LU_Decomposition.ipynb

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   "source": [
    "import scipy.linalg as lg\n",
    "import numpy as np"
   ]
  },
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   "execution_count": 4,
   "outputs": [
    {
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     "text": [
      "2723.033419254051\n"
     ],
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   "source": [
    "A = 3*np.random.random((10,10))\n",
    "print(lg.det(A))"
   ],
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   "outputs": [
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     "text": [
      "[[ 1.00000000e+00  7.62813705e-17 -1.60768895e-16  1.14091754e-16\n  -2.90039184e-16 -1.04688526e-16  4.16411595e-16  2.22176220e-16\n  -1.29650062e-16  2.74771083e-16]\n [ 5.87117304e-16  1.00000000e+00 -2.09360873e-16  1.31173254e-16\n   2.27013540e-16  4.43634510e-16 -1.67044991e-16 -1.32146955e-15\n  -4.15563905e-16  1.51059139e-16]\n [ 1.86642131e-16  1.40036410e-16  1.00000000e+00 -3.73279771e-16\n  -1.49497525e-17 -4.39069317e-17 -2.00759080e-16  1.50559730e-16\n   1.98620493e-16 -2.52110914e-16]\n [ 4.37746697e-17 -8.98012087e-18 -2.22857557e-16  1.00000000e+00\n  -2.62739006e-16 -2.55393368e-17  3.58717228e-16  5.07305587e-17\n  -4.98597216e-17  1.51746178e-16]\n [ 4.34590153e-16  6.37534283e-16  3.29795376e-16 -2.86790625e-16\n   1.00000000e+00  9.26263287e-17 -2.25862069e-16 -6.18448146e-16\n  -9.83556457e-17 -1.55994869e-17]\n [ 8.16492489e-17 -7.48518841e-18 -2.31812107e-17 -3.72885745e-16\n  -3.93792298e-16  1.00000000e+00  2.65678178e-16  3.90371435e-17\n   1.84311460e-16  1.55321668e-16]\n [ 2.60051977e-16  4.11914594e-16  6.67549132e-17  2.18093683e-17\n  -1.18583899e-16  8.96806468e-17  1.00000000e+00 -5.60095482e-16\n  -8.63049205e-17 -7.09442317e-18]\n [ 2.96721223e-16  6.55556173e-16 -2.93650753e-16 -2.05655270e-17\n   1.38362741e-17  1.87811731e-16 -8.10777426e-17  1.00000000e+00\n  -3.26050509e-16  1.08890552e-16]\n [ 4.04716703e-16  1.29479266e-16 -4.26871275e-17 -2.85864333e-16\n   1.56739553e-16  7.12681982e-17  2.72387653e-16 -6.90786044e-16\n   1.00000000e+00  1.07098762e-16]\n [ 8.00932229e-16  9.02099484e-16  3.13489857e-16 -6.44610795e-16\n  -2.66826779e-16  1.75158239e-16 -4.65034904e-16 -6.23971814e-16\n  -9.93353942e-17  1.00000000e+00]]\n"
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   "source": [
    "A_inv = lg.inv(A)\n",
    "I = np.dot(A,A_inv)\n",
    "print(I)"
   ],
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   "cell_type": "code",
   "execution_count": 7,
   "outputs": [
    {
     "name": "stdout",
     "text": [
      "[[ 5.10479595e-01]\n [ 1.55236702e-01]\n [-4.16721388e-02]\n [-5.25037208e-01]\n [-3.31265431e-04]\n [ 1.54326975e-01]\n [ 1.94969796e-01]\n [ 1.98862703e-01]\n [ 3.00062064e-01]\n [-2.10826450e-01]]\n"
     ],
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    }
   ],
   "source": [
    "b = np.random.random((10,1))\n",
    "x = lg.solve(A,b)\n",
    "print(x)"
   ],
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   "cell_type": "code",
   "execution_count": 9,
   "outputs": [
    {
     "data": {
      "text/plain": "3"
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     "metadata": {},
     "output_type": "execute_result",
     "execution_count": 9
    }
   ],
   "source": [
    "lu = lg.lu(A,permute_l=False)\n",
    "len(lu)"
   ],
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   "execution_count": 10,
   "outputs": [],
   "source": [
    "P = lu[0]\n",
    "L = lu[1]\n",
    "U = lu[2]"
   ],
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   "cell_type": "code",
   "execution_count": 11,
   "outputs": [
    {
     "name": "stdout",
     "text": [
      "[[-2.22044605e-16  0.00000000e+00  0.00000000e+00 -1.11022302e-16\n   0.00000000e+00  0.00000000e+00  0.00000000e+00 -4.44089210e-16\n   3.33066907e-16  0.00000000e+00]\n [ 0.00000000e+00  0.00000000e+00  0.00000000e+00  0.00000000e+00\n   0.00000000e+00  0.00000000e+00  0.00000000e+00  0.00000000e+00\n   0.00000000e+00  0.00000000e+00]\n [ 0.00000000e+00  0.00000000e+00  0.00000000e+00  0.00000000e+00\n   0.00000000e+00 -5.55111512e-17  0.00000000e+00  0.00000000e+00\n   0.00000000e+00  1.11022302e-16]\n [ 0.00000000e+00  0.00000000e+00  0.00000000e+00  0.00000000e+00\n   2.22044605e-16  1.11022302e-16 -1.11022302e-16  0.00000000e+00\n  -1.11022302e-16  4.44089210e-16]\n [ 0.00000000e+00  5.55111512e-17  0.00000000e+00  0.00000000e+00\n  -2.22044605e-16  0.00000000e+00  0.00000000e+00  0.00000000e+00\n   0.00000000e+00  0.00000000e+00]\n [ 0.00000000e+00  0.00000000e+00  0.00000000e+00  0.00000000e+00\n   0.00000000e+00  0.00000000e+00  0.00000000e+00  0.00000000e+00\n   0.00000000e+00  0.00000000e+00]\n [ 0.00000000e+00  2.22044605e-16  0.00000000e+00  0.00000000e+00\n   0.00000000e+00 -2.77555756e-17 -9.71445147e-17 -2.22044605e-16\n  -4.44089210e-16 -2.22044605e-16]\n [ 0.00000000e+00  2.22044605e-16  0.00000000e+00  2.22044605e-16\n   0.00000000e+00  1.11022302e-16 -2.22044605e-16  0.00000000e+00\n   3.33066907e-16  0.00000000e+00]\n [ 0.00000000e+00  0.00000000e+00  0.00000000e+00  0.00000000e+00\n   2.22044605e-16  0.00000000e+00  0.00000000e+00 -4.44089210e-16\n   0.00000000e+00  2.22044605e-16]\n [ 0.00000000e+00  0.00000000e+00  0.00000000e+00  0.00000000e+00\n  -2.22044605e-16  0.00000000e+00  4.44089210e-16  0.00000000e+00\n  -8.88178420e-16  0.00000000e+00]]\n"
     ],
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   ],
   "source": [
    "A1 = np.dot(np.dot(P,L),U)\n",
    "print(A-A1)"
   ],
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     "name": "#%%\n",
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   "cell_type": "code",
   "execution_count": 12,
   "outputs": [
    {
     "name": "stdout",
     "text": [
      "[[0.]\n [0.]\n [0.]\n [0.]\n [0.]\n [0.]\n [0.]\n [0.]\n [0.]\n [0.]]\n"
     ],
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    }
   ],
   "source": [
    "y = lg.solve(np.dot(P,L),b)\n",
    "x1 = lg.solve(U,y)\n",
    "print(x1-x)"
   ],
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   "execution_count": 13,
   "outputs": [
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     "name": "stdout",
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      "[[ 0.78707579  0.05954258  0.45804034  1.          0.          0.\n   0.          0.          0.          0.        ]\n [ 0.18376838  1.          0.          0.          0.          0.\n   0.          0.          0.          0.        ]\n [ 0.16559747  0.25217767  1.          0.          0.          0.\n   0.          0.          0.          0.        ]\n [ 0.24649863  0.67202089  0.66466062  0.52882061 -0.84799078  0.21472193\n  -0.36162373 -0.65970046  1.          0.        ]\n [ 0.30139913  0.07327398  0.40162532 -0.21572541  1.          0.\n   0.          0.          0.          0.        ]\n [ 1.          0.          0.          0.          0.          0.\n   0.          0.          0.          0.        ]\n [ 0.26717943  0.46561081  0.3514701   0.04319724  0.22343755 -0.26903203\n  -0.93125814  1.          0.          0.        ]\n [ 0.40640392  0.58855853  0.05119614  0.37788399 -0.13441697 -0.08807\n   0.25656502 -0.39643431 -0.89607948  1.        ]\n [ 0.79387172  0.66097921  0.70604647  0.17999145 -0.74745931  1.\n   0.          0.          0.          0.        ]\n [ 0.73860975 -0.02276046  0.6410004  -0.26914769  0.82080233  0.63781159\n   1.          0.          0.          0.        ]]\n"
     ],
     "output_type": "stream"
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   ],
   "source": [
    "print(np.dot(P,L))"
   ],
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