initial commit

.gitattributes

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+[attr] lfs-file filter=lfs diff=lfs merge=lfs -text
+
+*.pdf lfs-file

.gitignore

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+# Python-generated files
+__pycache__/
+*.py[oc]
+build/
+dist/
+wheels/
+*.egg-info
+
+# Virtual environments
+.venv
+
+# 
+.vscode

.python-version

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+3.12

1.ipynb

@@ -0,0 +1,273 @@
+{
+ "cells": [
+  {
+   "cell_type": "code",
+   "execution_count": 4,
+   "id": "8f8630bf",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "import numpy as np\n",
+    "import matplotlib.pyplot as plt"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "9751d328",
+   "metadata": {},
+   "source": [
+    "# Floating Point\n",
+    "\n",
+    "**Double Precision Floating Point Number Representation**\n",
+    "\n",
+    "$$(-1)^s \\times 2^{c-1023} \\times (1+f)$$\n",
+    "\n",
+    "* $s$: sign bit\n",
+    "* $c$: 11 bit exponent\n",
+    "* $f$: 52 bit fraction"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "9c0b5413",
+   "metadata": {},
+   "source": []
+  },
+  {
+   "cell_type": "markdown",
+   "id": "964d247f",
+   "metadata": {},
+   "source": []
+  },
+  {
+   "cell_type": "markdown",
+   "id": "6f7aef85",
+   "metadata": {},
+   "source": [
+    "# Root Finding\n",
+    "\n",
+    "When there is no or hard to find analytic solution.\n",
+    "\n",
+    "* General procedure of root finding\n",
+    "\n",
+    "\n",
+    "## Incremental Search\n",
+    "\n",
+    "Detecting a sign change in the interval $[x, dx]$\n",
+    "\n",
+    "\n",
+    "## Bracketing Method\n",
+    "\n",
+    "* there must be a solution in a given interval $[a,b]$\n",
+    "* reducing the interval systematically\n",
+    "* [Merit]Convergence to an exact solution\n",
+    "\n",
+    "Methods:\n",
+    "* bisection\n",
+    "* linear interpolation\n",
+    "\n",
+    "## Open Method\n",
+    "\n",
+    "* starting from an arbitrary point, find the root based on a regular iteration\n",
+    "* [Risk] divergence\n",
+    "* [Merit] faster than _bracketing_\n",
+    "\n",
+    "Methods:\n",
+    "* fixed-point iter\n",
+    "* Newton-Raphson method\n",
+    "* secant method\n",
+    "* Muller method\n"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "2649c3d4",
+   "metadata": {},
+   "source": [
+    "## Bisection Method\n",
+    "based on the principle: **A solution must exists in $[x_1, x_2]$ if $f(x_1)f(x_2) < 0$**\n",
+    "\n",
+    "so interval reduces by half at each iteration.\n",
+    "\n",
+    "* simple\n",
+    "* maximum error $<$ (initial interval/$2^n$)\n",
+    "* slow converge\n",
+    "* cannot cope with multiple roots"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "ef80f29d",
+   "metadata": {},
+   "source": [
+    "## Linear Interpolation Method(False Position Method)\n",
+    "\n",
+    "An interval $[a_{n+1}, b_{n+1}]$, containing an approximation of $f(x) = 0$, is found from an interval $[a_n, b_n]$\n",
+    "\n",
+    "$$p_{n+1} = a_n - \\frac{f(a_n)(b_n-a_n)}{f(b_n) - f(a_n)}$$\n",
+    "\n",
+    "then set $$\\begin{cases}\n",
+    "a_{n+1} = a_n, b_{n+1} = p_{n+1} &\\quad \\text{if} f(a_n)f(p_{n+1}) < 0\\\\\n",
+    "a_{n+1} = p_{n+1}, b_{n+1} = b_n &\\quad \\text{else}\n",
+    "\\end{cases}$$\n",
+    "\n",
+    "* faster convergence than bisection\n",
+    "* convergence speed depends on **curvature**\n",
+    "\n",
+    "\n",
+    "\n",
+    "### Modified Linear Interpolation\n",
+    "\n"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "856f5bbf",
+   "metadata": {},
+   "source": [
+    "## Newton-Raphson Method\n",
+    "\n",
+    "\n",
+    "* **fast convergence**: quadratic convergence\n",
+    "* Inefficient if **the derivative calculation is complex**\n",
+    "* Initial guess matter\n",
+    "  * Cycling\n",
+    "  * Wandering\n",
+    "  * Overshooting(a root the methods get is not the root I wanted)"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "cf78e848",
+   "metadata": {},
+   "source": [
+    "## Secant Method\n",
+    "\n",
+    "$$p_{n+1} = p_n - \\frac{f(p_n)(p_n - p_{n-1})}{f(p_n) - f(p_{n-1})}$$\n",
+    "\n",
+    "Similar to linear interpolation but update rule is different.\n",
+    "\n",
+    "* fast convergence than **linear interpolation**\n",
+    "* more stable than Newton-Raphson\n",
+    "  * but convergence is slower than Newton-Raphson"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 36,
+   "id": "167cae2f",
+   "metadata": {},
+   "outputs": [
+    {
+     "data": {
+      "image/png": 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truncated
+      "text/plain": [
+       "<Figure size 640x480 with 1 Axes>"
+      ]
+     },
+     "metadata": {},
+     "output_type": "display_data"
+    }
+   ],
+   "source": [
+    "x = np.linspace(0, 4)\n",
+    "\n",
+    "f = lambda t: t**2 - 5\n",
+    "y = f(x)\n",
+    "\n",
+    "p = [0.5, 3.5]\n",
+    "\n",
+    "for _ in range(2):\n",
+    "    p.append(p[-1] - f(p[-1]) * (p[-1] - p[-2]) / (f(p[-1]) - f(p[-2])))\n",
+    "\n",
+    "p = np.array(p)\n",
+    "plt.plot(\n",
+    "    p,\n",
+    "    np.zeros_like(p),\n",
+    "    \"ro\",\n",
+    ")\n",
+    "for i, p0 in enumerate(p):\n",
+    "    plt.text(p0 + 0.05, 0.1, f\"$p_{i}$\", size=16, color=\"black\", va=\"bottom\")\n",
+    "\n",
+    "plt.plot(p, f(p), \"ro\")\n",
+    "\n",
+    "for pos in p:\n",
+    "    _y = [0, f(pos)]\n",
+    "    plt.vlines(pos, min(_y), max(_y), color=\"red\")\n",
+    "\n",
+    "for p0, p1 in zip(p[:-1], p[1:]):\n",
+    "    plt.plot([p0, p1], [f(p0), f(p1)], color=\"green\")\n",
+    "\n",
+    "plt.title(\"Secant Method\")\n",
+    "plt.plot(x, y, color=\"blue\")\n",
+    "plt.hlines(0, 0, 4, color=\"black\")\n",
+    "plt.show()"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "17e94fbf",
+   "metadata": {},
+   "source": [
+    "## Fixed Point Iteration\n",
+    "\n",
+    "principle: $f(x) = 0 \\to x = g(x)$\n",
+    "\n",
+    "$$x_{k+1} = g(x_k)$$\n"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "4cc731ad",
+   "metadata": {},
+   "source": [
+    "## Muller Method"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "c3702bdd",
+   "metadata": {},
+   "source": [
+    "## Error Analysis"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "c0e38d45",
+   "metadata": {},
+   "source": [
+    "## Accelerating Convergence\n"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "1e7717a7",
+   "metadata": {},
+   "source": [
+    "## Fail-safe Methods\n"
+   ]
+  }
+ ],
+ "metadata": {
+  "kernelspec": {
+   "display_name": "Python 3",
+   "language": "python",
+   "name": "python3"
+  },
+  "language_info": {
+   "codemirror_mode": {
+    "name": "ipython",
+    "version": 3
+   },
+   "file_extension": ".py",
+   "mimetype": "text/x-python",
+   "name": "python",
+   "nbconvert_exporter": "python",
+   "pygments_lexer": "ipython3",
+   "version": "3.12.11"
+  }
+ },
+ "nbformat": 4,
+ "nbformat_minor": 5
+}

pyproject.toml

@@ -0,0 +1,11 @@
+[project]
+name = "2025-02-numerical"
+version = "0.1.0"
+description = "Add your description here"
+readme = "README.md"
+requires-python = ">=3.12"
+dependencies = [
+    "jupyter>=1.1.1",
+    "matplotlib>=3.10.6",
+    "numpy>=2.3.2",
+]