&gt;mongo_network&lt;/code&gt;.&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="na"&gt;version&lt;/span&gt;&lt;span class="pi"&gt;:&lt;/span&gt; &lt;span class="s1"&gt;'&lt;/span&gt;&lt;span class="s"&gt;3'&lt;/span&gt;

&lt;span class="na"&gt;networks&lt;/span&gt;&lt;span class="pi"&gt;:&lt;/span&gt;
  &lt;span class="na"&gt;mongo_network&lt;/span&gt;&lt;span class="pi"&gt;:&lt;/span&gt;
    &lt;span class="na"&gt;driver&lt;/span&gt;&lt;span class="pi"&gt;:&lt;/span&gt; &lt;span class="s"&gt;bridge&lt;/span&gt;

&lt;span class="na"&gt;services&lt;/span&gt;&lt;span class="pi"&gt;:&lt;/span&gt;
  &lt;span class="na"&gt;mongo1&lt;/span&gt;&lt;span class="pi"&gt;:&lt;/span&gt;
    &lt;span class="na"&gt;hostname&lt;/span&gt;&lt;span class="pi"&gt;:&lt;/span&gt; &lt;span class="s"&gt;mongo1&lt;/span&gt;
    &lt;span class="na"&gt;image&lt;/span&gt;&lt;span class="pi"&gt;:&lt;/span&gt; &lt;span class="s"&gt;mongo&lt;/span&gt;
    &lt;span class="na"&gt;ports&lt;/span&gt;&lt;span class="pi"&gt;:&lt;/span&gt;
      &lt;span class="pi"&gt;-&lt;/span&gt; &lt;span class="s"&gt;27017:27017&lt;/span&gt;
    &lt;span class="na"&gt;restart&lt;/span&gt;&lt;span class="pi"&gt;:&lt;/span&gt; &lt;span class="s"&gt;always&lt;/span&gt;
    &lt;span class="na"&gt;entrypoint&lt;/span&gt;&lt;span class="pi"&gt;:&lt;/span&gt; &lt;span class="pi"&gt;[&lt;/span&gt; &lt;span class="s2"&gt;"&lt;/span&gt;&lt;span class="s"&gt;/usr/bin/mongod"&lt;/span&gt;&lt;span class="pi"&gt;,&lt;/span&gt; &lt;span class="s2"&gt;"&lt;/span&gt;&lt;span class="s"&gt;--config"&lt;/span&gt;&lt;span class="pi"&gt;,&lt;/span&gt; &lt;span class="s2"&gt;"&lt;/span&gt;&lt;span class="s"&gt;/data/config/ssl.conf"&lt;/span&gt;&lt;span class="pi"&gt;,&lt;/span&gt; &lt;span class="s2"&gt;"&lt;/span&gt;&lt;span class="s"&gt;--bind_ip_all"&lt;/span&gt;&lt;span class="pi"&gt;,&lt;/span&gt; &lt;span class="s2"&gt;"&lt;/span&gt;&lt;span class="s"&gt;--replSet"&lt;/span&gt;&lt;span class="pi"&gt;,&lt;/span&gt; &lt;span class="s2"&gt;"&lt;/span&gt;&lt;span class="s"&gt;dbrs"&lt;/span&gt; &lt;span class="pi"&gt;]&lt;/span&gt;
    &lt;span class="na"&gt;networks&lt;/span&gt;&lt;span class="pi"&gt;:&lt;/span&gt;
      &lt;span class="pi"&gt;-&lt;/span&gt; &lt;span class="s"&gt;mongo_network&lt;/span&gt;
    &lt;span class="na"&gt;volumes&lt;/span&gt;&lt;span class="pi"&gt;:&lt;/span&gt;
      &lt;span class="pi"&gt;-&lt;/span&gt; &lt;span class="s"&gt;./.db/mongo1:/data/db&lt;/span&gt;
      &lt;span class="pi"&gt;-&lt;/span&gt; &lt;span class="s"&gt;./wait-for-mongodb.sh:/scripts/wait-for-mongodb.sh&lt;/span&gt;
      &lt;span class="pi"&gt;-&lt;/span&gt; &lt;span class="s"&gt;./init.sh:/scripts/init.sh&lt;/span&gt;
      &lt;span class="pi"&gt;-&lt;/span&gt; &lt;span class="s"&gt;./ssl:/data/ssl&lt;/span&gt;
      &lt;span class="pi"&gt;-&lt;/span&gt; &lt;span class="s"&gt;./config:/data/config&lt;/span&gt;
    &lt;span class="na"&gt;links&lt;/span&gt;&lt;span class="pi"&gt;:&lt;/span&gt;
      &lt;span class="pi"&gt;-&lt;/span&gt; &lt;span class="s"&gt;mongo2&lt;/span&gt;
      &lt;span class="pi"&gt;-&lt;/span&gt; &lt;span class="s"&gt;mongo3&lt;/span&gt;

  &lt;span class="na"&gt;mongo2&lt;/span&gt;&lt;span class="pi"&gt;:&lt;/span&gt;
    &lt;span class="na"&gt;hostname&lt;/span&gt;&lt;span class="pi"&gt;:&lt;/span&gt; &lt;span class="s"&gt;mongo2&lt;/span&gt;
    &lt;span class="na"&gt;image&lt;/span&gt;&lt;span class="pi"&gt;:&lt;/span&gt; &lt;span class="s"&gt;mongo&lt;/span&gt;
    &lt;span class="na"&gt;ports&lt;/span&gt;&lt;span class="pi"&gt;:&lt;/span&gt;
      &lt;span class="pi"&gt;-&lt;/span&gt; &lt;span class="s"&gt;27018:27017&lt;/span&gt;
    &lt;span class="na"&gt;restart&lt;/span&gt;&lt;span class="pi"&gt;:&lt;/span&gt; &lt;span class="s"&gt;always&lt;/span&gt;
    &lt;span class="na"&gt;entrypoint&lt;/span&gt;&lt;span class="pi"&gt;:&lt;/span&gt; &lt;span class="pi"&gt;[&lt;/span&gt; &lt;span class="s2"&gt;"&lt;/span&gt;&lt;span class="s"&gt;/usr/bin/mongod"&lt;/span&gt;&lt;span class="pi"&gt;,&lt;/span&gt; &lt;span class="s2"&gt;"&lt;/span&gt;&lt;span class="s"&gt;--bind_ip_all"&lt;/span&gt;&lt;span class="pi"&gt;,&lt;/span&gt; &lt;span class="s2"&gt;"&lt;/span&gt;&lt;span class="s"&gt;--replSet"&lt;/span&gt;&lt;span class="pi"&gt;,&lt;/span&gt; &lt;span class="s2"&gt;"&lt;/span&gt;&lt;span class="s"&gt;dbrs"&lt;/span&gt; &lt;span class="pi"&gt;]&lt;/span&gt;
    &lt;span class="na"&gt;networks&lt;/span&gt;&lt;span class="pi"&gt;:&lt;/span&gt;
      &lt;span class="pi"&gt;-&lt;/span&gt; &lt;span class="s"&gt;mongo_network&lt;/span&gt;
    &lt;span class="na"&gt;volumes&lt;/span&gt;&lt;span class="pi"&gt;:&lt;/span&gt;
      &lt;span class="pi"&gt;-&lt;/span&gt; &lt;span class="s"&gt;./.db/mongo2:/data/db&lt;/span&gt;
      &lt;span class="pi"&gt;-&lt;/span&gt; &lt;span class="s"&gt;./wait-for-mongodb.sh:/scripts/wait-for-mongodb.sh&lt;/span&gt;

  &lt;span class="na"&gt;mongo3&lt;/span&gt;&lt;span class="pi"&gt;:&lt;/span&gt;
    &lt;span class="na"&gt;hostname&lt;/span&gt;&lt;span class="pi"&gt;:&lt;/span&gt; &lt;span class="s"&gt;mongo3&lt;/span&gt;
    &lt;span class="na"&gt;image&lt;/span&gt;&lt;span class="pi"&gt;:&lt;/span&gt; &lt;span class="s"&gt;mongo&lt;/span&gt;
    &lt;span class="na"&gt;ports&lt;/span&gt;&lt;span class="pi"&gt;:&lt;/span&gt;
      &lt;span class="pi"&gt;-&lt;/span&gt; &lt;span class="s"&gt;27019:27017&lt;/span&gt;
    &lt;span class="na"&gt;restart&lt;/span&gt;&lt;span class="pi"&gt;:&lt;/span&gt; &lt;span class="s"&gt;always&lt;/span&gt;
    &lt;span class="na"&gt;entrypoint&lt;/span&gt;&lt;span class="pi"&gt;:&lt;/span&gt; &lt;span class="pi"&gt;[&lt;/span&gt; &lt;span class="s2"&gt;"&lt;/span&gt;&lt;span class="s"&gt;/usr/bin/mongod"&lt;/span&gt;&lt;span class="pi"&gt;,&lt;/span&gt; &lt;span class="s2"&gt;"&lt;/span&gt;&lt;span class="s"&gt;--bind_ip_all"&lt;/span&gt;&lt;span class="pi"&gt;,&lt;/span&gt; &lt;span class="s2"&gt;"&lt;/span&gt;&lt;span class="s"&gt;--replSet"&lt;/span&gt;&lt;span class="pi"&gt;,&lt;/span&gt; &lt;span class="s2"&gt;"&lt;/span&gt;&lt;span class="s"&gt;dbrs"&lt;/span&gt; &lt;span class="pi"&gt;]&lt;/span&gt;
    &lt;span class="na"&gt;networks&lt;/span&gt;&lt;span class="pi"&gt;:&lt;/span&gt;
      &lt;span class="pi"&gt;-&lt;/span&gt; &lt;span class="s"&gt;mongo_network&lt;/span&gt;
    &lt;span class="na"&gt;volumes&lt;/span&gt;&lt;span class="pi"&gt;:&lt;/span&gt;
      &lt;span class="pi"&gt;-&lt;/span&gt; &lt;span class="s"&gt;./.db/mongo3:/data/db&lt;/span&gt;
      &lt;span class="pi"&gt;-&lt;/span&gt; &lt;span class="s"&gt;./wait-for-mongodb.sh:/scripts/wait-for-mongodb.sh&lt;/span&gt;
&lt;/code&gt;&lt;/pre&gt;

&lt;/div&gt;



&lt;p&gt;Before composing, it is recommended to update the &lt;code&gt;/etc/hosts&lt;/code&gt; file to include the replicant mappings:&lt;br&gt;
&lt;/p&gt;

&lt;div class="highlight js-code-highlight"&gt;
&lt;pre class="highlight plaintext"&gt;&lt;code&gt;# Replica set mappings
127.0.0.1 mongo1
127.0.0.1 mongo2
127.0.0.1 mongo3
&lt;/code&gt;&lt;/pre&gt;

&lt;/div&gt;



&lt;h3&gt;
  
  
  init.sh
&lt;/h3&gt;

&lt;p&gt;This shell file is used to initialize the replica set and will be ran in the docker container for the primary node (i.e. &lt;code&gt;mongo1&lt;/code&gt;, defined in the "docker-compose.yml" section).&lt;br&gt;
&lt;/p&gt;

&lt;div class="highlight js-code-highlight"&gt;
&lt;pre class="highlight shell"&gt;&lt;code&gt;&lt;span class="c"&gt;#!/bin/bash&lt;/span&gt;

mongosh &lt;span class="nt"&gt;--tls&lt;/span&gt; &lt;span class="se"&gt;\&lt;/span&gt;
  &lt;span class="nt"&gt;--tlsCAFile&lt;/span&gt; /data/ssl/ca.pem &lt;span class="se"&gt;\&lt;/span&gt;
  &lt;span class="nt"&gt;--tlsCertificateKeyFile&lt;/span&gt; /data/ssl/server.pem &lt;span class="o"&gt;&amp;lt;&amp;lt;&lt;/span&gt;&lt;span class="no"&gt;EOF&lt;/span&gt;&lt;span class="sh"&gt;
var config = {
    "_id": "dbrs",
    "version": 1,
    "members": [
        {
            "_id": 1,
            "host": "mongo1:27017",
            "priority": 3
        }
    ]
};
rs.initiate(config, { force: true });
rs.status();

var config = {
    "_id": "dbrs",
    "version": 1,
    "members": [
        {
            "_id": 2,
            "host": "mongo2:27017",
            "priority": 2
        },
        {
            "_id": 3,
            "host": "mongo3:27017",
            "priority": 1
        }
    ]
};
rs.initiate(config, { force: true });
rs.status();
&lt;/span&gt;&lt;span class="no"&gt;EOF
&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;

&lt;/div&gt;



&lt;h3&gt;
  
  
  wait-for-mongo.sh
&lt;/h3&gt;

&lt;p&gt;This script will wait for a mongo server to come up&lt;br&gt;
&lt;/p&gt;

&lt;div class="highlight js-code-highlight"&gt;
&lt;pre class="highlight shell"&gt;&lt;code&gt;&lt;span class="c"&gt;#!/bin/bash&lt;/span&gt;

&lt;span class="nv"&gt;NAME&lt;/span&gt;&lt;span class="o"&gt;=&lt;/span&gt;&lt;span class="nv"&gt;$1&lt;/span&gt;
&lt;span class="nv"&gt;OPTS&lt;/span&gt;&lt;span class="o"&gt;=&lt;/span&gt;&lt;span class="nv"&gt;$2&lt;/span&gt;

&lt;span class="k"&gt;while&lt;/span&gt; &lt;span class="o"&gt;[&lt;/span&gt; &lt;span class="nt"&gt;-z&lt;/span&gt; &lt;span class="sb"&gt;`&lt;/span&gt;mongosh &lt;span class="nt"&gt;--eval&lt;/span&gt; &lt;span class="s1"&gt;'db.runCommand("ping").ok'&lt;/span&gt; &lt;span class="nt"&gt;--quiet&lt;/span&gt; &lt;span class="nt"&gt;--host&lt;/span&gt; &lt;span class="nv"&gt;$OPTS&lt;/span&gt; 2&amp;gt;/dev/null&lt;span class="sb"&gt;`&lt;/span&gt; &lt;span class="o"&gt;]&lt;/span&gt;&lt;span class="p"&gt;;&lt;/span&gt; &lt;span class="k"&gt;do
    &lt;/span&gt;&lt;span class="nb"&gt;echo&lt;/span&gt; &lt;span class="s2"&gt;"Waiting for MongoDB &lt;/span&gt;&lt;span class="nv"&gt;$NAME&lt;/span&gt;&lt;span class="s2"&gt; to start..."&lt;/span&gt;
    &lt;span class="nb"&gt;sleep &lt;/span&gt;1
&lt;span class="k"&gt;done

&lt;/span&gt;&lt;span class="nb"&gt;echo&lt;/span&gt; &lt;span class="s2"&gt;"MongoDB &lt;/span&gt;&lt;span class="nv"&gt;$NAME&lt;/span&gt;&lt;span class="s2"&gt; is up and running"&lt;/span&gt;
&lt;/code&gt;&lt;/pre&gt;

&lt;/div&gt;



&lt;h3&gt;
  
  
  run.sh
&lt;/h3&gt;

&lt;p&gt;&lt;code&gt;run.sh&lt;/code&gt; will (1) start the replicants, (2) wait for the replicants to come up, and (3) define primary and secondary replicants on &lt;code&gt;mongo1&lt;/code&gt;.&lt;br&gt;
&lt;/p&gt;

&lt;div class="highlight js-code-highlight"&gt;
&lt;pre class="highlight shell"&gt;&lt;code&gt;&lt;span class="c"&gt;#!/bin/bash&lt;/span&gt;

&lt;span class="nb"&gt;set&lt;/span&gt; &lt;span class="nt"&gt;-e&lt;/span&gt;

&lt;span class="c"&gt;# remove volumes&lt;/span&gt;
&lt;span class="nb"&gt;rm&lt;/span&gt; &lt;span class="nt"&gt;-rf&lt;/span&gt; .db

&lt;span class="c"&gt;# drop existing containers&lt;/span&gt;
docker compose &lt;span class="nt"&gt;-f&lt;/span&gt; &lt;span class="s2"&gt;"docker-compose.yml"&lt;/span&gt; down

&lt;span class="c"&gt;# prune containers&lt;/span&gt;
docker system prune &lt;span class="nt"&gt;--force&lt;/span&gt;

docker-compose &lt;span class="nt"&gt;-f&lt;/span&gt; &lt;span class="s2"&gt;"docker-compose.yml"&lt;/span&gt; up &lt;span class="nt"&gt;-d&lt;/span&gt; &lt;span class="se"&gt;\&lt;/span&gt;
    &lt;span class="nt"&gt;--remove-orphans&lt;/span&gt; &lt;span class="se"&gt;\&lt;/span&gt;
    &lt;span class="nt"&gt;--force-recreate&lt;/span&gt; &lt;span class="se"&gt;\&lt;/span&gt;
    &lt;span class="nt"&gt;--build&lt;/span&gt; mongo1

docker-compose &lt;span class="nt"&gt;-f&lt;/span&gt; &lt;span class="s2"&gt;"docker-compose.yml"&lt;/span&gt; up &lt;span class="nt"&gt;-d&lt;/span&gt; &lt;span class="se"&gt;\&lt;/span&gt;
    &lt;span class="nt"&gt;--remove-orphans&lt;/span&gt; &lt;span class="se"&gt;\&lt;/span&gt;
    &lt;span class="nt"&gt;--force-recreate&lt;/span&gt; &lt;span class="se"&gt;\&lt;/span&gt;
    &lt;span class="nt"&gt;--build&lt;/span&gt; mongo2

docker-compose &lt;span class="nt"&gt;-f&lt;/span&gt; &lt;span class="s2"&gt;"docker-compose.yml"&lt;/span&gt; up &lt;span class="nt"&gt;-d&lt;/span&gt; &lt;span class="se"&gt;\&lt;/span&gt;
    &lt;span class="nt"&gt;--remove-orphans&lt;/span&gt; &lt;span class="se"&gt;\&lt;/span&gt;
    &lt;span class="nt"&gt;--force-recreate&lt;/span&gt; &lt;span class="se"&gt;\&lt;/span&gt;
    &lt;span class="nt"&gt;--build&lt;/span&gt; mongo3

docker-compose &lt;span class="nt"&gt;-f&lt;/span&gt; &lt;span class="s2"&gt;"docker-compose.yml"&lt;/span&gt; &lt;span class="nb"&gt;exec&lt;/span&gt; &lt;span class="nt"&gt;-T&lt;/span&gt; mongo3 /scripts/wait-for-mongodb.sh &lt;span class="s2"&gt;"mongo3:27017"&lt;/span&gt;
docker-compose &lt;span class="nt"&gt;-f&lt;/span&gt; &lt;span class="s2"&gt;"docker-compose.yml"&lt;/span&gt; &lt;span class="nb"&gt;exec&lt;/span&gt; &lt;span class="nt"&gt;-T&lt;/span&gt; mongo2 /scripts/wait-for-mongodb.sh &lt;span class="s2"&gt;"mongo2:27017"&lt;/span&gt;

&lt;span class="nv"&gt;M1_OPTS&lt;/span&gt;&lt;span class="o"&gt;=&lt;/span&gt;&lt;span class="s2"&gt;"--tls --tlsCAFile /data/ssl/ca.pem --tlsCertificateKeyFile /data/ssl/server.pem --tlsAllowInvalidCertificates"&lt;/span&gt;
docker-compose &lt;span class="nt"&gt;-f&lt;/span&gt; &lt;span class="s2"&gt;"docker-compose.yml"&lt;/span&gt; &lt;span class="nb"&gt;exec&lt;/span&gt; &lt;span class="nt"&gt;-T&lt;/span&gt; mongo1 /scripts/wait-for-mongodb.sh &lt;span class="s2"&gt;"mongo1:27017"&lt;/span&gt; &lt;span class="s2"&gt;"&lt;/span&gt;&lt;span class="nv"&gt;$M1_OPTS&lt;/span&gt;&lt;span class="s2"&gt;"&lt;/span&gt;

&lt;span class="nb"&gt;echo&lt;/span&gt; &lt;span class="s2"&gt;"Creating replica set..."&lt;/span&gt;
docker-compose &lt;span class="nt"&gt;-f&lt;/span&gt; &lt;span class="s2"&gt;"docker-compose.yml"&lt;/span&gt; &lt;span class="nb"&gt;exec&lt;/span&gt; &lt;span class="nt"&gt;-T&lt;/span&gt; mongo1 /scripts/init.sh
&lt;/code&gt;&lt;/pre&gt;

&lt;/div&gt;



&lt;p&gt;Putting everything together, we run the above shell script and then test our setup by opening a mongo shell:&lt;br&gt;
&lt;/p&gt;

&lt;div class="highlight js-code-highlight"&gt;
&lt;pre class="highlight plaintext"&gt;&lt;code&gt;mongosh --tls --tlsCAFile &amp;lt;path-to-ca-file&amp;gt; --tlsCertificateKeyFile &amp;lt;path-to-client-file&amp;gt;
&lt;/code&gt;&lt;/pre&gt;

&lt;/div&gt;



</description>
      <category>mongodb</category>
      <category>docker</category>
      <category>security</category>
      <category>productivity</category>
    </item>
  </channel>
</rss>
