feat: halley's method + pdf -> img

.husky/pre-commit

@@ -2,5 +2,5 @@
 . "$(dirname "$0")/_/husky.sh"
 
 npx pretty-quick --staged
-node .husky/scripts/cleanMaths.js
+.husky/scripts/pdfToPng.sh
 git add .

.husky/scripts/cleanMaths.js

@@ -1,73 +0,0 @@
-const fs = require('fs');
-const path = require('path');
-const { pdfToPng } = require('pdf-to-png-converter');
-const Jimp = require('jimp');
-
-const mathsDir = path.resolve(__dirname, '..', '..', 'maths');
-
-const cropBuffer = buffer =>
-    new Promise((resolve, reject) => {
-        Jimp.read(buffer)
-            .then(cropped => {
-                cropped.autocrop();
-
-                new Jimp(
-                    cropped.getWidth() + 100,
-                    cropped.getHeight() + 100,
-                    '#ffffff',
-                    async (err, image) => {
-                        if (err) reject(err);
-
-                        image.blit(cropped, 50, 50);
-
-                        resolve(await image.getBufferAsync(Jimp.MIME_PNG));
-                    }
-                );
-            })
-            .catch(err => reject(err));
-    });
-
-const crawlDirectory = async dir => {
-    const dirents = fs.readdirSync(dir, { withFileTypes: true });
-
-    const files = await Promise.all(
-        dirents.map(dirent => {
-            const res = path.resolve(dir, dirent.name);
-            return dirent.isDirectory() ? crawlDirectory(res) : res;
-        })
-    );
-
-    return files.flat();
-};
-
-(async () => {
-    const files = await crawlDirectory(mathsDir);
-
-    files.forEach(file => {
-        const [filePath, fileDirectory, fileNameWithExt] = file.match(/(.*)\/((?:.(?!\/))+)$/);
-        const [fileName, fileExtension] = fileNameWithExt.split('.');
-
-        if (file.endsWith('.pdf')) {
-            pdfToPng(filePath).then(async output => {
-                const pageCount = output.length;
-
-                if (pageCount > 1)
-                    output.forEach(async (page, i) =>
-                        fs.writeFileSync(
-                            path.join(fileDirectory, `${fileName}-${i + 1}.png`),
-                            await cropBuffer(page.content)
-                        )
-                    );
-                else
-                    fs.writeFileSync(
-                        path.join(fileDirectory, `${fileName}.png`),
-                        await cropBuffer(output[0].content)
-                    );
-            });
-        }
-
-        if (fileExtension !== 'tex' && fileExtension !== 'png' && fileExtension !== 'cls') {
-            fs.rmSync(filePath);
-        }
-    });
-})();

.husky/scripts/pdfToPng.sh

@@ -0,0 +1,22 @@
+#!/bin/bash
+find . -print0 | while IFS= read -r -d '' file
+do
+  if [ -f "$file" ]; then
+	if [[ $file == *pdf ]]; then
+		mkdir temp
+		gm convert "$file" +adjoin temp/temp%02d.png
+
+		for temp in ./temp/*.png
+		do
+			gm convert "$temp" -fuzz 80% -trim +repage -bordercolor white -border 50x25 "$temp"
+		done
+
+		readarray -d . -t arr <<< $file
+
+		gm convert -append ./temp/*.png "${arr[1]:1}.png"
+		
+		rm -rf temp
+		rm "$file" "${arr[1]:1}.synctex.gz"
+	fi
+fi
+done

maths/halley's method.tex

@@ -0,0 +1,53 @@
+\documentclass{style}
+\usepackage{amsmath}
+\usepackage{breqn}
+\usepackage{mathtools}
+
+\begin{document}
+\begin{center}
+	Suppose we have an $n$th degree polynomial, such that $f(x) = c_{n}x^{n} + c_{n-1}x^{n-1} + c_{n-2}x^{n-2} + ... + c_{0} = \sum_{i=0}^{n}c_{n-i}x^{n-i}$
+
+	\hfill
+
+	By the power rule of differentiation, we can conclude that the first derivative of $f(x)$ is as follows
+
+	\begin{dmath*}
+		f'(x) = nc_{n}x^{n-1} + (n-1)c_{n-1}x^{n-2} + (n-2)c_{n-2}x^{n-3} + ... + c_{1} = \sum_{i=0}^{n-1}(n-i)c_{n-i}x^{n-i-1}
+	\end{dmath*}
+
+	And the second derivative is
+
+	\begin{dmath*}
+		f''(x) = n(n-1)c_{n}x^{n-2} + (n-1)(n-2)c_{n-1}x^{n-3} + (n-2)(n-3)c_{n-2}x^{n-4} + ...+ c_{2} = \sum_{i=0}^{n-2}(i-n)(i-n+1)a_{n-i}x^{n-i-2}
+	\end{dmath*}
+
+	After starting with an initial guess, the next iteration of Halley's method is given by $x_k - \frac{2f(x_k)f'(x_k)}{2[f'(x_k)]^2 - f(x_k)f''(x_k)}$.
+	
+	This means that we must first find $f(x)f'(x)$, $[f'(x)]^2$, and $f(x)f''(x)$. These all consist of the multiplication of two series - there is a nice general form to this problem stated below
+
+	\begin{dmath*}
+		(\sum_{i=0}^{n}x_{i})(\sum_{j=0}^{m}y_{j}) = \sum_{i=0}^{n}\sum_{j=0}^{m}x_iy_j
+	\end{dmath*}
+
+	\newpage
+	From this, we can conclude that:
+
+	\begin{dmath*}
+		f(x)f'(x) = (\sum_{i=0}^{n}c_{n-i}x^{n-i})(\sum_{i=0}^{n-1}(n-i)c_{n-i}x^{n-i-1}) = \sum_{i=0}^{n}\sum_{j=0}^{n-1}(n-j)c_{n-i}c_{n-j}x^{2n-i-j-1}
+	\end{dmath*}
+
+	\begin{dmath*}
+		[f'(x)]^2 = (\sum_{i=0}^{n-1}(n-i)c_{n-i}x^{n-i-1})(\sum_{i=0}^{n-1}(n-i)c_{n-i}x^{n-i-1}) = \sum_{i=0}^{n-1}\sum_{j=0}^{n-1}(n-i)(n-j)c_{n-i}c_{n-j}x^{2(n-1)-i-j}
+	\end{dmath*}
+
+	\begin{dmath*}
+		f(x)f''(x) = (\sum_{i=0}^{n}c_{n-i}x^{n-i})(\sum_{i=0}^{n-2}(i-n)(i-n+1)a_{n-i}x^{n-i-2}) = \sum_{i=0}^{n}\sum_{j=0}^{n-2}(j-n)(j-n+1)c_{n-i}c_{n-j}x^{2(n-1)-i-j}
+	\end{dmath*}
+
+	And all that is left to do is to plug it into the formula for Halley's method, leaving us with the following:
+
+	\begin{align*}
+		x_{k+1} = x_{k} - \dfrac{2\sum_{i=0}^{n}\sum_{j=0}^{n-1}(n-j)c_{n-i}c_{n-j}x^{2n-i-j-1}}{\splitdfrac{2\sum_{i=0}^{n-1}\sum_{j=0}^{n-1}(n-i)(n-j)c_{n-i}c_{n-j}x^{2(n-1)-i-j}}{- \sum_{i=0}^{n}\sum_{j=0}^{n-2}(j-n)(j-n+1)c_{n-i}c_{n-j}x^{2(n-1)-i-j}}}
+	\end{align*}
+\end{center}
+\end{document}

maths/style.cls

@@ -1,6 +1,9 @@
 \LoadClass[17pt]{extarticle}
 \pagenumbering{gobble}
 
+\usepackage{geometry}
+\geometry{a4paper, portrait, margin=1in}
+
 \newcommand{\euler}{\begin{gather*}
 	\text{By Euler's formula:} \\
 	e^{i\theta} = cos(\theta) + i\sin(\theta) \\

package.json

@@ -1,16 +1,13 @@
 {
     "name": "the-honk",
     "scripts": {
-        "prepare": "husky install",
-        "maths": "node .husky/scripts/cleanMaths.js"
+        "prepare": "husky install"
     },
     "devDependencies": {
         "@types/node": "^17.0.6",
         "commitizen": "^4.2.4",
         "cz-conventional-changelog": "^3.3.0",
         "husky": "^7.0.4",
-        "jimp": "^0.16.1",
-        "pdf-to-png-converter": "^1.0.0",
         "prettier": "^2.5.1",
         "pretty-quick": "^3.1.3"
     }