2025-02-Numerical/notes/2.ipynb

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   "source": [
    "import numpy as np\n",
    "import matplotlib.pyplot as plt\n",
    "\n",
    "plt.rcParams[\"font.family\"] = \"D2Coding\""
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    "# Linear Equation\n",
    "\n",
    "$$A\\cdot x = b$$\n",
    "\n",
    "* $A$: [m, n]\n",
    "* $b$: [m, 1]"
   ]
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    "## Gauss Elimination\n",
    "\n",
    "## Gauss-Jordan Elimination"
   ]
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    "## 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",
    "$$\\begin{align*}\n",
    "l_{i1} &= a_{i1} & (i = 1, 2, \\cdots, n) \\\\\n",
    "u_{1j} &= a_{1j} / l_{11} & (j = 2, 3, \\cdots, n) \\\\\n",
    "l_{ij} &= a_{ij} - \\sum_{k=1}^{j-1} {l_{ik} u_{kj}} & (j \\leq i, i = 2,3,\\cdots, n) \\\\\n",
    "u_{ij} &= (a_{ij} - \\sum_{k=1}^{i-1} {l_{ik} u_{kj}}) / l_{ii} & (j > i, j = 3,4,\\cdots, n)\n",
    "\\end{align*}$$"
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   "source": [
    "# SVD(Singular Value Decomposition)\n",
    "\n",
    "**why?**\n",
    "\n",
    "**what?**\n",
    "\n",
    "$$A = UWV^T$$\n",
    "\n",
    "* $U$: [m, n]\n",
    "* $W$: [n, n] diagonal with positive or zero\n",
    "* $V$: [n, n] orthogonal\n",
    "\n",
    "\n",
    "**properties**\n",
    "\n",
    "* orthonormality\n",
    "* uniqueness"
   ]
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     "output_type": "stream",
     "text": [
      "(4, 3)\n"
     ]
    },
    {
     "data": {
      "text/plain": [
       "array([[ 1.,  0., -0.],\n",
       "       [ 1.,  1., -1.],\n",
       "       [ 0.,  1., -1.],\n",
       "       [-1.,  1.,  1.]])"
      ]
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   "source": [
    "A = np.array([[1, 0, 0], [1, 1, -1], [0, 1, -1], [-1, 1, 1]])\n",
    "print(A.shape)\n",
    "U, S, Vh = np.linalg.svd(A, False)"
   ]
  },
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   "source": [
    "## Homogeneous\n",
    "\n",
    "\n",
    "## Nonhomogeneous\n",
    "\n",
    "\n"
   ]
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