add and fix some

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": {