add and fix some

2025-11-10 17:24:53 +09:00
parent 49355f8f54
commit 12e68927ad
12 changed files with 595 additions and 10 deletions
--- a/notes/1.ipynb
+++ b/notes/1.ipynb
@@ -18,7 +18,9 @@
   "source": [
    "# Floating Point\n",
    "\n",
-    "**Double Precision Floating Point Number Representation**\n",
+    "$$s\\times M \\times B^{e-E}$$\n",
+    "\n",
+    "**IEEE-754 Double Precision Floating Point Number Representation**\n",
    "\n",
    "$$(-1)^s \\times 2^{c-1023} \\times (1+f)$$\n",
    "\n",
@@ -31,13 +33,35 @@
   "cell_type": "markdown",
   "id": "9c0b5413",
   "metadata": {},
-   "source": []
+   "source": [
+    "## Machine Accuracy\n",
+    "\n",
+    "$$\\epsilon = b^{1-m}$$\n",
+    "\n",
+    "(typically $b = 2$, $m: \\text {bit for matissa}$)\n"
+   ]
  },
  {
   "cell_type": "markdown",
   "id": "964d247f",
   "metadata": {},
-   "source": []
+   "source": [
+    "# Errors\n",
+    "\n",
+    "* **Round-off Error**\n",
+    "\n",
+    "* **Truncation Error**\n",
+    "\n",
+    "Taylor Series\n",
+    "\n",
+    "approximations: zero order, first order(recommended), second order, etc..\n",
+    "\n",
+    "let $v_t$: true value, and $v_a$: approximation\n",
+    "Absolute error: $(v_t - v_a)$\n",
+    "Relative error: $(v_t - v_a) / v_t$\n",
+    "\n",
+    "so stopping condition is $e_a < e_s$ where $e_s$ is desired.\n"
+   ]
  },
  {
   "cell_type": "markdown",
@@ -48,8 +72,7 @@
    "\n",
    "When there is no or hard to find analytic solution.\n",
    "\n",
-    "* General procedure of root finding\n",
-    "\n",
+    "General procedure of root finding\n",
    "\n",
    "## Incremental Search\n",
    "\n",
--- a/notes/2.ipynb
+++ b/notes/2.ipynb
@@ -29,18 +29,75 @@
  {
   "cell_type": "code",
   "execution_count": null,
-   "id": "3bed3d44",
+   "id": "e7357276",
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
-   "cell_type": "code",
-   "execution_count": null,
+   "cell_type": "markdown",
+   "id": "1113de3c",
+   "metadata": {},
+   "source": []
+  },
+  {
+   "cell_type": "markdown",
+   "id": "3bed3d44",
+   "metadata": {},
+   "source": [
+    "## Gauss Elimination\n",
+    "\n",
+    "## Gauss-Jordan Elimination"
+   ]
+  },
+  {
+   "cell_type": "markdown",
   "id": "a5239ed9",
   "metadata": {},
-   "outputs": [],
-   "source": []
+   "source": [
+    "## LU decomposition\n",
+    "\n",
+    "$$A = LU$$\n",
+    "\n",
+    "$$Ax = b \\Rightarrow LUx = b \\Rightarrow \\\\\n",
+    "L^{-1}LUx = L^{-1}b \n",
+    "$$\n",
+    "\n",
+    "At that time, $Ux = c$ is easy to compute.\n",
+    "and compute $Lc = b$ to easily get $x$.\n",
+    "\n",
+    "\n",
+    "Various LUdcmps\n",
+    "* Doolittle $L_{ii} = 1$\n",
+    "* Crout $U_{ii} = 1$\n",
+    "* Cholesky $L_{ii} = U_{ii}$, which is appropriate for symmetric, positive-definite matrix\n",
+    "\n",
+    "### Crout Implementation\n",
+    "\n",
+    "$$\\begin{bmatrix}\n",
+    "a_{11} & a_{12} & \\cdots a_{1n} \\\\\n",
+    "a_{21} & a_{22} & \\cdots a_{2n} \\\\\n",
+    "\\vdots & \\vdots & \\ddots \\vdots \\\\\n",
+    "a_{n1} & a_{n2} & \\cdots a_{nn}\n",
+    "\\end{bmatrix} = \\begin{bmatrix}\n",
+    "l_{11} & 0 & \\cdots 0 \\\\\n",
+    "l_{21} & l_{22} & \\cdots 0 \\\\\n",
+    "\\vdots & \\vdots & \\ddots \\vdots \\\\\n",
+    "l_{n1} & l_{n2} & \\cdots l_{nn}\n",
+    "\\end{bmatrix}\\begin{bmatrix}\n",
+    "1 & u_{12} & \\cdots u_{1n} \\\\\n",
+    "0 & 1 & \\cdots u_{2n} \\\\\n",
+    "\\vdots & \\vdots & \\ddots \\vdots \\\\\n",
+    "0 & 0 & \\cdots 1\n",
+    "\\end{bmatrix}$$\n",
+    "\n",
+    "$$\\begin{align*}\n",
+    "l_{i1} &= a_{i1} & (i = 1, 2, \\cdots, n) \\\\\n",
+    "u_{1j} &= a_{1j} / l_{11} & (j = 2, 3, \\cdots, n) \\\\\n",
+    "l_{ij} &= a_{ij} - \\sum_{k=1}^{j-1} {l_{ik} u_{kj}} & (j \\leq i, i = 2,3,\\cdots, n) \\\\\n",
+    "u_{ij} &= (a_{ij} - \\sum_{k=1}^{i-1} {l_{ik} u_{kj}}) / l_{ii} & (j > i, j = 3,4,\\cdots, n)\n",
+    "\\end{align*}$$"
+   ]
  },
  {
   "cell_type": "markdown",
@@ -111,6 +168,14 @@
    "\n",
    "\n"
   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "733012d2",
+   "metadata": {},
+   "outputs": [],
+   "source": []
  }
 ],
 "metadata": {
--- a/notes/3.ipynb
+++ b/notes/3.ipynb
@@ -0,0 +1,73 @@
+{
+ "cells": [
+  {
+   "cell_type": "code",
+   "execution_count": 3,
+   "id": "af0a5571",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "import sys\n",
+    "\n",
+    "sys.path.append(\"../modules\")\n",
+    "\n",
+    "import fixed"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 5,
+   "id": "8a865027",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "from fixed.c_fixedpoint import FixedPoint"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 10,
+   "id": "802380cd",
+   "metadata": {},
+   "outputs": [
+    {
+     "ename": "SystemError",
+     "evalue": "invalid format string: %.8f, frac_bits=%d)",
+     "output_type": "error",
+     "traceback": [
+      "\u001b[31m---------------------------------------------------------------------------\u001b[39m",
+      "\u001b[31mSystemError\u001b[39m                               Traceback (most recent call last)",
+      "\u001b[36mFile \u001b[39m\u001b[32m~/uworkspace/2025-02-Numerical/.venv/lib/python3.12/site-packages/IPython/core/formatters.py:770\u001b[39m, in \u001b[36mPlainTextFormatter.__call__\u001b[39m\u001b[34m(self, obj)\u001b[39m\n\u001b[32m    763\u001b[39m stream = StringIO()\n\u001b[32m    764\u001b[39m printer = pretty.RepresentationPrinter(stream, \u001b[38;5;28mself\u001b[39m.verbose,\n\u001b[32m    765\u001b[39m     \u001b[38;5;28mself\u001b[39m.max_width, \u001b[38;5;28mself\u001b[39m.newline,\n\u001b[32m    766\u001b[39m     max_seq_length=\u001b[38;5;28mself\u001b[39m.max_seq_length,\n\u001b[32m    767\u001b[39m     singleton_pprinters=\u001b[38;5;28mself\u001b[39m.singleton_printers,\n\u001b[32m    768\u001b[39m     type_pprinters=\u001b[38;5;28mself\u001b[39m.type_printers,\n\u001b[32m    769\u001b[39m     deferred_pprinters=\u001b[38;5;28mself\u001b[39m.deferred_printers)\n\u001b[32m--> \u001b[39m\u001b[32m770\u001b[39m \u001b[43mprinter\u001b[49m\u001b[43m.\u001b[49m\u001b[43mpretty\u001b[49m\u001b[43m(\u001b[49m\u001b[43mobj\u001b[49m\u001b[43m)\u001b[49m\n\u001b[32m    771\u001b[39m printer.flush()\n\u001b[32m    772\u001b[39m \u001b[38;5;28;01mreturn\u001b[39;00m stream.getvalue()\n",
+      "\u001b[36mFile \u001b[39m\u001b[32m~/uworkspace/2025-02-Numerical/.venv/lib/python3.12/site-packages/IPython/lib/pretty.py:411\u001b[39m, in \u001b[36mRepresentationPrinter.pretty\u001b[39m\u001b[34m(self, obj)\u001b[39m\n\u001b[32m    400\u001b[39m                         \u001b[38;5;28;01mreturn\u001b[39;00m meth(obj, \u001b[38;5;28mself\u001b[39m, cycle)\n\u001b[32m    401\u001b[39m                 \u001b[38;5;28;01mif\u001b[39;00m (\n\u001b[32m    402\u001b[39m                     \u001b[38;5;28mcls\u001b[39m \u001b[38;5;129;01mis\u001b[39;00m \u001b[38;5;129;01mnot\u001b[39;00m \u001b[38;5;28mobject\u001b[39m\n\u001b[32m    403\u001b[39m                     \u001b[38;5;66;03m# check if cls defines __repr__\u001b[39;00m\n\u001b[32m   (...)\u001b[39m\u001b[32m    409\u001b[39m                     \u001b[38;5;129;01mand\u001b[39;00m \u001b[38;5;28mcallable\u001b[39m(_safe_getattr(\u001b[38;5;28mcls\u001b[39m, \u001b[33m\"\u001b[39m\u001b[33m__repr__\u001b[39m\u001b[33m\"\u001b[39m, \u001b[38;5;28;01mNone\u001b[39;00m))\n\u001b[32m    410\u001b[39m                 ):\n\u001b[32m--> \u001b[39m\u001b[32m411\u001b[39m                     \u001b[38;5;28;01mreturn\u001b[39;00m \u001b[43m_repr_pprint\u001b[49m\u001b[43m(\u001b[49m\u001b[43mobj\u001b[49m\u001b[43m,\u001b[49m\u001b[43m \u001b[49m\u001b[38;5;28;43mself\u001b[39;49m\u001b[43m,\u001b[49m\u001b[43m \u001b[49m\u001b[43mcycle\u001b[49m\u001b[43m)\u001b[49m\n\u001b[32m    413\u001b[39m     \u001b[38;5;28;01mreturn\u001b[39;00m _default_pprint(obj, \u001b[38;5;28mself\u001b[39m, cycle)\n\u001b[32m    414\u001b[39m \u001b[38;5;28;01mfinally\u001b[39;00m:\n",
+      "\u001b[36mFile \u001b[39m\u001b[32m~/uworkspace/2025-02-Numerical/.venv/lib/python3.12/site-packages/IPython/lib/pretty.py:786\u001b[39m, in \u001b[36m_repr_pprint\u001b[39m\u001b[34m(obj, p, cycle)\u001b[39m\n\u001b[32m    784\u001b[39m \u001b[38;5;250m\u001b[39m\u001b[33;03m\"\"\"A pprint that just redirects to the normal repr function.\"\"\"\u001b[39;00m\n\u001b[32m    785\u001b[39m \u001b[38;5;66;03m# Find newlines and replace them with p.break_()\u001b[39;00m\n\u001b[32m--> \u001b[39m\u001b[32m786\u001b[39m output = \u001b[38;5;28;43mrepr\u001b[39;49m\u001b[43m(\u001b[49m\u001b[43mobj\u001b[49m\u001b[43m)\u001b[49m\n\u001b[32m    787\u001b[39m lines = output.splitlines()\n\u001b[32m    788\u001b[39m \u001b[38;5;28;01mwith\u001b[39;00m p.group():\n",
+      "\u001b[31mSystemError\u001b[39m: invalid format string: %.8f, frac_bits=%d)"
+     ]
+    }
+   ],
+   "source": [
+    "FixedPoint(1.0, frac_bits=4)"
+   ]
+  }
+ ],
+ "metadata": {
+  "kernelspec": {
+   "display_name": "2025-02-Numerical (3.12.11)",
+   "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
+}
--- a/notes/4.ipynb
+++ b/notes/4.ipynb
@@ -0,0 +1,41 @@
+{
+ "cells": [
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "db67ff8c",
+   "metadata": {},
+   "outputs": [],
+   "source": []
+  },
+  {
+   "cell_type": "markdown",
+   "id": "6e4ebb4e",
+   "metadata": {},
+   "source": [
+    "# Eigen\n",
+    "\n",
+    "$$Ax = \\lambda x$$\n",
+    "\n",
+    "$$(A-\\lambda I) x = 0$$\n",
+    "\n",
+    "$$\\lambda$$"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "2a5a8e9b",
+   "metadata": {},
+   "outputs": [],
+   "source": []
+  }
+ ],
+ "metadata": {
+  "language_info": {
+   "name": "python"
+  }
+ },
+ "nbformat": 4,
+ "nbformat_minor": 5
+}
--- a/notes/7.ipynb
+++ b/notes/7.ipynb
--- a/notes/8.ipynb
+++ b/notes/8.ipynb
@@ -0,0 +1,70 @@
+{
+ "cells": [
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "62ac3700",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "import sys\n",
+    "sys.path.append(\"../modules\")\n",
+    "import numpy as np"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "3d7375ae",
+   "metadata": {},
+   "source": [
+    "# Data Fitting\n",
+    "\n",
+    "It is approximate fit, which allows some errors depending on noise model.\n",
+    "\n",
+    "## Least-Square Data Fitting\n",
+    "\n",
+    "Given $N$ data points and a models has $M$ adjustable parameters.\n",
+    "\n",
+    "find $\\mathbf{a} = \\begin{bmatrix}a_1, a_2, \\cdots, a_M\\end{bmatrix}$ that minimizes $$S = \\sum_{i=1}^N{[y_i - y(x_i; \\mathbf{a})]^2}$$\n",
+    "\n",
+    "It is also equivalent to **maximum likelihood estimation** if $e_i$ is **independently distributed Gaussian**.\n",
+    "\n",
+    "\n",
+    "### Line Fitting\n",
+    "\n",
+    "$y = a + bx, \\quad e_i = y_i - (a + bx)$\n",
+    "\n",
+    "$S(a, b) = \\sum_{i = 1}^{N}e_i^2$\n",
+    "\n",
+    "$$\\begin{align*}\n",
+    "\\frac{\\partial S}{\\partial a} &= 2\\sum_{i = 1}^N [y_i-a-bx_i](-1) = 0\\\\\n",
+    "\\frac{\\partial S}{\\partial b} &= 2\\sum_{i = 1}^N [y_i =a - bx_i](-x_i) = 0\\\\\n",
+    "\\end{align*}$$\n",
+    "\n",
+    "$$\\begin{align*}\n",
+    "a\\sum 1 + b\\sum x_i &= \\sum y_i\\\\\n",
+    "a\\sum x_i + b \\sum x_i^2 &= \\sum x_iy_i\n",
+    "\\end{align*}$$\n",
+    "\n",
+    "$$\\therefore \\begin{bmatrix}\n",
+    "a \\\\ b\n",
+    "\\end{bmatrix} = \\begin{bmatrix}\n",
+    "\\sum 1 & \\sum x_i \\\\\n",
+    "\\sum x_i & \\sum x_i^2 \\\\\n",
+    "\\end{bmatrix}^{-1} \\begin{bmatrix}\n",
+    "    \\sum y_i \\\\ \\sum x_i y_i\n",
+    "\\end{bmatrix}$$\n",
+    "\n",
+    "\n",
+    "\n"
+   ]
+  }
+ ],
+ "metadata": {
+  "language_info": {
+   "name": "python"
+  }
+ },
+ "nbformat": 4,
+ "nbformat_minor": 5
+}