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diff --git a/docs/feed.rss b/docs/feed.rss index 12e9f8d..12b90df 100644 --- a/docs/feed.rss +++ b/docs/feed.rss @@ -4,8 +4,8 @@ <title>Navan's Archive</title> <description>Rare Tips, Tricks and Posts</description> <link>https://web.navan.dev/</link><language>en</language> - <lastBuildDate>Thu, 21 Mar 2024 14:27:28 -0000</lastBuildDate> - <pubDate>Thu, 21 Mar 2024 14:27:28 -0000</pubDate> + <lastBuildDate>Tue, 26 Mar 2024 18:20:37 -0000</lastBuildDate> + <pubDate>Tue, 26 Mar 2024 18:20:37 -0000</pubDate> <ttl>250</ttl> <atom:link href="https://web.navan.dev/feed.rss" rel="self" type="application/rss+xml"/> @@ -557,9 +557,7 @@ creating<span class="w"> </span>a<span class="w"> </span>DOS<span class="w"> </s <script src="https://cdn.jsdelivr.net/npm/mathjax@4.0.0-beta.4/input/tex/extensions/noerrors.js" charset="UTF-8"></script> -<p>$$ -y = ax^3 + bx^2 + cx + d -$$</p> +<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mrow><mi>y</mi><mo>=</mo><mi>a</mi><msup><mi>x</mi><mn>3</mn></msup><mo>+</mo><mi>b</mi><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mi>c</mi><mi>x</mi><mo>+</mo><mi>d</mi></mrow></math> <h3>Optimizer Selection & Training</h3> @@ -584,9 +582,7 @@ $$</p> <p>Our loss function is Mean Squared Error (MSE):</p> -<p>$$ -= \frac{1}{n} \sum_{i=1}^{n}{(Y_i - \hat{Y_i})^2} -$$</p> +<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mrow><mo>=</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mi>n</mi></mrow></mfrac><msubsup><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mrow><mi>n</mi></mrow></msubsup><mrow><mo stretchy="false">(</mo><mi>Y</mi><mi>_</mi><mi>i</mi><mo>−</mo><mover><mrow><mi>Y</mi><mi>_</mi><mi>i</mi></mrow><mo stretchy="false">^</mo></mover><msup><mo stretchy="false">)</mo><mn>2</mn></msup></mrow></mrow></math> <p>Where <math xmlns="http://www.w3.org/1998/Math/MathML"><mover><msub><mi>Y</mi><mi>i</mi></msub><mo stretchy="false" style="math-style:normal;math-depth:0;">^</mo></mover></math> is the predicted value and <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>Y</mi><mi>i</mi></msub></math> is the actual value</p> @@ -726,7 +722,7 @@ $$</p> <p>How would you modify this code to use another type of nonlinear regression? Say, </p> -<p>$$ y = ab^x $$</p> +<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mrow><mi>y</mi><mo>=</mo><mi>a</mi><msup><mi>b</mi><mi>x</mi></msup></mrow></math> <p>Hint: Your loss calculation would be similar to:</p> @@ -3188,6 +3184,52 @@ values using the X values. We then plot it to compare the actual data and predic <item> <guid isPermaLink="true"> + https://web.navan.dev/posts/2024-03-26-Derivation-of-the-Quadratic-Equation.html + </guid> + <title> + Quadratic Formula Derivation + </title> + <description> + Quick derivation of the quadratic equation by completing the square + </description> + <link>https://web.navan.dev/posts/2024-03-26-Derivation-of-the-Quadratic-Equation.html</link> + <pubDate>Tue, 26 Mar 2024 15:36:00 -0000</pubDate> + <content:encoded><![CDATA[<h1>Quadratic Formula Derivation</h1> + +<p>The standard form of a quadratic equation is:</p> + +<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mrow><mi>a</mi><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mi>b</mi><mi>x</mi><mo>+</mo><mi>c</mi><mo>=</mo><mn>0</mn></mrow></math> + +<p>Here, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mi>a</mi><mo>,</mo><mi>b</mi><mo>,</mo><mi>c</mi><mo>∈</mo><mi>ℝ</mi></mrow></math>, and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mi>a</mi><mo>≠</mo><mn>0</mn></mrow></math></p> + +<p>We begin by first dividing both sides by the coefficient <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mi>a</mi></mrow></math></p> + +<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mrow><mi>⟹</mi><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mfrac><mrow><mi>b</mi></mrow><mrow><mi>a</mi></mrow></mfrac><mi>x</mi><mo>+</mo><mfrac><mrow><mi>c</mi></mrow><mrow><mi>a</mi></mrow></mfrac><mo>=</mo><mn>0</mn></mrow></math> + +<p>We can rearrange the equation:</p> + +<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mfrac><mrow><mi>b</mi></mrow><mrow><mi>a</mi></mrow></mfrac><mi>x</mi><mo>=</mo><mo>−</mo><mfrac><mrow><mi>c</mi></mrow><mrow><mi>a</mi></mrow></mfrac></mrow></math> + +<p>We can then use the method of completing the square. (<a rel="noopener" target="_blank" href="https://www.mathsisfun.com/algebra/completing-square.html">Maths is Fun</a> has a really good explanation for this technique)</p> + +<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mfrac><mrow><mi>b</mi></mrow><mrow><mi>a</mi></mrow></mfrac><mi>x</mi><mo>+</mo><mo stretchy="false">(</mo><mfrac><mrow><mi>b</mi></mrow><mrow><mn>2</mn><mi>a</mi></mrow></mfrac><msup><mo stretchy="false">)</mo><mn>2</mn></msup><mo>=</mo><mfrac><mrow><mo>−</mo><mi>c</mi></mrow><mrow><mi>a</mi></mrow></mfrac><mo>+</mo><mo stretchy="false">(</mo><mfrac><mrow><mi>b</mi></mrow><mrow><mn>2</mn><mi>a</mi></mrow></mfrac><msup><mo stretchy="false">)</mo><mn>2</mn></msup></mrow></math> + +<p>On our LHS, we can clearly recognize that it is the expanded form of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mo stretchy="false">(</mo><mi>x</mi><mo>+</mo><mi>d</mi><msup><mo stretchy="false">)</mo><mn>2</mn></msup></mrow></math> i.e <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>2</mn><mi>x</mi><mi>·</mi><mi>d</mi><mo>+</mo><msup><mi>d</mi><mn>2</mn></msup></mrow></math></p> + +<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mrow><mi>⟹</mi><mo stretchy="false">(</mo><mi>x</mi><mo>+</mo><mfrac><mrow><mi>b</mi></mrow><mrow><mn>2</mn><mi>a</mi></mrow></mfrac><msup><mo stretchy="false">)</mo><mn>2</mn></msup><mo>=</mo><mfrac><mrow><mo>−</mo><mi>c</mi></mrow><mrow><mi>a</mi></mrow></mfrac><mo>+</mo><mfrac><mrow><msup><mi>b</mi><mn>2</mn></msup></mrow><mrow><mn>4</mn><msup><mi>a</mi><mn>2</mn></msup></mrow></mfrac><mo>=</mo><mfrac><mrow><mo>−</mo><mn>4</mn><mi>a</mi><mi>c</mi><mo>+</mo><msup><mi>b</mi><mn>2</mn></msup></mrow><mrow><mn>4</mn><msup><mi>a</mi><mn>2</mn></msup></mrow></mfrac></mrow></math> + +<p>Taking the square root of both sides</p> + +<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mrow><mtable displaystyle="true" rowspacing="3pt" columnspacing="0em 2em"><mtr><mtd columnalign="right"><mi>x</mi><mo>+</mo><mfrac><mrow><mi>b</mi></mrow><mrow><mn>2</mn><mi>a</mi></mrow></mfrac></mtd><mtd columnalign="left"><mi /><mo>=</mo><mfrac><mrow><msqrt><mrow><mo>−</mo><mn>4</mn><mi>a</mi><mi>c</mi><mo>+</mo><msup><mi>b</mi><mn>2</mn></msup></mrow></msqrt></mrow><mrow><mn>2</mn><mi>a</mi></mrow></mfrac></mtd></mtr><mtr><mtd columnalign="right"><mi>x</mi></mtd><mtd columnalign="left"><mi /><mo>=</mo><mfrac><mrow><mi>±</mi><msqrt><mrow><mo>−</mo><mn>4</mn><mi>a</mi><mi>c</mi><mo>+</mo><msup><mi>b</mi><mn>2</mn></msup></mrow></msqrt><mo>−</mo><mi>b</mi></mrow><mrow><mn>2</mn><mi>a</mi></mrow></mfrac></mtd></mtr><mtr><mtd columnalign="right" /><mtd columnalign="left"><mi /><mo>=</mo><mfrac><mrow><mo>−</mo><mi>b</mi><mi>±</mi><msqrt><mrow><msup><mi>b</mi><mn>2</mn></msup><mo>−</mo><mn>4</mn><mi>a</mi><mi>c</mi></mrow></msqrt></mrow><mrow><mn>2</mn><mi>a</mi></mrow></mfrac></mtd></mtr></mtable></mrow></math> + +<p>This gives you the world famous quadratic formula:</p> + +<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mrow><mi>x</mi><mo>=</mo><mfrac><mrow><mo>−</mo><mi>b</mi><mi>±</mi><msqrt><mrow><msup><mi>b</mi><mn>2</mn></msup><mo>−</mo><mn>4</mn><mi>a</mi><mi>c</mi></mrow></msqrt></mrow><mrow><mn>2</mn><mi>a</mi></mrow></mfrac></mrow></math> +]]></content:encoded> + </item> + + <item> + <guid isPermaLink="true"> https://web.navan.dev/posts/2022-08-05-Why-You-No-Host.html </guid> <title> |