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farzadab/Intro to Machine Learning Tutorial 1 - Fundamentals of Linear Algebra.ipynb

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Linear Algebra Tutorial notebook [by Wilder Lavington]
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Intro to Machine Learning Tutorial 1 - Fundamentals of Linear Algebra.ipynb
{
"cells": [
{
"metadata": {
"trusted": true
},
"cell_type": "code",
"source": "# lets get all of our imports \nimport numpy as np\nfrom scipy import linalg",
"execution_count": 5,
"outputs": []
},
{
"metadata": {
"trusted": true
},
"cell_type": "code",
"source": "# scaler - scaler multipliction\nx = 1\ny = 10\nx*y",
"execution_count": 6,
"outputs": [
{
"output_type": "execute_result",
"execution_count": 6,
"data": {
"text/plain": "10"
},
"metadata": {}
}
]
},
{
"metadata": {
"trusted": true
},
"cell_type": "code",
"source": "# other functions \nx*y + 1 - np.sin(x)\nx*y + 1 + np.log(x)\nx*y + 1 + np.exp(x)",
"execution_count": 7,
"outputs": [
{
"output_type": "execute_result",
"execution_count": 7,
"data": {
"text/plain": "13.718281828459045"
},
"metadata": {}
}
]
},
{
"metadata": {
"trusted": true
},
"cell_type": "code",
"source": "# generating arrays (random)\nnp.random.rand(2)",
"execution_count": 8,
"outputs": [
{
"output_type": "execute_result",
"execution_count": 8,
"data": {
"text/plain": "array([0.05119316, 0.95945369])"
},
"metadata": {}
}
]
},
{
"metadata": {
"trusted": true
},
"cell_type": "code",
"source": "# generating arrays (linspace)\nnp.linspace(2.0, 3.0, num=20)",
"execution_count": 9,
"outputs": [
{
"output_type": "execute_result",
"execution_count": 9,
"data": {
"text/plain": "array([2. , 2.05263158, 2.10526316, 2.15789474, 2.21052632,\n 2.26315789, 2.31578947, 2.36842105, 2.42105263, 2.47368421,\n 2.52631579, 2.57894737, 2.63157895, 2.68421053, 2.73684211,\n 2.78947368, 2.84210526, 2.89473684, 2.94736842, 3. ])"
},
"metadata": {}
}
]
},
{
"metadata": {
"trusted": true
},
"cell_type": "code",
"source": "# generating matrices (random)\nnp.random.rand(2,3)",
"execution_count": 10,
"outputs": [
{
"output_type": "execute_result",
"execution_count": 10,
"data": {
"text/plain": "array([[0.95843142, 0.93179404, 0.83591989],\n [0.26030427, 0.82862825, 0.55423052]])"
},
"metadata": {}
}
]
},
{
"metadata": {
"trusted": true
},
"cell_type": "code",
"source": "# generating a array of zeros\nnp.zeros(5)",
"execution_count": 11,
"outputs": [
{
"output_type": "execute_result",
"execution_count": 11,
"data": {
"text/plain": "array([0., 0., 0., 0., 0.])"
},
"metadata": {}
}
]
},
{
"metadata": {
"trusted": true
},
"cell_type": "code",
"source": "# generating a matrix of zeros\nnp.zeros((2, 4))",
"execution_count": 8,
"outputs": [
{
"output_type": "execute_result",
"execution_count": 8,
"data": {
"text/plain": "array([[0., 0., 0., 0.],\n [0., 0., 0., 0.]])"
},
"metadata": {}
}
]
},
{
"metadata": {
"trusted": true
},
"cell_type": "code",
"source": "# generating a matrix of ones\nnp.ones((2, 4))",
"execution_count": 13,
"outputs": [
{
"output_type": "execute_result",
"execution_count": 13,
"data": {
"text/plain": "array([[1., 1., 1., 1.],\n [1., 1., 1., 1.]])"
},
"metadata": {}
}
]
},
{
"metadata": {
"trusted": true
},
"cell_type": "code",
"source": "v = np.random.rand(10)\ns = 10",
"execution_count": 15,
"outputs": []
},
{
"metadata": {
"trusted": true
},
"cell_type": "code",
"source": "# scaler - vector multipliction\ns * v",
"execution_count": 30,
"outputs": [
{
"output_type": "execute_result",
"execution_count": 30,
"data": {
"text/plain": "array([3.37087199, 3.19873035, 5.16301149, 1.81301167, 9.50608079,\n 7.85936299, 3.50233133, 2.06077056, 5.43221286, 2.12565731])"
},
"metadata": {}
}
]
},
{
"metadata": {
"trusted": true
},
"cell_type": "code",
"source": "# vector - vector multipliction: element-wise multiplication\nv1 = np.random.rand(5)\nv2 = np.random.rand(5)\n\nv1 * v2",
"execution_count": 34,
"outputs": [
{
"output_type": "execute_result",
"execution_count": 34,
"data": {
"text/plain": "array([0.88838432, 0.74905216, 0.27255124, 0.12250431, 0.04525032])"
},
"metadata": {}
}
]
},
{
"metadata": {
"trusted": true
},
"cell_type": "code",
"source": "# element-wise multiplication: matching shapes\nv2 = np.random.rand(4)\n\nv1 * v2",
"execution_count": 32,
"outputs": [
{
"output_type": "error",
"ename": "ValueError",
"evalue": "operands could not be broadcast together with shapes (5,) (4,) ",
"traceback": [
"\u001b[0;31m---------------------------------------------------------------------------\u001b[0m",
"\u001b[0;31mValueError\u001b[0m Traceback (most recent call last)",
"\u001b[0;32m<ipython-input-32-ec75d435b1b5>\u001b[0m in \u001b[0;36m<module>\u001b[0;34m()\u001b[0m\n\u001b[1;32m 1\u001b[0m \u001b[0;31m# element-wise multiplication: matching shapes\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m 2\u001b[0m \u001b[0mv2\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mnp\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mrandom\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mrand\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;36m4\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m----> 3\u001b[0;31m \u001b[0mv1\u001b[0m\u001b[0;34m*\u001b[0m\u001b[0mv2\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m",
"\u001b[0;31mValueError\u001b[0m: operands could not be broadcast together with shapes (5,) (4,) "
]
}
]
},
{
"metadata": {
"trusted": true
},
"cell_type": "code",
"source": "# make a two norm (l2)\nv = np.random.rand(100)\nnp.sqrt(sum(v*v))",
"execution_count": 33,
"outputs": [
{
"output_type": "execute_result",
"execution_count": 33,
"data": {
"text/plain": "5.6292714435169"
},
"metadata": {}
}
]
},
{
"metadata": {
"trusted": true
},
"cell_type": "code",
"source": "# vector - matrix multipliction: broadcast\nv1 = np.array([1,100])\nv2 = np.array([1,2,3,4,5,6]).reshape(3,2)\n\nv1 * v2",
"execution_count": 36,
"outputs": [
{
"output_type": "execute_result",
"execution_count": 36,
"data": {
"text/plain": "array([[ 1, 200],\n [ 3, 400],\n [ 5, 600]])"
},
"metadata": {}
}
]
},
{
"metadata": {
"trusted": true
},
"cell_type": "code",
"source": "# \"element-wise\" matrix - matrix multipliction: \nv1 = np.random.rand(3,3)\nv2 = np.random.rand(3,3)\nv1 * v2",
"execution_count": 46,
"outputs": [
{
"output_type": "execute_result",
"execution_count": 46,
"data": {
"text/plain": "array([[0.0349701 , 0.31071419, 0.3563422 ],\n [0.40592048, 0.19475094, 0.37454803],\n [0.18862149, 0.84884285, 0.90960497]])"
},
"metadata": {}
}
]
},
{
"metadata": {
"trusted": true
},
"cell_type": "code",
"source": "# \"element-wise\" matrix - matrix multipliction: : size mismatch\nm1 = np.random.rand(3,8)\nm2 = np.random.rand(8,3)\nm1 * m2",
"execution_count": 47,
"outputs": [
{
"output_type": "error",
"ename": "ValueError",
"evalue": "operands could not be broadcast together with shapes (3,8) (8,3) ",
"traceback": [
"\u001b[0;31m---------------------------------------------------------------------------\u001b[0m",
"\u001b[0;31mValueError\u001b[0m Traceback (most recent call last)",
"\u001b[0;32m<ipython-input-47-d4ac30799317>\u001b[0m in \u001b[0;36m<module>\u001b[0;34m()\u001b[0m\n\u001b[1;32m 2\u001b[0m \u001b[0mv1\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mnp\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mrandom\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mrand\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;36m3\u001b[0m\u001b[0;34m,\u001b[0m\u001b[0;36m8\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m 3\u001b[0m \u001b[0mv2\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mnp\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mrandom\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mrand\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;36m8\u001b[0m\u001b[0;34m,\u001b[0m\u001b[0;36m3\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m----> 4\u001b[0;31m \u001b[0mv1\u001b[0m \u001b[0;34m*\u001b[0m \u001b[0mv2\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m",
"\u001b[0;31mValueError\u001b[0m: operands could not be broadcast together with shapes (3,8) (8,3) "
]
}
]
},
{
"metadata": {
"trusted": true
},
"cell_type": "code",
"source": "# matrix - matrix multipliction\nm1 = np.random.rand(3,8)\nm2 = np.random.rand(8,3)\nnp.matmul(m1, m2)",
"execution_count": 88,
"outputs": [
{
"output_type": "execute_result",
"execution_count": 88,
"data": {
"text/plain": "array([[2.38098437, 2.70527655, 2.36424057],\n [2.76582467, 2.15053236, 2.25971462],\n [2.85678593, 2.32536773, 1.49983105]])"
},
"metadata": {}
}
]
},
{
"metadata": {
"trusted": true
},
"cell_type": "code",
"source": "# option 2 for matrix multiplication: @\nm1 @ m2",
"execution_count": 89,
"outputs": [
{
"output_type": "execute_result",
"execution_count": 89,
"data": {
"text/plain": "array([[2.38098437, 2.70527655, 2.36424057],\n [2.76582467, 2.15053236, 2.25971462],\n [2.85678593, 2.32536773, 1.49983105]])"
},
"metadata": {}
}
]
},
{
"metadata": {
"trusted": true
},
"cell_type": "code",
"source": "# matrix - matrix multipliction: size mismatch\nm1 = np.random.rand(3,8)\nm2 = np.random.rand(3,3)\nnp.matmul(m1, m2)",
"execution_count": 85,
"outputs": [
{
"output_type": "error",
"ename": "ValueError",
"evalue": "shapes (3,8) and (3,3) not aligned: 8 (dim 1) != 3 (dim 0)",
"traceback": [
"\u001b[0;31m---------------------------------------------------------------------------\u001b[0m",
"\u001b[0;31mValueError\u001b[0m Traceback (most recent call last)",
"\u001b[0;32m<ipython-input-85-062e974cab63>\u001b[0m in \u001b[0;36m<module>\u001b[0;34m()\u001b[0m\n\u001b[1;32m 2\u001b[0m \u001b[0mm1\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mnp\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mrandom\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mrand\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;36m3\u001b[0m\u001b[0;34m,\u001b[0m\u001b[0;36m8\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m 3\u001b[0m \u001b[0mm2\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mnp\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mrandom\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mrand\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;36m3\u001b[0m\u001b[0;34m,\u001b[0m\u001b[0;36m3\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m----> 4\u001b[0;31m \u001b[0mnp\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mmatmul\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mm1\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mm2\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m",
"\u001b[0;31mValueError\u001b[0m: shapes (3,8) and (3,3) not aligned: 8 (dim 1) != 3 (dim 0)"
]
}
]
},
{
"metadata": {
"trusted": true
},
"cell_type": "code",
"source": "# option 3 for matrix multiplication: np.matrix\nm1 = np.matrix(np.random.rand(5,4))\nm2 = np.matrix(np.random.rand(4,5))\nm1 * m2",
"execution_count": 49,
"outputs": [
{
"output_type": "execute_result",
"execution_count": 49,
"data": {
"text/plain": "matrix([[1.52944453, 0.52535152, 1.10522876, 1.84089603, 1.67559165],\n [1.42694103, 0.75291401, 1.09411873, 1.3024963 , 1.51751165],\n [1.10570363, 0.44129516, 0.64177185, 1.19396572, 1.01129319],\n [0.98769848, 0.27332851, 0.66693344, 0.86839107, 1.15408258],\n [0.72794788, 0.47725534, 0.60221488, 0.46389334, 0.79510672]])"
},
"metadata": {}
}
]
},
{
"metadata": {
"trusted": true
},
"cell_type": "code",
"source": "# matrix - vector multipliction\nm = np.matrix(np.random.rand(5,4))\nv = np.matrix(np.random.rand(4,1))\nm * v",
"execution_count": 50,
"outputs": [
{
"output_type": "execute_result",
"execution_count": 50,
"data": {
"text/plain": "matrix([[1.46877674],\n [0.54569547],\n [1.05901183],\n [1.13739921],\n [0.66695699]])"
},
"metadata": {}
}
]
},
{
"metadata": {
"trusted": true
},
"cell_type": "code",
"source": "# matrix - vector multipliction: size mismatch\nm = np.matrix(np.random.rand(5,4))\nv = np.matrix(np.random.rand(4))\nm * v",
"execution_count": 52,
"outputs": [
{
"output_type": "error",
"ename": "ValueError",
"evalue": "shapes (5,4) and (1,4) not aligned: 4 (dim 1) != 1 (dim 0)",
"traceback": [
"\u001b[0;31m---------------------------------------------------------------------------\u001b[0m",
"\u001b[0;31mValueError\u001b[0m Traceback (most recent call last)",
"\u001b[0;32m<ipython-input-52-7195480c4f51>\u001b[0m in \u001b[0;36m<module>\u001b[0;34m()\u001b[0m\n\u001b[1;32m 2\u001b[0m \u001b[0mm\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mnp\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mmatrix\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mnp\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mrandom\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mrand\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;36m5\u001b[0m\u001b[0;34m,\u001b[0m\u001b[0;36m4\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m 3\u001b[0m \u001b[0mv\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mnp\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mmatrix\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mnp\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mrandom\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mrand\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;36m4\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m----> 4\u001b[0;31m \u001b[0mm\u001b[0m \u001b[0;34m*\u001b[0m \u001b[0mv\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m",
"\u001b[0;32m~/anaconda3_420/lib/python3.5/site-packages/numpy/matrixlib/defmatrix.py\u001b[0m in \u001b[0;36m__mul__\u001b[0;34m(self, other)\u001b[0m\n\u001b[1;32m 213\u001b[0m \u001b[0;32mif\u001b[0m \u001b[0misinstance\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mother\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0;34m(\u001b[0m\u001b[0mN\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mndarray\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mlist\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mtuple\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m)\u001b[0m \u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m 214\u001b[0m \u001b[0;31m# This promotes 1-D vectors to row vectors\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m--> 215\u001b[0;31m \u001b[0;32mreturn\u001b[0m \u001b[0mN\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mdot\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mself\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0masmatrix\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mother\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m 216\u001b[0m \u001b[0;32mif\u001b[0m \u001b[0misscalar\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mother\u001b[0m\u001b[0;34m)\u001b[0m \u001b[0;32mor\u001b[0m \u001b[0;32mnot\u001b[0m \u001b[0mhasattr\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mother\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0;34m'__rmul__'\u001b[0m\u001b[0;34m)\u001b[0m \u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m 217\u001b[0m \u001b[0;32mreturn\u001b[0m \u001b[0mN\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mdot\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mself\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mother\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
"\u001b[0;31mValueError\u001b[0m: shapes (5,4) and (1,4) not aligned: 4 (dim 1) != 1 (dim 0)"
]
}
]
},
{
"metadata": {
"trusted": true
},
"cell_type": "code",
"source": "# matrix norms \nfrom numpy.linalg import norm\n\nm = np.matrix([1,2,3,4]).reshape(2,2)\n\nprint(norm(m))\nprint(norm(m, 'fro'))\nprint(norm(m, np.inf))\nprint(norm(m, -np.inf))\nprint(norm(m, 2))\nprint(norm(m, -2))",
"execution_count": 76,
"outputs": [
{
"output_type": "stream",
"text": "5.477225575051661\n5.477225575051661\n7.0\n3.0\n5.464985704219043\n0.36596619062625746\n",
"name": "stdout"
}
]
},
{
"metadata": {
"trusted": true
},
"cell_type": "code",
"source": "# solving a linear system\n# from numpy.linalg import solve\n\nA = np.matrix(np.random.rand(5,5))\nb = np.matrix(np.random.rand(5,1))\nnp.linalg.solve(A, b)",
"execution_count": 78,
"outputs": [
{
"output_type": "execute_result",
"execution_count": 78,
"data": {
"text/plain": "matrix([[ 0.98458229],\n [-0.69050311],\n [ 0.65658468],\n [ 0.05219171],\n [-0.18769165]])"
},
"metadata": {}
}
]
},
{
"metadata": {
"trusted": true
},
"cell_type": "code",
"source": "import scipy.linalg as sla\n\n# solving a linear system (LU)\nA = np.matrix(np.random.rand(5,5))\nb = np.matrix(np.random.rand(5,1))\nP, L, U = sla.lu(A)\n# LU x = P^{-1} * b\nPb = P.transpose()*b\n# Ly = Pb\ny = sla.solve_triangular(U, Pb)\n# Ux = y\nx = sla.solve_triangular(L, y, lower=True)\nx",
"execution_count": 98,
"outputs": [
{
"output_type": "execute_result",
"execution_count": 98,
"data": {
"text/plain": "array([[ 0.19169271],\n [ 2.02041356],\n [-1.78465993],\n [ 0.10167166],\n [-1.89017998]])"
},
"metadata": {}
}
]
},
{
"metadata": {
"trusted": true
},
"cell_type": "code",
"source": "# finding a determinant\nimport numpy.linalg as la\n\nA = np.matrix(np.random.rand(5,5))\nla.det(A)",
"execution_count": 105,
"outputs": [
{
"output_type": "execute_result",
"execution_count": 105,
"data": {
"text/plain": "0.009537127775052836"
},
"metadata": {}
}
]
},
{
"metadata": {
"trusted": true
},
"cell_type": "code",
"source": "# finding a determinant: only for square matrices\nA = np.matrix(np.random.rand(5,2))\nla.det(A)",
"execution_count": 107,
"outputs": [
{
"output_type": "error",
"ename": "LinAlgError",
"evalue": "Last 2 dimensions of the array must be square",
"traceback": [
"\u001b[0;31m---------------------------------------------------------------------------\u001b[0m",
"\u001b[0;31mLinAlgError\u001b[0m Traceback (most recent call last)",
"\u001b[0;32m<ipython-input-107-f3ed28bdff00>\u001b[0m in \u001b[0;36m<module>\u001b[0;34m()\u001b[0m\n\u001b[1;32m 1\u001b[0m \u001b[0;31m# finding a determinant: only for square matrices\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m 2\u001b[0m \u001b[0mA\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mnp\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mmatrix\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mnp\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mrandom\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mrand\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;36m5\u001b[0m\u001b[0;34m,\u001b[0m\u001b[0;36m2\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m----> 3\u001b[0;31m \u001b[0mla\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mdet\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mA\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m",
"\u001b[0;32m~/anaconda3_420/lib/python3.5/site-packages/numpy/linalg/linalg.py\u001b[0m in \u001b[0;36mdet\u001b[0;34m(a)\u001b[0m\n\u001b[1;32m 2017\u001b[0m \u001b[0ma\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0masarray\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0ma\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m 2018\u001b[0m \u001b[0m_assertRankAtLeast2\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0ma\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m-> 2019\u001b[0;31m \u001b[0m_assertNdSquareness\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0ma\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m 2020\u001b[0m \u001b[0mt\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mresult_t\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0m_commonType\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0ma\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m 2021\u001b[0m \u001b[0msignature\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0;34m'D->D'\u001b[0m \u001b[0;32mif\u001b[0m \u001b[0misComplexType\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mt\u001b[0m\u001b[0;34m)\u001b[0m \u001b[0;32melse\u001b[0m \u001b[0;34m'd->d'\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
"\u001b[0;32m~/anaconda3_420/lib/python3.5/site-packages/numpy/linalg/linalg.py\u001b[0m in \u001b[0;36m_assertNdSquareness\u001b[0;34m(*arrays)\u001b[0m\n\u001b[1;32m 213\u001b[0m \u001b[0mm\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mn\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0ma\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mshape\u001b[0m\u001b[0;34m[\u001b[0m\u001b[0;34m-\u001b[0m\u001b[0;36m2\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m]\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m 214\u001b[0m \u001b[0;32mif\u001b[0m \u001b[0mm\u001b[0m \u001b[0;34m!=\u001b[0m \u001b[0mn\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m--> 215\u001b[0;31m \u001b[0;32mraise\u001b[0m \u001b[0mLinAlgError\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m'Last 2 dimensions of the array must be square'\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m 216\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m 217\u001b[0m \u001b[0;32mdef\u001b[0m \u001b[0m_assertFinite\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m*\u001b[0m\u001b[0marrays\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
"\u001b[0;31mLinAlgError\u001b[0m: Last 2 dimensions of the array must be square"
]
}
]
},
{
"metadata": {
"trusted": true
},
"cell_type": "code",
"source": "# solving least squares normal equations\nA = np.matrix(np.random.rand(10,3))\nb = np.matrix(np.random.rand(10,1))\n# Ax = b --> x = (A^t A)^-1 A^t b --> (A^t A) x = A^t b\nla.solve(A.transpose(1,0)*A, A.transpose(1,0)*b)",
"execution_count": 230,
"outputs": [
{
"output_type": "execute_result",
"execution_count": 230,
"data": {
"text/plain": "matrix([[-0.28516054],\n [ 0.98967105],\n [-0.20840818]])"
},
"metadata": {}
}
]
},
{
"metadata": {
"trusted": true
},
"cell_type": "code",
"source": "# solving least squares normal equations: QR\n\n# A = QR\n# (A^t A) x = A^t b --> (R^t Q^t Q R) x = R^t Q^t b --> (R^t R) x = R^t Q^t b --> R x = Q^t b\nQ, R = np.linalg.qr(A)\nnp.linalg.solve(R, Q.transpose()*b)",
"execution_count": 231,
"outputs": [
{
"output_type": "execute_result",
"execution_count": 231,
"data": {
"text/plain": "matrix([[-0.28516054],\n [ 0.98967105],\n [-0.20840818]])"
},
"metadata": {}
}
]
},
{
"metadata": {
"trusted": true
},
"cell_type": "code",
"source": "# solving least squares SVD\n# A = U S V\n# |Ax-b| = |U^t (Ax - b)| = |SVx - U^t b| = |Sz - U^t b| --> S^t S x = S^t U^t b --> x = (S^t S)^-1 S^t U^t b\nU, s, V = la.svd(A)\nS = np.matrix(np.append(np.diag(s), np.zeros((7,3)), axis=0))\nV.transpose() * np.diag(np.power(1/s, 2)) * S.transpose() * U.transpose() * b",
"execution_count": 232,
"outputs": [
{
"output_type": "execute_result",
"execution_count": 232,
"data": {
"text/plain": "matrix([[-0.28516054],\n [ 0.98967105],\n [-0.20840818]])"
},
"metadata": {}
}
]
},
{
"metadata": {
"trusted": true
},
"cell_type": "code",
"source": "# finding eigen values cholesky decomposition\nB = np.matrix(np.random.rand(4,4))\nA = B * np.transpose(B) \nL = la.cholesky(A)\nL",
"execution_count": 254,
"outputs": [
{
"output_type": "execute_result",
"execution_count": 254,
"data": {
"text/plain": "matrix([[ 1.33128599, 0. , 0. , 0. ],\n [ 0.98050935, 0.43465579, 0. , 0. ],\n [ 0.88163049, -0.04960794, 0.7781799 , 0. ],\n [ 0.5850072 , -0.09671825, 0.39623734, 0.25361176]])"
},
"metadata": {}
}
]
},
{
"metadata": {
"trusted": true
},
"cell_type": "code",
"source": "# eigen decomposition LL' = A -> ED*D'E' -> eigs = D.^2\n# print(np.sqrt(np.linalg.eigvals(A)))\n# np.linalg.norm(L, axis=1)",
"execution_count": 255,
"outputs": []
},
{
"metadata": {
"trusted": false
},
"cell_type": "code",
"source": "# finding eigen values QR algorithm (this ones wierd!)\nA = np.matrix(np.random.rand(4,4))\nA = np.transpose(A)*A\nA_iter = A\nQ, R = la.qr(A)\nfor i in range(10**4):\n A_iter = R*Q\n Q, R = la.qr(A_iter)\nprint(np.transpose(Q)*A_iter)\nprint(la.eig(A)[0])",
"execution_count": 414,
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": "[[ 5.62337093e+00 7.30621903e-16 -5.58503854e-16 -1.45752362e-16]\n [ 0.00000000e+00 4.81876877e-01 -7.85316443e-17 3.50929779e-17]\n [ 0.00000000e+00 0.00000000e+00 6.86708026e-02 -4.31502051e-17]\n [ 0.00000000e+00 0.00000000e+00 0.00000000e+00 1.38738068e-03]]\n[5.62337093e+00 4.81876877e-01 6.86708026e-02 1.38738068e-03]\n"
}
]
},
{
"metadata": {
"trusted": false
},
"cell_type": "code",
"source": "# finding eigen vectors power method\nA = np.matrix(np.random.rand(5,5))\nA = A*np.transpose(A)\nx = np.matrix(np.random.rand(5,1))\nfor i in range(10**3):\n x = A*x/la.norm(A*x)\nAx = A*x\nprint(Ax[0,0]/x[0,0])\nprint(la.eig(A)[0])",
"execution_count": 326,
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": "8.332651802211897\n[8.3326518 0.83269214 0.37356706 0.17681973 0.024805 ]\n"
}
]
},
{
"metadata": {
"trusted": false
},
"cell_type": "code",
"source": "# finding eigen vectors inverse power method\nA = np.matrix(np.random.rand(5,5))\nA = A*np.transpose(A)\nx = np.matrix(np.random.rand(5,1))\n# pick an eigen value\neig3 = la.eig(A)[0][3]\nfor i in range(10**4):\n x = la.solve(A,x)/la.norm(la.solve(A,x))\nAx = A*x\nprint(Ax[0,0]/x[0,0])\nprint(eig3)",
"execution_count": 406,
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": "0.025304822560150934\n0.025304822560150524\n"
}
]
},
{
"metadata": {
"trusted": false
},
"cell_type": "code",
"source": "# finding eigen vectors shifted inverse power method\nA = np.matrix(np.random.rand(5,5))\nA = A*np.transpose(A)\nx = np.matrix(np.random.rand(5,1))\n# pick an eigen value\neig3 = la.eig(A)[0][3]\nfor i in range(10**4):\n x = la.solve(A-eig3*np.eye(5),x)/la.norm(la.solve(A-eig3*np.eye(5),x))\nAx = A*x\nprint(Ax[0,0]/x[0,0])\nprint(eig3)",
"execution_count": 405,
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": "0.16029749338013227\n0.16029749338013202\n"
}
]
},
{
"metadata": {
"trusted": false
},
"cell_type": "code",
"source": "import matplotlib.pyplot as plt\n# plotting stuff\nplt.plot([1, 2, 3, 4])\nplt.ylabel('some numbers')\nplt.show()",
"execution_count": 15,
"outputs": [
{
"data": {
"image/png": 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\n",
"text/plain": "<Figure size 432x288 with 1 Axes>"
},
"metadata": {},
"output_type": "display_data"
}
]
},
{
"metadata": {
"trusted": false
},
"cell_type": "code",
"source": "plt.plot([1, 2, 3, 4], [1, 4, 9, 16])\nplt.ylabel('some numbers')\nplt.show()",
"execution_count": 16,
"outputs": [
{
"data": {
"image/png": 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C3ZdAJAm8t3Qjv35lIRt37uWmk7vwi+/3pFE9TRstRxbNqaEM4If83/sRjK7uSs3sv4CbCI9GWgDc4O463SRSDUW7y/jd64uY8eU6erRtwiM3D+eErJZBlyVJJJqvC68CHwPvAhVHu0Iz6wj8DOjj7nvMbDJwBfD00X62SDpxd17PX8/dry2ieO8+fn5GD358WjfdN0CqLJogaOTut8VgvQ3NbB/QCFhXw58vktI27NjLb2Ys5N0lGxnYqTkPjBxK73Zq7ZHqiSYI3jCzc9z9zZpYobuvNbOHgDXAHuBtd3+7Jj5bJNW5Oy/nFvCHmUvYFwrx63OOY/TJXait6SHkKEQTBD8HfmVmpcA+wAi3FFTr64eZtSQ86qgLsB2YYmZXu/vzByw3BhgDkJWledFFvtm6m9unLeCzlVsZ1rUV918ygM6aJE5qQJU6i2vI94BV7r4ZwMymA8OBbwWBu08AJgBkZ2f7gR8iki4qQs5T/1rFQ28vo26tWvzh4v5ckZOpSeKkxgQxtmwNMMzMGhE+NXQGkBdAHSIJb9mG8CRx8wu2c0bvttxzcT/aN9ckcVKz4h4E7j7bzKYC8wg3qH1B5Ju/iISVlYd45IMV/P39FTRtUJe/XnkC5w9or0niJCYC6TZx97vQfQ5EDurLgu3cNjWfZRuLufD4Dtx1fl9aNa4XdFmSwqIKAjM7Gejh7k9FGsyauPuq2JYmkl72lFXwp7eXMfFfq2jbtAFPXpfNGcdpkjiJvWg6i+8iPNFcL+ApwvcjeB44KbaliaSPT7/ewu3TFrCmqISrhmZx+9m9adZAk8RJfERzRHAxcALhc/q4+zozq+mRRCJpaefefdz35hJemlPAsa0b8dIPh3FiN908XuIrmiAoc3c3MwcwMw1cFqkB7y7eyK9nLGBzcSljvtuV//peTxrW0/QQEn/RBMFkM3sMaGFmPwRGA4/HtiyR1LV1Vyl3v76Y1+evo3e7pky4JpuBmS2CLkvSWDQNZQ+Z2ZnATsLXCX7r7u/EvDKRFOPuvDZ/HXe/tohdpeX84syejD2lG/Xq1Aq6NElzUY0acvd3zGz2/uXNrJW7F8W0MpEUsm77Hn4zYyHvLd3E8ZkteHDkAHoeo0ttkhiiGTX0I+C/Cd+eMkRkriGga2xLE0l+oZDz4pw13D9rKRUh587z+nD98M6aJE4SSjRHBLcA/dx9S6yLEUklq7bs5vZp+cxeVcRJ3Vtz38UDyGrdKOiyRP6PaILga6Ak1oWIpIryihBPfrKKP7/zFfXq1OKBS/szKjtT00NIwoomCO4APo1cIyjd/6K7/yxmVYkkqSXrd3LbtHzyC3dwZp9juOeifhzTrEHQZYkcVjRB8BjwHuF7C4diW45Iciotr+Dv763gkQ++pkWjuvz9qkGc07+djgIkKUQTBHXd/Rcxr0QkSc39Zhu3TctnxaZdXHJCR+48rw8tNUmcJJFogmBW5G5hr/PtU0MaPippraSsnD++tYynP11N+2YNeOqGHE7r1TboskSqLJoguDLy+x2VXtPwUUlrnyzfwu3T8ynctodrhh3LuBG9aKpJ4iRJRdNZ3CUehYgkgx179nHvzMVMziukS5vGTBozjKFdNUmcJLdoGsrqAjcD34289AHwmLvvi2FdIgnnrUUbuHPGQrbuLmPsKd34z+/1oEFdTRInyS+aU0OPEr4HwSOR59dEXrupuis1sxbAE0A/wqeZRrv7Z9X9PJFY2lxcyt2vLWLmgvUc174ZT16XQ/9OzYMuS6TGRBMEOe4+sNLz98xs/lGu92HgH+4+0szqAWq3lITj7rzyxVp+98ZiSkoruPWsXoz5blfq1tYkcZJaogmCCjPr5u5fA5hZV6Ciuis0s+aETzNdD+DuZUBZdT9PJBbWbt/Dr6Yv4MOvNjMoKzxJXPe2miROUlM0QXAr8L6ZrSQ84dyxwA1Hsc4uwGbgKTMbCMwFfu7uu4/iM0VqRCjkPD/7Gx6YtRQH7j6/D9ecqEniJLVFM2ron2bWg/C9CACWuXvp4f5MFOscBPzU3Web2cPA7cCdlReK9C6MAcjKyjqK1YlE5+vNu7h9Wj65q7fxnR5t+MPF/clspbOWkvqOeLLTzC4D6rl7PnAB8JKZDTqKdRYChe4+O/J8KuFg+BZ3n+Du2e6enZGRcRSrEzm88ooQj3ywgrMf/phlG4r548gBPDt6iEJA0kY0p4budPcpZnYycAbwEOFRQ0Ors0J332BmBWbWy92XRT5zcXU+S+RoLVq3g9um5bNw7U5G9G3H7y7qS9ummiRO0ktUF4sjv58LPO7uM83snqNc70+BFyIjhlZydNccRKps774K/t97yxn/4UpaNqrHoz8YxNn92wddlkggogmCtZGb158JPGBm9YnilNLhuPuXQPbRfIZIdeWtLmLctHxWbt7NpYM6ced5x9GikSaJk/QVTRCMAkYAD7n7djNrT3gkkUhS2V0aniTumc9W06F5Q54ZPYRTeur6k0g0o4ZKgOmVnq8H1seyKJGa9tFXm7lj+gLW7djDdSd25pazetGkfjTfg0RSn34SJKVtLynjnplLmDq3kK4ZjZnyoxPJ7twq6LJEEoqCQFLWrAXrufPVRWwrKeMnp3Xjp6drkjiRg1EQSMrZtHMvv311Ef9YtIG+HZrxzOgc+nbQJHEih6IgkJTh7kydW8jv31jM3vIQ40b04off0SRxIkeiIJCUUFBUwq9eWcDHy7eQ07kl9186gG4ZTYIuSyQpKAgkqYVCzrOfrebBt5ZhwO8v7MsPhh5LLU0SJxI1BYEkrRWbirlt2gLmfrONU3pmcO/F/ejUUvMDiVSVgkCSzr6KEBM+WsnD7y6nUf3a/HnUQC4+oSNmOgoQqQ4FgSSVhWt3cOvUfJas38m5/dtz9wV9yWhaP+iyRJKagkCSwt59Ffzl3eU8/vFKWjWux/irBzOiX7ugyxJJCQoCSXhzVhVx+7R8Vm7ZzajsTvz6nD40b1Q36LJEUoaCQBLWrtJyHpi1lOc+/4ZOLRvy/I1DOblHm6DLEkk5CgJJSO8v28Svpy9g/c69jD6pC7ec1ZNG9fTPVSQW9JMlCWXb7jJ+/8Zipn+xlu5tmzB17HAGH9sy6LJEUpqCQBKCuzNzwXruenURO/bs42end+cnp3enfh1NEicSawoCCdzGnXu5c8ZC3l68kf4dm/PcjUPp06FZ0GWJpI3AgsDMagN5wFp3Py+oOiQ47s7kvALumbmEsvIQd5zdmxtP7kIdTRInEldBHhH8HFgC6KtfGlqztYQ7XsnnXyu2MqRLKx64dABd2jQOuiyRtBRIEJhZJ+Bc4F7gF0HUIMEoKSvnxdlr+NPbX1G7lnHPRf24akiWJokTCVBQRwR/AcYBTQ+1gJmNAcYAZGVlxaksiQV3Z37hDiblFvD6/HXsKi3ntF4Z3Htxfzq0aBh0eSJpL+5BYGbnAZvcfa6ZnXqo5dx9AjABIDs72+NUntSg7SVlvPLFWiblFrB0QzEN6tbi3P4duDwnk5zOLTVJnEiCCOKI4CTgAjM7B2gANDOz59396gBqkRoWCjmfrdzKy7kFvLVoA2XlIQZ0as69F/fj/IEdaNZAU0OIJJq4B4G73wHcARA5IrhFIZD8NuzYy9S5BUzKK6CgaA/NGtThqiFZjMrO1FBQkQSnPgKptn0VId5buolJuQV8sGwTIYfh3Vpzy/d7cVbfdjSoq2YwkWQQaBC4+wfAB0HWIFW3cvMuJuUVMG3uWrbsKqVt0/rcfGo3RmVncmxrDQEVSTY6IpCo7CmrYNbC9bycW8CcVUXUrmWc3rstV+RkckrPDDWBiSQxBYEc1sK1O3g5dw2vfrGO4tJyjm3diHEjejFyUCfaNmsQdHkiUgMUBPJ/7CjZx6vzw8M+F63bSf06tTinf3tGZWcyrGsrDfsUSTEKAgHCTV+zVxUxKbeANxesp7Q8RJ/2zfjdhX25cGBH3RFMJIUpCNLcpp17mTqvkMm5BazeWkLT+nW4LLsTV+Rk0a9j86DLE5E4UBCkofKKEB8s28ykvALeW7qJipAzpEsrfnp6D87p356G9TTsUySdKAjSyDdbdzM5r4ApeYVsKi6lTZN63PSdLozKzqRbRpOgyxORgCgIUtzefRW8tWgDL88p4LOVW6llcGqvtlyek8npvdtSV8M+RdKegiBFLV63k8l5BbzyxVp27NlHZquG/PLMnozM7kT75prxU0T+TUGQQor37uO1+euYlFtAfuEO6tWuxVn92nFFTiYndm2tOf9F5KAUBEnO3cn7ZhsvzwkP+9yzr4JexzTlrvP7cNHxHWnZuF7QJYpIglMQJKktu0qZPq+Ql3MLWLl5N43r1eaiEzpweU4WAzs1V9OXiERNQZBEKkLOR8s3M2lOAe8u2Uh5yBl8bEseHNmNc/u3p3F97U4RqTr9z5EECopKmJJXwJS5hazfsZdWjetxw0mduTwnk+5tD3m3TxGRqCgIElRpeQXvLN7IpNwCPlmxBYDv9MjgzvP68L3jjqFeHQ37FJGaoSBIMMs2FDMpt4BXvihkW8k+OrZoyM/P6MFl2Zl01I3eRSQGFAQJYFdpOW/MX8ekvAK+WLOdurWN7/dpx6icTE7u3obaGvYpIjEU9yAws0zgWeAYwIEJ7v5wvOsImrvzRcF2Js0p4PX8dZSUVdC9bRN+c+5xXHxCR1o3qR90iSKSJoI4IigHfunu88ysKTDXzN5x98UB1BJ3RbvLmD6vkEm5BSzftIuGdWtz/sD2XJ6TxaCsFhr2KSJxF/cgcPf1wPrI42IzWwJ0BFI2CEIh55MVW5iUV8Dbizawr8I5PrMF913Sn/MGtKdpA831LyLBCfQagZl1Bk4AZgdZR6ys276HKXmFTM4rYO32PbRoVJerhx3L5TmZ9G7XLOjyRESAAIPAzJoA04D/dPedB3l/DDAGICsrK87VVV9ZeYh/LtnIy7kFfLR8M+5wcvc23H52b87scwwN6mqufxFJLIEEgZnVJRwCL7j79IMt4+4TgAkA2dnZHsfyqmXFpl1Mzitg2txCtu4uo12zBvzHad0ZlZ1JZqtGQZcnInJIQYwaMuBJYIm7/zne669JJWXlzMxfz6TcAvK+2UadWsYZx7XlipwsvtszQ8M+RSQpBHFEcBJwDbDAzL6MvPYrd38zgFqqzN3JL9zBy7kFvD5/HbtKy+napjF3nN2bSwZ1IqOphn2KSHIJYtTQJ0DSfVXeXlLGjC/W8nJuAUs3FNOgbi3O6d+eK3KyyOncUsM+RSRpqbP4MEIh5/OVW3k5t4B/LNpAWXmI/h2bc89F/bjg+A4007BPEUkBCoKD2LBjL1PnFjA5r5A1RSU0a1CHK3MyGZWTSd8OzYMuT0SkRikIIvZVhHh/6SYm5Rbw/rJNhByGdW3FL87syYh+7TTsU0RSVtoHwaotu5mUW8DUuYVs2VVKRtP6jD2lG6OyM+ncpnHQ5YmIxFxaBsGesgpmLQwP+5y9qojatYzTerXl8pxMTuuVQZ3amutfRNJHWgXBwrU7mJRbwIwv11K8t5xjWzfi1rN6MXJwJ45p1iDo8kREApHyQbBjzz5e+zI87HPRup3Uq1OLc/q14/KcLIZ2aUUtNX2JSJpL6SD46z+X8/f3V1BaHuK49s347wv6ctHxHWneSMM+RUT2S+kg6NCiISMHd+KKnCz6dWympi8RkYNI6SAYObgTIwd3CroMEZGEpuExIiJpTkEgIpLmFAQiImlOQSAikuYUBCIiaU5BICKS5hQEIiJpTkEgIpLmzN2DruGIzGwz8E01/3gbYEsNlhMkbUviSZXtAG1LojqabTnW3TOOtFBSBMHRMLM8d88Ouo6aoG1JPKmyHaBtSVTx2BadGhIRSXMKAhGRNJcOQTAh6AJqkLYl8aTKdoC2JVHFfFtS/hqBiIgcXjocEYiIyGGkRBCY2UQz22RmCw/xvpnZX81shZnlm9mgeNcYrSi25VQz22FmX0Z+/TbeNUbDzDLN7H0zW2xmi8zs5wdZJin2S5Tbkiz7pYGZzTGz+ZFt+e+DLFPfzCZF9stsM+sc/0qPLMptud7MNlfaLzcFUWs0zKy2mX1hZm8c5L3Y7hN3T/pfwHeBQcDCQ7x/DjALMGDuNMkDAAAC90lEQVQYMDvomo9iW04F3gi6zii2oz0wKPK4KfAV0CcZ90uU25Is+8WAJpHHdYHZwLADlvkxMD7y+ApgUtB1H8W2XA/8Lehao9yeXwAvHuzfUaz3SUocEbj7R0DRYRa5EHjWwz4HWphZ+/hUVzVRbEtScPf17j4v8rgYWAJ0PGCxpNgvUW5LUoj8Xe+KPK0b+XXghcILgWcij6cCZ1gC3uc1ym1JCmbWCTgXeOIQi8R0n6REEEShI1BQ6XkhSfqDHHFi5HB4lpn1DbqYI4kcxp5A+BtbZUm3Xw6zLZAk+yVyCuJLYBPwjrsfcr+4ezmwA2gd3yqjE8W2AFwaOfU41cwy41xitP4CjANCh3g/pvskXYIglcwj3DY+EPh/wIyA6zksM2sCTAP+0913Bl3P0TjCtiTNfnH3Cnc/HugEDDGzfkHXVF1RbMvrQGd3HwC8w7+/VScMMzsP2OTuc4OqIV2CYC1Q+ZtAp8hrScfdd+4/HHb3N4G6ZtYm4LIOyszqEv6P8wV3n36QRZJmvxxpW5Jpv+zn7tuB94ERB7z1v/vFzOoAzYGt8a2uag61Le6+1d1LI0+fAAbHu7YonARcYGargZeB083s+QOWiek+SZcgeA24NjJKZRiww93XB11UdZhZu/3nBs1sCOF9mHA/pJEanwSWuPufD7FYUuyXaLYlifZLhpm1iDxuCJwJLD1gsdeA6yKPRwLveeQqZSKJZlsOuOZ0AeHrOwnF3e9w907u3pnwheD33P3qAxaL6T6pU1MfFCQze4nwqI02ZlYI3EX4whHuPh54k/AIlRVACXBDMJUeWRTbMhK42czKgT3AFYn4Q0r4W841wILIOVyAXwFZkHT7JZptSZb90h54xsxqEw6rye7+hpn9Dshz99cIh95zZraC8MCFK4Ir97Ci2ZafmdkFQDnhbbk+sGqrKJ77RJ3FIiJpLl1ODYmIyCEoCERE0pyCQEQkzSkIRETSnIJARCTNKQhERNKcgkBEJM0pCERE0tz/B3wrYdjs3/9VAAAAAElFTkSuQmCC\n",
"text/plain": "<Figure size 432x288 with 1 Axes>"
},
"metadata": {},
"output_type": "display_data"
}
]
},
{
"metadata": {
"trusted": false
},
"cell_type": "code",
"source": "# function interpolation by solving polynomial basis vandermonde\n# 1 x x^2 x^3 ...\nx = np.linspace(0,1,10)\ny = 1.5*x**3 + 1 - 0.5*x**2 + 2*x\nN = 4\nV = np.array([[x_i**i for i in range(0,4)] for x_i in x])\ny = np.transpose(np.matrix(y))\n# lets get the coefficients V b = y \nb = la.solve(np.matmul(np.transpose(V),V), np.transpose(V)*y)\nf = lambda x: b[0,0] + b[1,0]*x + b[2,0]*x**2 + b[3,0]*x**3\ny_hat = np.array([f(x_i) for x_i in x])\n# lets see\nplt.plot(x,y)\nplt.plot(x,y_hat)\nprint(b)",
"execution_count": 60,
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": "[[ 1. ]\n [ 2. ]\n [-0.5]\n [ 1.5]]\n"
},
{
"data": {
"image/png": 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hrxjsismKuZWkW/6k2boAYM65ygeZNQZmAx875/4awPitQIpzbt/ZxqSkpLjc3NwqRBWR/883W7+XtAMfsC0ilhPDJtIz+RKvY0kdMLOlzrmUysZVOnM3MwNeBNadrdjNrBOwxznnzCwN31E4+6uYWUQCkLfgbTp+NkGzdTmnQHbLXATcAqwysxX++x4C4gCcc5OB64C7zawU+Bq4wQXyJ4GIBKzkwD42TL2XtINz2BoRy6ZhL5OR/H2vY0k9FcjRMl8AVsmYicDEYIUSkW9aueAtOn82gcHuAFkxt5F865903nU5J71DVaQe++ZsPY6Dw17RbF0ConIXqadWzn+Tzot+TbI7SFbX0STf8kfN1iVgKneReqbkwD6+mjqe1IMfsiUinpJrppKRdLHXsSTEqNxF6pGV82fQedGDJLmDZHW9neRbntRsXapF5S5SD5QUF/lm6yUf+Wbrw6eRoUvfSQ2o3EU89o3ZeuxPSb75D5qtS42p3EU84putjyO15GO2RCRwaMSrZAz6rtexJEyo3EU8sOLTGcR8/huSXAnZsWNIvuVJmjRt5nUsCSMqd5E65Jut30NqyVw2RyRQMuI10jVbl1qgchepIys+eZ2YLx4k0R0iK+5/GHzzHzRbl1qjchepZSX797Bx6jhSDs3zz9ankzHoIq9jSZhTuYvUohXz/o+u/36QQe4wWXF3MPjmJzRblzqhchepBb7Z+j2kHPqETZHdKBnxOhkDv+N1LGlAVO4iQVReVsbSWZPptuLPvtl6/FgG//fjmq1LnVO5iwTJ+py5RM57iNTSjXzVqCclwzVbF++o3EVqqGDrBna/PYHBRxayl/bkJv+J5KvHEhEZ6XU0acBU7iLVdOTQAVa98XuSd06nHUZW/FgGjXqElFZtvI4monIXqaqy0lKWZU6iW95fyeAguW0uJ3bUU2R0vdDraCL/oXIXqYI1X86h6aePkFq2ifWN+lB81VRSki/xOpbIt6jcRQKwa/M69sycQPLRRRQSRW7qXxg8dAwWEeF1NJEKqdxFzuFwSTGrZzzK4IIZtCOSrIS7SLr+UTq1aOV1NJFzUrmLVKCstJSl7z3HhaufJYMSlrQbSsKoP5HRJcHraCIBUbmLnGH1F5m0mP8oaeVbWde4H8VXTydVV0WSEKNyF/Hbkb+Kfe9MIOnYlxRYB5YNeZakH92m/eoSkiotdzOLBaYBHQEHTHHO/e2MMQb8DbgKOAaMds4tC35ckeArObCPdTMeIbnwTdrTmKwLxpM06iG6NG/pdTSRagtk5l4KPOCcW2ZmrYGlZjbPObf2tDFDgR7+jyHA3/3/itRbpadOsvTdZ+m59nnS3GFy21/NBdf/kYxOcV5HE6mxSsvdObcb2O1fPmxm64AY4PRyHwFMc845INvM2ppZZ//nitQ7qz57h9af/Y4h5dtZ02QgxcOeIk3ngZEwUqV97maWACQBOWc8FAPsOO32Tv99KnepV7ZtWMHB9yYw6Oscdlonln9nEok/vEn71SXsBFzuZtYKmAnc55w7VJ2VmdlYYCxAXJz+9JW6U7J/D+tmPMzgve/QniZk97iPpOt+TddmLbyOJlIrAip3M2uMr9inO+feqWDILiD2tNtd/fd9g3NuCjAFICUlxVU5rUgVnTp5gqUzn6bPhomkumPkRg2n+6gnSe/Y1etoIrUqkKNlDHgRWOec++tZhmUC481sBr4XUku0v128tnLBW7T9/Pekl+9kVdMkWg1/iiH99Dq/NAyBzNwvAm4BVpnZCv99DwFxAM65ycAcfIdB5uM7FPL24EcVCczWdbkcev/XDDqeyw7rworv/YNBPxil/erSoARytMwXgFUyxgHjghVKpDqKCray+Z3/ZXDRexyz5mT3/CXJP/4VsbrEnTRAeoeqhLy9u7aw5d3HSSzKJJlylkZfS8/rnyQ9urPX0UQ8o3KXkFW4I59t7/+BpKJMknEsbz+UrsMfZUi33l5HE/Gcyl1CTuGOfLa99zhJ+2ZzPo5l519N7PBHSEvo5XU0kXpD5S4ho3D7Rra9/wRJ+2bRHlgeNYy44Q8zJF6lLnImlbvUe7u3bWD7+0+QtP8Df6lfQ/yIRxgS18PraCL1lspd6q2CrRvYmfk4SfvncD7G8ugRxI94mCGx3b2OJlLvqdyl3tm1eR27Zj1OUvFHRGEsix5BwrWPMKTrhV5HEwkZKnepN3ZtXseuzMdIPvARUUSyLPpaEq59WKUuUg0qd/HczvzV7J79BEkHPiaKSHI7/JgLRz7CEF2vVKTaVO7imR35qyic9QRJB+f6Sr3jdVx47cOkq9RFakzlLnVux8aVFM5+guSD84iiMbmdRtF9xEOkd4n3OppI2FC5S53Z/tUK9s5+gqSST4iiMUs63UD3kQ+RrsvaiQSdyl1q3bYNKyj64HGSSj4liiYs6XwT3a99kPROsZV/sohUi8pdas229cso+uAJkg/NJ9pf6j1GPqQLZYjUAZW7BN3Wdbnsn/MESYcWEk0TcrrcQq+RvyG9Q4zX0UQaDJW7BM2WtUsonvMESYc/I5qm5MTcQu+RD5GhU++K1DmVu9TY+txPObbgWZKPLqKDa0ZO19voM/JBMqI6eR1NpMFSuUu1lJWWkvfpdJrnTqb3qbUcoiVZMbfTZ+RvVOoi9YDKXark6OGDrJr9ArFfvUKS20OBdSS71wQGDBtHRuu2XscTET+VuwSkqGAr+bOfpm/BTNI5yoZGvVk2+EEGXX4LXRrp20ikvtFPpZzT5tU57J/3NIMOfkIa5axs9T1aXPJzeqf+0OtoInIOKnf5FldezqrP3sGyJzLgxHI6uaYs7zCSrkN/SfIFfbyOJyIBULnLf5w4foy8Of8kevW/GFi+nb20J6vbePpecx9D2kd7HU9EqkDlLhzcV8i6Wc/QY9sMUjnI5ogEliQ+yaChY8ho2szreCJSDZWWu5m9BAwD9jrn+lfw+CXA+8AW/13vOOceC2ZIqR078ldR8OFfGLhvDhl2kpXNUtn9nXvp/91ruCAiwut4IlIDgczcXwEmAtPOMeZz59ywoCSSWuXKy1m3eC4nFv2NQUez6EgkK9r/iA5X/IJBfVK8jiciQVJpuTvnFplZQu1HkdpUeuokK+ZOpc3yf9C3dCMHaM3i2J/Sfdj9pOnsjCJhJ1j73DPMbCVQAPzSObcmSF9XauhwSTFrZj9PwsZXSaGIHdaFnL6PMPDqu0hv2drreCJSS4JR7suAeOfcETO7CngP6FHRQDMbC4wFiIvTBRpqU+GOfLbO/gv9Ct8j3b5mbZMBFKY+xsBLryc2MtLreCJSy2pc7s65Q6ctzzGzF8wsyjm3r4KxU4ApACkpKa6m65Zv27jic0o+fYbEQwuIAlaedwnnXXo/fZMu9jqaiNShGpe7mXUC9jjnnJmlARHA/honk4CVl5WRt+BNGi+eRL+TqzjimpPb6XoSrn6AwXEV/hElImEukEMhXwcuAaLMbCfwO6AxgHNuMnAdcLeZlQJfAzc45zQrrwNHDx9k9YdT6Lz+FRLLd1FIFNnd76fvsHtJb3u+1/FExEOBHC1zYyWPT8R3qKTUkZ35q9k59zn67p3FEI6xsVEPcgf/hUGX30qnJk29jici9YDeoRoiysvKWLXoHVg8hQHHltCBCPLa/IBW37ubXoMvxfSmIxE5jcq9njt0cD9r5/ydmI3TGeQK2EdbcuLvoMeV40npEu91PBGpp1Tu9dS2dUsp/HQiA4o+IN1OsL5RH3IT72fg5bfqfC8iUimVez1SVlpK3vwZNF76T/qfWEEn15i8dj+k7SXj6J34Pa/jiUgIUbnXAwf3FbJ+ziTiN79OEkUUEkVWt3H0GjqO1A4xXscTkRCkcvfQprws9i94noHFc0m3U6xpMpDCwY8y4NIb6dS4idfxRCSEqdzr2KmTJ8j75DVaLH+RPqfW0Nk1ZeX5Q+lw2Xj69RvidTwRCRMq9zqyf88OvpozkQu3vclgitllHcnu8Qv6DL1HVzkSkaBTudeyr5Yt5NDCiQwsWUCGlZLXLIWC1D8y4PvXEdNI//0iUjvULrXgxPFjrJr7CuflvUzP0q844pqzvMO1dL78Xgb2TPQ6nog0ACr3INq7awubPnyOnjtnkkIJ2yNiyOn9G/oOvZMhbdp7HU9EGhCVew258nLWL5nLsc//zsDDnzOEcvJaplOQfif9LhpBXKROCyAidU/lXk3Hjx0h78N/cf7aV+hTtoVDtGRpp1HE/ujnJF7Qx+t4ItLAqdyrqGDrBrZ//Dd6736fNI6wOSKBnP6/Y8DQ/yG95XlexxMRAVTuAXHl5az59yxOZU1m4NEsOmCsbPVdml10N33Tr+QCnZFRROoZlfs5+C6G8Q86rX+V/uU7OMB5LOl6KwlX/ozBsd29jiciclYq9wrsyF/FrrnP0W/PLIbY12yM7M6SxD8w4Ee3k968pdfxREQqpXL3810MYyaWM4WBx5fQ0UX6LoZx8T30Sv6BLoYhIiGlwZe772IYL9B143QGud3soy1ZcWPpMfReUjrHeR1PRKRaGmy5b1u3lMJPnmPAvg99F8No3JfcxAcY+MNbdDEMEQl5Darcz3YxjHY/GE/vQd/1Op6ISNA0iHKv+GIY4+l91XhSozt7HU9EJOjCutw35X3J/vnPM/DAPP/FMAbpYhgi0iCEXblXeDGMqKvocNm99Oub6nU8EZE6UWm5m9lLwDBgr3OufwWPG/A34CrgGDDaObcs2EErs69wBxs/1MUwREQgsJn7K8BEYNpZHh8K9PB/DAH+7v+3TmzInc/hRS9842IYu9P+yIDv/4SYyMi6iiEiUq9UWu7OuUVmlnCOISOAac45B2SbWVsz6+yc2x2kjN9y4vgx8j5+hTarXqaX/2IYyzqMpMvl43UxDBERgrPPPQbYcdrtnf77aqXcV86fQddFE0ilhG0RXcnp8yD9ht5J+nntamN1IiIhqU5fUDWzscBYgLi46r37s21Mb3a06EtB+p30/+4I4nVaABGRbwlGue8CYk+73dV/37c456YAUwBSUlJcdVYW3yuR+AkfVedTRUQajGBMezOBW80nHSipzf3tIiJSuUAOhXwduASIMrOdwO+AxgDOucnAHHyHQebjOxTy9toKKyIigQnkaJkbK3ncAeOClkhERGpMr0aKiIQhlbuISBhSuYuIhCGVu4hIGFK5i4iEIfMd7OLBis2KgG3V/PQoYF8Q44QCbXPDoG1uGGqyzfHOuUpPdetZudeEmeU651K8zlGXtM0Ng7a5YaiLbdZuGRGRMKRyFxEJQ6Fa7lO8DuABbXPDoG1uGGp9m0Nyn7uIiJxbqM7cRUTkHOp1uZvZlWa2wczyzew3FTze1Mze8D+eU8nlAENCANv8CzNba2Z5ZvapmcV7kTOYKtvm08b92MycmYX8kRWBbLOZjfI/12vM7P/qOmOwBfC9HWdmC8xsuf/7+yovcgaLmb1kZnvNbPVZHjcze87//5FnZslBDeCcq5cfQCSwCbgAaAKsBPqeMeYeYLJ/+QbgDa9z18E2/wBo4V++uyFss39ca2ARkA2keJ27Dp7nHsByoJ3/dgevc9fBNk8B7vYv9wW2ep27htt8MZAMrD7L41cBHwIGpAM5wVx/fZ65pwH5zrnNzrmTwAx8F+M+3Qhgqn/5beAyM7M6zBhslW6zc26Bc+6Y/2Y2vitfhbJAnmeAx4GngON1Ga6WBLLNdwCTnHMHAJxze+s4Y7AFss0OOM+/3AYoqMN8QeecWwQUn2PICGCa88kG2ppZ52Ctvz6X+9kuvF3hGOdcKVACnF8n6WpHINt8ujH4fvOHskq32f/naqxz7oO6DFaLAnmeewI9zezfZpZtZlfWWbraEcg2/x642X9RoDnAvXUTzTNV/Xmvkjq9QLYEj5ndDKQA3/c6S20yswjgr8Boj6PUtUb4ds1cgu+vs0VmNsA5d9DTVLXrRuAV59zTZpYBvGpm/Z1z5V4HC0X1eeYeyIW3/zPGzBrh+1Nuf52kqx0BXWzczH4IPAwMd86dqKNstaWybW4N9AcWmtlWfPsmM0P8RdVAnuedQKZz7pRzbgvwFb6yD1WBbPMY4E0A51wW0AzfOVjCVUA/79VVn8t9CdDDzLqZWROtUJxSAAABHElEQVR8L5hmnjEmE7jNv3wdMN/5X6kIUZVus5klAf/AV+yhvh8WKtlm51yJcy7KOZfgnEvA9zrDcOdcrjdxgyKQ7+338M3aMbMofLtpNtdlyCALZJu3A5cBmFkffOVeVKcp61YmcKv/qJl0oMQ5tztoX93rV5QrebX5Knwzlk3Aw/77HsP3ww2+J/8tfBfnXgxc4HXmOtjmT4A9wAr/R6bXmWt7m88Yu5AQP1omwOfZ8O2OWgusAm7wOnMdbHNf4N/4jqRZAVzhdeYabu/rwG7gFL6/xMYAdwF3nfYcT/L/f6wK9ve13qEqIhKG6vNuGRERqSaVu4hIGFK5i4iEIZW7iEgYUrmLiIQhlbuISBhSuYuIhCGVu4hIGPp/CF5i2v39ZkkAAAAASUVORK5CYII=\n",
"text/plain": "<Figure size 432x288 with 1 Axes>"
},
"metadata": {},
"output_type": "display_data"
}
]
},
{
"metadata": {
"trusted": false
},
"cell_type": "code",
"source": "# function interpolation by solving polynomial basis vandermonde\n# 1 x x^2 x^3 ...\nx = np.linspace(0,1,10)\ny = 3*x**5+2*x**4+1.5*x**3 + 1 - 0.5*x**2 + 2*x\nN = 4\nV = np.array([[x_i**i for i in range(0,4)] for x_i in x])\ny = np.transpose(np.matrix(y))\n# lets get the coefficients V b = y \nb = la.solve(np.matmul(np.transpose(V),V), np.transpose(V)*y)\nf = lambda x: b[0,0] + b[1,0]*x + b[2,0]*x**2 + b[3,0]*x**3\ny_hat = np.array([f(x_i) for x_i in x])\n# lets see\nplt.plot(x,y)\nplt.plot(x,y_hat)\nprint(b)",
"execution_count": 63,
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": "[[ 0.94049688]\n [ 4.06797744]\n [-10.0308642 ]\n [ 13.95679012]]\n"
},
{
"data": {
"image/png": 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Nc7EPfsUbgeNZ1+1inj1dmyJEgopbRL6bvVsJTL+G7a4Tv024iWmXDdWmCBGiv2URabqSNdg/zqBmfyWTq2/l15cdT06GNkWIFM24RaRpti3C/nkRlXVw6YH7OPu00zi+V5bXqVoVzbhFJHSFc7CnzmF3bSLjq37MSSeN5eaTe3udqtXRjFtEQrNyJjbjWjbTmUkH7uLW80Zzme745wkVt4g0bvHT2Gu3sYI+TPbfxYNXjuaUvjlep2q1VNwi8u0+eQTe/Smf2GDuS7yLv183igFdtSGCl1TcInJwZvDe/fDJI8zyH8djmXfy3DXH06VdG6+TtXoqbhH5T/56bNbtuCXP8Ez9qbyb9wOeu3w4GcnaL7IlUHGLyNfVVROYcR2+1a/xSP35FA26nScuHESCLq5pMVTcIvJvNRXUPzuJ+M0f8/O679F+zG38dkxvbTvWwqi4RSSoqpTap84nrng5P6y/mRMuuoXzh+R6nUoOIuTids7FAQXANjMbH75IIhJx5UVU/2MClG/lB/yQy66+keN6dfA6lRxCU2bctwGrgIwwZRERL5SspfrJCdRV7eVHST/ljmuv0j6RLVxIrzY453KBccDj4Y0jIhG1bTHVU06nsqqKH2c+xM9uvV6lHQVCnXE/DNwF6DsqEiMCGz6k7l8T2e1P4S/d/sivrziHlES97BUNQtnlfTxQbGaLGhk32TlX4JwrKCkpabaAItL8alfMxP/MhWyqb8/zAx7ngWvOU2lHkVCWSk4AJjjnNgHPA2Occ//85iAzm2Jm+WaWn52d3cwxRaS5VM6bStz0K1kROIKFJ/+TH1x0MnE+ne4XTRotbjO7x8xyzSwPmAi8b2aXhz2ZiDS70nd+R9rbt/FpYAAl57/A98YM0TnaUUi/G4m0BmbseOkeOi9/jHfc8WRfPZVRPbSpb7RqUnGb2RxgTliSiEh4BPxsfmoyR2yezsz4Mxh8w+Mcka2zeqOZZtwiMczqqin822X02j2bGakTGXPTn8hMS/I6lhwmFbdIjKo/sI+Nj55Pn8oCZmTfzLjJvyQ5Ic7rWNIMVNwiMahyTzG7HjuHHjVrmdXrfs6//A58OnMkZqi4RWJMybZCDjwxgVz/TuYO+yPjJ1zldSRpZipukRiyYfVSUp6/kEyrYuWpT3LyqHO8jiRhoOIWiRFLFsyh+xvfwznYdcEMhgw+wetIEiba0kIkBnzw1gz6vDGRel8SdVe+SW+VdkxTcYtEMTPj1eencPy8G9ibkEPKTbPp2GOA17EkzLRUIhKlausDTH/it1yy/SG2pfSjy82vkZCe5XUsiQAVt0gUKt9fx6t/vYcr9v2dLZnH0v2mGbgk3XW5tVBxi0SZT9buZPO0e7mibgZFXc6g+zXPQLyuhmxNVNwiUaKwpJJpL73IhG1/4ATfVnb1mUTupEfBp6shWxsVt0gLV76/jsffWkCPJb/hR3EfU9GmE7Xjn6bjgAmgW7K2SipukRaqzh/g2U83sP39v3BL4HlS4mupGnEb6WN/BImpXscTD6m4RVoYM2POmhJemvkSN1b+haN9m6nMHUX8+X8kPquP1/GkBVBxi7Qga3dV8PCrn3LSlr/wv/FzqE7tiI2bStrR52lZRL7UaHE755KBj4CkhvHTzez+cAcTaU1KK2t4+N1VWMFT/DrhBdITavCPvJXkk++GpDSv40kLE8qMuwYYY2aVzrkEYK5z7k0zmx/mbCIxr6bez1OfbuL999/i3sDjDEoopK7bCfjO+T3kHOV1PGmhGi1uMzOgsuHdhIY3C2cokVhnZrz9xS4efWMBk/ZN5dn4Dwik5cBZT5Aw4EIti8i3CmmN2zkXBywCegOPmtmCg4yZDEwG6N69e3NmFIkpK7aV86tZK+i+5WX+mfg8GQn7ccfejO/kuyFZe0FK40IqbjPzA8c459oBLzvnBpjZim+MmQJMAcjPz9eMXOQbivdV87t31rBy8cc8mDiVQQnrsNyRuHG/h066MZSErqm7vO91zn0AnAmsaGy8iEB1nZ8n5m7kmQ+Wcou9wG8S34OUDnD6X3GDJ2pZRJoslLNKsoG6htJuA5wGPBT2ZCJRzsx4bdkOfvvGSkZWvsM7SS+Qbvtww6+DU+6DNu28jihRKpQZd2fgqYZ1bh/wopnNCm8skei2dOteHpi1kqotn/O3lKc5OmEldBkO434PnQd7HU+iXChnlSwDhkQgi0jU2773AL99azWzl67n3pSXuTT5LVxSOzj1z3DMf4FPe5fI4dOVkyLNoKqmnr99uIEpH29gHJ8wL/05UuvKcPlXw5ifQEp7ryNKDFFxixyGQMB4ack2/uft1WRUbOC1dv+iz4HPIXsIjJsGXYd5HVFikIpb5DtauLGMB2atpHDbTn6Z+TrnJc/EWRqMfxiGXqH7ZEvYqLhFmmhr2X5+/eYq3li+g8vTFvFiu3/S5kAxDPkenPpzSO3gdUSJcSpukRCt3rmPFz7byr/mb6G3bzsfdXqe7nsXQtYgGPcsdBvudURpJVTcIt+itLKGV5duZ8biIr7Yvo90Xw2PdXmPMWUv4g6kwNm/g/xrtCwiEaXiFvmGmno/H6wuZvqibcxZU0x9IMAFHYv5nz4L6bf7HXy7S2HwZXDazyEtx+u40gqpuEUIXuW4rKicGYuLmPn5dvbur2NAWgWP91zKcZXvkrR3PVQmQd+zYORN0H2k15GlFVNxS6u2s7yal5dsY8biItYXV9I+voYf5K5hvH1Iu13zcUUG3Y+DUbdC//N0mbq0CCpuaXUO1Pp5Z+VOpi8q4pP1u8ECXN1pE4/3nMcRxe/jdh6AzB5w8j0w6BJo38PryCJfo+KWVsHMWLixjBmLi3hj+U4qa+oZ3baYF3ouYsjed4nfuxOq28LgiTB4EnQbobv2SYul4paYtqV0PzMWF/HSkiK2lh2ge2IFD3Zeztja90ndsxJ2xEOf04OF3ecMSEj2OrJIo1TcEnMqqut4Y/kOZizaxsJNZSS7Wm7pvJaLj5hLx+JPcLv80GUIjPwtDLgQUrO8jizSJCpuiQn+gPHJ+t3MWFzE21/spKaunnMzt/BG3gL6lb2Pr6wCMrrCCbcFZ9fZfb2OLPKdqbglqq0vrmD6om28smQbO/dVMzB5N491WczxVe+RVFkE/jQ4akKwrPNG6baqEhNC2QGnG/A00JHg7u5TzOyRcAcTOZQ9VbW8tmw7MxYV8XlROe19VdzRaTnj0z4ks+xzKPZBz5Ph9Puh3zhITPU6skizCmXGXQ/8wMwWO+fSgUXOuXfNbGWYs4l8qbY+wIdrS5ixqIjZq3eBv47LO6zl4W6fklc2F1dWCzn94bRfwMCLIaOL15FFwiaUHXB2ADsaHlc451YBXQEVt4RNnT/A8m3lzC8sZd6GUgo27eFAXT0npW7hX10XMbR8NvFVe4BsGH59cCmk00CdwietQpPWuJ1zeQS3MVsQjjDSetX7A3yxfR/zvizqMqpq/XSkjLPbF3FD560MOfApqRUboSwZ+p4dPN+61xiI00s10rqE/BPvnEsDZgC3m9m+g3x+MjAZoHv37s0WUGKTP2Cs3L4vOKMuLOWzjWUEaioY5CtkbPpW7srcRK+a1SRXF8N+oDoBuh0Lp3wf+p8LyW29PgQRzzgza3yQcwnALOBtM/tDY+Pz8/OtoKCgGeJJrAgEjFU79zFvQynzC0sp2FhC15qNDPZtYFSbTQyNLySnehOOhp/H9j2ha35w66/cfOg4QBfHSExzzi0ys/xQxoZyVokDngBWhVLaIhAs6jW7KoIz6vW72bpxDb1qVzPYt4HbEjfS1xWSmFQTHJzYIVjQXSc1/HeoNtcV+RahLJWcAHwPWO6cW9rwsXvN7I3wxZJoY2asK65k3oZSlq3bxIHNn9GrJljU58YV0p5ySASLS8Z1GQxdr20o6WGQmacXFUWaIJSzSuYC+lclX2NmbCipYsH6HWxbtRArWkSf+jWc6DZwpW9HcEyCo759bxK6jw/Oorvm4zoeDXEJHqcXiW56OV5CYmZsLKlkxYol7Fk7j6TipfSrX8NFbjNJrh6AAylZWNdh0ON66DoM12UICXoRUaTZqbjloCrKy9i8bgW7N6+kevsXtC1bxlGBdUxwVQDUuGTKswZQnXc6ib1G4nLzaZPRVUseIhGg4m7NqvdhZRso37aG0i2rqS1eR0L5JtpXb6U95QxoGObHx46kHpTknEld75Fk9T2epJyjyNEGuSKeUHHHuupyKCuE0g0ESgup3LGG+pINJO3bRGr9HhzQruFth7VnZ1xndmacgK9DT9K79KNLz/6079aPXN3vQ6TFUHHHgupyKN0QLOiGt8Du9fhLC0moLv1ymA+osvZsCnRiizuGqtQjiM/qRUbXfnTr1Z8ju3VkSLJeOBRp6VTc0eLA3q8V87+LegPsL/3a0GKXxQZ/DhsDg9hsndiZ0JWErF5kdutL39yO9O+cwfk5aSTG6xanItFIxd2S1NcEy3j32oa39cFiLiv8j3KuSMyhyNeFNbVDWVmXxWbrxEbrRG16d3p3zaZ/5wz6d2nLqC4Z5Ga2welFQ5GYoeKONLNgCe9e95WCbni8dzNY4Muhdamd2ZPcja0pJ7IqLovPKjJZVZPNZutIXU0ivbLTODovg/5dMhjduS39u2TQPjXRw4MTkUhQcYeLvx72bILSgxT0gT1fDrP4ZKozelDapi+bU8awsq4Tn1V04NO97aisDt6bo01CHP06p9O/ZwZXdQkWdL9O6SQn6KwOkdZIxX24DuyF0vVfL+bd64LLG4G6L4cFUnOoSu/BruxTKbQuLK/pyLy97Vm8L41AZXCtOd7n6JGVSp/cNK4Zkkbvjun075xBj6xU4nxa6hCRIBV3KAIBKN/672IuXffvx5W7vhxmvnj87XqwL7UH2484kbX+Tiw9kMPcPZkUlsZDwzJ1UryPXtlp9OmZxknZafTpmEbvnHSO6JBCQpxeMBSRb6fi/qaKnbD5UyhZ3VDO64JFXV/95RBLbkddZm/KckaxtVNXVtV1ZlFVFp+WplKy/d+3yU1LiqdXThpD+6ZxaU4avXPS6JOTTtfMNppBi8h3puKu2AWbPoZNc4NvpeuCH3c+rN0RVLftRXHPfDa5rqyo6cj8iiyW7PZRudf/5Zdol5JAn5w0Th2YTp//K+iOaXTKSNbZHCLS7FpfcVeWfKWoPw4udwD+xHR2tx/G2h7jWBDox8flOazZXUv1jn+f5ZGTnkTvnDQuHBpcf+7dsMzRITVRBS0iERP7xV21+8vZtL/wI+JK1wBQ40thTdIAPk28kreq+rC8+gj8+4JnaXRt14beOWlc3vv/1p/T6J2dTtsUXVUoIt6LueK2qt3sXTWH/es+JGXbp2RWrgdgP8ks9PdlfmAi8wL9WevrRff0DHrlpDI6O42rc9LolZ1Gz+xUUhJj7q9FRGJIKFuX/QMYDxSb2YDGxkdKbX2ALWVVbC4qombDXNK2z6PbvsX08G8kE0iyJAoCR7I4bhI7ModD52Po0bEd+dnBFwq7ZbYhXmdwiEgUCmVqORX4M/B0eKMcXEV1HRtKqlhfXMmGkkq279hBevFCelUuYYRbySluCz5nVJPI+qSjmZ0zmbruJ9K213D6dW7PqLQkrT+LSEwJZeuyj5xzeeEO4g8Y8wtL2VBS+WVJry+uZP++PQz3reY430rG+VbR37cJH0ZdfBJ7OhxD8REXknHUWFLyhjMgPokW8yuBiEiYNNtirnNuMjAZoHv37k3/88B1TxUQV1fB6KT1XJSylnzfF3RNXoePAOZLxLoNx9djIuSNIqHrMHISkpsrvohI1Gi24jazKcAUgPz8fGtk+H/w+Wso6PggKaXLcRaAmgTIHQ49zoO8E3G5w3EJbZorrohI1Go5p08kJJPapR/0Px3yToTcEZCY4nUqEZEWp+UUN8AFU7xOICLS4jV6Ppxz7jlgHtDXOVfknLs2/LFERORQQjmrZFIkgoiISGh0BYqISJRRcYuIRBkVt4hIlFFxi4hEGRW3iEiUUXGLiEQZZ9bkq9Mb/6LOlQCbv+MfzwJ2N2OcaKBjjn2t7XhBx9xUR5hZdigDw1Lch8M5V2Bm+V7niCQdc+xrbccLOuZw0lKJiEiUUXGlv59HAAADoklEQVSLiESZlljcrfFOUzrm2Nfajhd0zGHT4ta4RUTk27XEGbeIiHwLz4rbOXemc26Nc269c+7ug3w+yTn3QsPnF0Ri38twCuF4v++cW+mcW+acm+2cO8KLnM2psWP+yrgLnXPmnIv6MxBCOWbn3CUN3+svnHPPRjpjcwvhZ7u7c+4D59yShp/vs73I2Vycc/9wzhU751Yc4vPOOfenhr+PZc65oc0ewswi/gbEARuAnkAi8DnQ/xtjbgb+2vB4IvCCF1kjeLynACkNj2+K5uMN9ZgbxqUDHwHzgXyvc0fg+9wHWAJkNryf43XuCBzzFOCmhsf9gU1e5z7MYx4NDAVWHOLzZwNvEtxKdySwoLkzeDXjHgGsN7NCM6sFngfO/caYc4GnGh5PB8Y651wEMzanRo/XzD4ws/0N784HciOcsbmF8j0GeAB4CKiOZLgwCeWYrwceNbM9AGZWHOGMzS2UYzYgo+FxW2B7BPM1OzP7CCj7liHnAk9b0HygnXOuc3Nm8Kq4uwJbv/J+UcPHDjrGzOqBcqBDRNI1v1CO96uuJfh/7GjW6DE3/ArZzcxej2SwMArl+3wkcKRz7hPn3Hzn3JkRSxceoRzzz4DLnXNFwBvArZGJ5pmm/ntvspa156TgnLscyAdO8jpLODnnfMAfgKs8jhJp8QSXS04m+FvVR865gWa219NU4TUJmGpmv3fOHQc845wbYGYBr4NFK69m3NuAbl95P7fhYwcd45yLJ/grVmlE0jW/UI4X59ypwH3ABDOriVC2cGnsmNOBAcAc59wmgmuBM6P8BcpQvs9FwEwzqzOzjcBagkUerUI55muBFwHMbB6QTPCeHrEqpH/vh8Or4v4M6OOc6+GcSyT44uPMb4yZCVzZ8Pgi4H1rWPmPQo0er3NuCPA3gqUd7eue0Mgxm1m5mWWZWZ6Z5RFc159gZgXexG0Wofxcv0Jwto1zLovg0klhJEM2s1COeQswFsA5dxTB4i6JaMrImglc0XB2yUig3Mx2NOszePjK7NkEZxsbgPsaPvYLgv94IfjNnQasBxYCPb1+NTnMx/sesAtY2vA20+vM4T7mb4ydQ5SfVRLi99kRXCJaCSwHJnqdOQLH3B/4hOAZJ0uB073OfJjH+xywA6gj+BvUtcCNwI1f+R4/2vD3sTwcP9e6clJEJMroykkRkSij4hYRiTIqbhGRKKPiFhGJMipuEZEoo+IWEYkyKm4RkSij4hYRiTL/H00vy20vYBloAAAAAElFTkSuQmCC\n",
"text/plain": "<Figure size 432x288 with 1 Axes>"
},
"metadata": {},
"output_type": "display_data"
}
]
},
{
"metadata": {
"trusted": false
},
"cell_type": "code",
"source": "# function interpolation by solving polynomial basis vandermonde\n# 1 x x^2 x^3 ...\nx = np.linspace(0,1,10)\ny = 3*x**5+2*x**4+1.5*x**3 + 1 - 0.5*x**2 + 2*x\nN = 3\nV = np.array([[x_i**i for i in range(0,3)] for x_i in x])\ny = np.transpose(np.matrix(y))\n# lets get the coefficients V b = y \nb = la.solve(np.matmul(np.transpose(V),V), np.transpose(V)*y)\nf = lambda x: b[0,0] + b[1,0]*x + b[2,0]*x**2\ny_hat = np.array([f(x_i) for x_i in x])\n# lets see\nplt.plot(x,y)\nplt.plot(x,y_hat)\nprint(b)",
"execution_count": 65,
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": "[[ 1.42295382]\n [-3.8753315 ]\n [10.90432099]]\n"
},
{
"data": {
"image/png": 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wcgqc/SsYcP1Rh0orqrjr7RU0Dg/l+fEDCAtVHHlKM24R8Y51s+DzP0HvMXDB7486ZK3l93PWkZ5TzJu3DKFlXKSfigxM+hEnInVv51J4705od6bTIjnmVqzT03Yxe8UefnFBCmen6DbQtaXgFpG6tX8rvDse4trAuKkQfvRsekNWIX94fz1nJSfwi+EpfioysCm4RaTulO6Ht8eCdcGEmRAVf9ThorJK7p66grjG4fxjXH9CQ7QpwulQj1tE6kZVOUy73rnQ5sb3Ib7LUYettTw4ay0795cy9bYzSIhu5KdCA59m3CLy41kLc++BHd/CVS9Ch2HHDZmyZAf/W7uXX13UjTM6x9fwQcRTHs24jTHbgSKgGqiy1ure3CLyg4WPw5ppcP7voe/Y4w6v3nWQP8/bwAXdk7j9nM5+KLBhqU2r5HxrbZ7XKhGRwLTqHfjyceg/Ac751XGHC0qdvnZSTCRPj+1HiPraP5p63CJy+rZ97bRIOp0Dl//juGV/1loemLGa7MIypt8+lGZREX4qtGHxtMdtgfnGmOXGmEneLEhEAsS+tTBtAjTvDNdOgbDjQ/k/X2/ls43ZPDSyBwPaN/NDkQ2TpzPus6y1e4wxScCnxphN1tqvjhzgDvRJAO3bt6/jMkWkXtmzHKaMhohomDAdGjc9bkja9v088fFmRvZuycSfdPR9jQ2YRzNua+0e9585wHvAkBrGTLbWplprUxMTdSWUSIO1cwm8eTVExsHED6FZx+OG5BeX8/OpK2nbrDFPjOmLMepr16VTBrcxJsoYE/P9Y+AiYJ23CxORemjrlzBllHM/7Ykf1RjaLpflvmmr2F9awQvXDSQ2Mtz3dTZwnrRKWgDvuX9ihgFTrbUfe7UqEal/0j9zetrNOjkX2MTUfO/sFxZk8HV6Ho+N6kPvNnE+LjI4nDK4rbVbgX4+qEVE6quN82DGzZDUA26Yc9yl7N9blJHHs59t4er+rRk/pJ1vawwiunJSRE5u3SxnB5tW/Zxtx04Q2jmFZfzi3VV0Sojir6P6qK/tRQpuETmxVVOdXdnbnwk3zqlx9QhAVbWLe95ZSUl5FS9dP4ioRrpExJsU3CJSs7RXYc6dzsU1E2ZCo5gTDn32sy0s3bafv47qTdcWJx4ndUPBLSLHW/yis1dkysUwfhpEnHjn9QWbc3hhQSbjBrdj9MC2PiwyeCm4ReRoXz8NnzwEPa6En7513EYIR8o6eIj7p62ie8sYHrmylw+LDG5qRImIw1pY8Bh89XfoMxaufhlCTxwR5VXV/HzqCqqqLS9OGEhkeKgPiw1uCm4RcUL70/+DRf+EATfAFc9ByImDuKC0kklT0lix8yD/um4AnROjfVisKLhFgp3LBR/9Bpb9Bwb/DEb+HUJO3EXdtb+Uia8vY2d+Kc+N68/lfVv7sFgBBbdIcHNVwwf3wsopMOweGPHn427NeqS1uwu45Y1llFdW8+atQzhTO9n4hYJbJFhVV8GcO2DtDDjnN3D+wycN7QWbcrh76gqaNYlg6m1nkKJlf36j4BYJRlUVMOtW2DgXhv8Bzn7gpMOnLt3J/72/ju4tY3jt5sEkxZ54pYl4n4JbJNhUljmXsKd/Ahf/DYbedcKh1lqemr+ZFxZkcl63RF64bqCuiqwH9BUQCSYVJfDudbB1IVz+LKTecsKh5VXV/HbmGuasymL8kHb8+arehIXq0o/6QMEtEizKi+Dta2HXErj6Jeh/3QmHFhyq5PYpaSzZup9fX9yNu87roptG1SMKbpFgcOggvHUNZK2Ea16B3teccOieg4e4+dXv2J5fwrM/7ceoAbqMvb5RcIs0dCX5MOVqyNkI174JPS4/4dB1ewq45fVlHKqs5o1bhjCsS4IPCxVPeRzcxphQIA3YY6098VdeROqPomx48yo4sA3GvwspF55w6MLNOdz99griGocz685hustfPVabGfe9wEYg1ku1iEhdKtgDb14JhVlw3XTofO4Jh7773U5+N2cd3VrE8NrEwbTQcr96zaOXiI0xbYHLgFe8W46I1IkDO+C1kc6M+4b3Thja1lqenr+ZB2ev5SfJCUy/Y6hCOwB4OuP+B/AbQL87idR3+ZnwxhXO0r+b3oc2g2ocVlHl4sFZa5i9cg/jBrfjz1f3JlzL/QLCKYPbGHM5kGOtXW6MOe8k4yYBkwDat29fZwWKSC3kbHLaI64quHketOxT47DCskrumLKcRZn5/Oqirtx9frKW+wUQT368/gS40hizHXgXuMAY89axg6y1k621qdba1MTExDouU0ROae8aeP1SwMDNH54wtLMOHmLsS4v5btt+nrm2Hz+/IEWhHWBOGdzW2oestW2ttR2BccAX1trrvV6ZiHhu93J443IIawwTP4Sk7jUOW59VwKgXvyXr4CHeuGWIthoLUFrHLRLodiyGt8dCVDzcOBeadahx2JdbcrnrreXENg5nxp1D6d5SC8QCVa2C21q7EFjolUpEpPa2LoR3xkNsG7hpLsTWvKnB9GW7eOi9tXRt4dzdr2WcVo4EMs24RQLVlvkw7XqIT4Yb50B00nFDrLU8+1k6z3+eztkpCbw4YSAxkeF+KFbqkoJbJBBt/ABmTIQWPeGGOdCk+XFDKqpcPDR7LbNW7GbsoLY8NrqPlvs1EApukUDicsE3z8CCvzrrsyfMhMZNjxtWWFbJXW+t4JuMPH45oiv3XKDlfg2JglskUJTuh/fucDZA6H0NXPE8NDp+d/W9BYeY+NoyMnKKeWpsP8YM0sqRhkbBLRII9qyAGTdB4V4Y+SQM+VmN+0Nu3FvIxNeWUVxexesTh3BWiu7u1xApuEXqM2th+Wvw0W8hKglu+RjaptY49Ov0XO58awXRjcKYccdQerTScr+GSsEtUl9VlMC8+2HNNEi+EEb/p8YXIQFmpO3iodlrSU6K5rWJg2kV19jHxYovKbhF6qO8dJh2A+RugvN/B2f/CkKOXxFireX5zzN49rMtWu4XRBTcIvXNutkw9x4IawQ3zIYuF9Q4rLLaxcOz1zJj+W7GDGrL37TcL2gouEXqi6oK+PT/YOnL0HYIjH0d4trUOLSorJK73l7B1+l53Ds8hfsu1I2igomCW6Q+KNgNM26G3cvgzLtgxKMQWnPLY/mO/Tw8ex2ZucX8fUxfrk1t59taxe8U3CL+lvEZzPoZVFfC2Deg19U1Dttz8BBPfLSJuauzaBHbiNcmDubsFN1CORgpuEX8xVUNX/4dvnwCknrAtVMgIfm4YaUVVby8MJN/f7UVgF9ckMzt53YhqpH++wYrfeVF/KEkH2bfBplfQL/xcNkzENHkqCEul2XOqj088fEmsgvLuaJfax4c2Z02TbXUL9gpuEV8bdcy5yrIkjy44jkYeNNxV0Eu37GfRz/YwOrdBfRrG8eLEwYyqEPNa7gl+Ci4RXzFWvhuMnzyO+e+2bfOh9b9jxpybB/7mWv7cXX/NoSEaMWI/MCTzYIjga+ARu7xM621f/R2YSINSnmRszZ7/XvQdSSMegkaNzt8uLSiipe/3MrkrzKxVn1sOTlPvivKgQustcXGmHDgG2PMR9baJV6uTaRhyNnoXAW5PxMu/BMM+8XhqyBr6mP/9pJutG3W5BQfVILZKYPbWmuBYve74e43682iRBqM1dNg3n0QEQ03fQAdzzp8aPmOAzw6bwOrdx2kb9s4XrhuIKkd1ceWU/Po9zBjTCiwHEgGXrDWLq1hzCRgEkD79u3rskaRwFNZBh8/6NzZr8NPYMyrENMSgKyDh3jc3cdOimnE02P7MWqA+tjiOY+C21pbDfQ3xjQF3jPG9LbWrjtmzGRgMkBqaqpm5BK8DuyA6TfC3lXwk/vggv+D0LDj+tj3XJDMHepjy2mo7S7vB40xC4BLgHWnGi8SdLZ8ArMnOStIxr0D3S/F5bK8v3I3T3y0mX2FZVzetxUPjuyuPracNk9WlSQCle7QbgyMAJ7wemUigaS6ChY+Bl8/DS37wrVvQvNOrNh5gEc/2MAqdx/7X9cNUB9bfjRPZtytgDfcfe4QYLq1dp53yxIJIMU5MPMW2P61czHNyL+TVWJ54t2VvL/K6WM/NbYfo9XHljriyaqSNcAAH9QiEnh2LIIZE6GsAK5+idKe1/LyQvWxxbv03SRyOqyFRf+Ezx6BZh1xTZjJ3H3NefypL9XHFq9TcIvUVlkBzLkLNs2DnlexauBfeGT2Llbt2qk+tviEglukNvaucZb6Feyi4JxH+UP22bz/ylr1scWnFNwinijdDwsfh2Wv4IpOYnqvl3lkQTQum83Pz0/mzvPUxxbf0XeayMlUV8Ky/8LCv2HLC8lsN5a7945k87IILu/bQn1s8QsFt0hNrIX0T7GfPIzJT2dzVCq/rv4pa7a0oU+bOGZM6Mlg9bHFTxTcIsfK2cSheb+l8c6F7DKteaTiV3xnB3N5v9b8YVBbBnVoph3Vxa8U3CJuxQey2Tvnj3TeMY0KG8mT1TewteM4RqV24oWeLWkcEervEkUABbcEOZfLsiR9LzlfvMD5+16jE4f4IOIS8gf/ktuG9Ka19neUekjBLUFpe14Js5bvYm/aXO4sf5VhIXvZEjOEygv/wlX9hqgVIvWagluCRmFZJR+u2cvM5bsp2LmW/wufwjkhaymK7UTFZdPo2v3i4zbtFamPFNzSoFW7LN9m5DFrxW4+XrePJlUHeSRmLlc0+hgaxcD5jxMz+DYIDfd3qSIeU3BLg5SZW8ys5buZvWIP+wrLaB4Jz7ZfzMV5bxBSWYIZchuc9xA00ZI+CTwKbmkwCkor+WBNFrNW7GblzoOEGDg3JYF/DtrHoM1PE5KVCckXwkV/haTu/i5X5LQpuCWgVVW7+Dojj1nLdzN/QzYVVS66tojm4Uu7c02bAuK//RMsXggJXWHCTEgZ4e+SRX40BbcEpPTsImau2M17K/aQU1RO0ybhjB/cjjGD2tG7aQVm4WOw4HVoFAsj/w6pt6iPLQ2GJ1uXtQPeBFoAFphsrX3O24WJHOtgaQVzV2cxa/luVu8uIDTEcH63RMYMasv53ZNoRDV892+Y8iRUFMOQSXDub9XHlgbHkxl3FfCAtXaFMSYGWG6M+dRau8HLtUmQq3ZZ1mcVsDgzn8Vb81mUkU9FtYvuLWP4/WU9uKp/GxJjGjn3Fdn8Icz/PezfCikXwUV/gcRu/j4FEa/wZOuyvcBe9+MiY8xGoA2g4JY65XJZNu0rYvHWfBZn5rN0Wz5FZVUAdE6M4vozO3DNoDb0ah33w1/atw4+eRi2fQkJ3WDCLEi50E9nIOIbtepxG2M64uw/ubSGY5OASQDt27evg9KkobPWkpFTzKLMH4L6QGklAB3im3BZn1YM7RLPmZ3jaREbefRfLs6FBX+FFW9AZByMfBJSJ6qPLUHB4+A2xkQDs4D7rLWFxx631k4GJgOkpqbaOqtQGgxrLdvySg7PqJds3U9ecTkAbZo2ZniPFgztHM/QLvEnvkdIVTks/Td89SRUlsKQ2+Hc36iPLUHFo+A2xoTjhPbb1trZ3i1JGpJd+0tZnJnPosw8lmzdz77CMgBaxDbirGQnpId2TqBd88Ynvz+ItbDpf04f+8A2SLnY3cfu6qMzEak/PFlVYoD/Ahuttc94vyQJZFkHDx1+MXFxZj57Dh4CICE6gjM6xzOsSzxDO8fTKSHKsxs5le6HdbNg5VuwdxUkdofrZzkX0ogEKU9m3D8BbgDWGmNWuZ972Fr7offKkkCRU1Tmbnvksygznx35pQA0bRLOmZ3imXROZ4Z2iSclKdrzO+5VV0Hm57Dqbdj8EVRXQIs+cPmzMOBGCNXlBxLcPFlV8g2gW6YJAPnF5SzZup/FW/NYnJlPZm4JADGRYZzRqTk3nNmBYV0S6N4ypva7nWevh1VTYc10KMmBJvEw+DboNx5a9fXC2YgEJk1d5KRyispYufPg4Vn1pn1FAERFhDK4U3OuTW3H0C7x9GodR2htgxqgJB/WzXRm13tXQ0gYdL0E+l8HySMgLKKOz0gk8Cm4BXDWUO/cX8r6rELWZxW4/yw8vOojMjyE1A7N+fXFrRnaJZ4+beIIDw05vU9WXQnpnzphveUTcFVCy75wyRPQZwxEJdThmYk0PAruIFTZayLCAAAKkUlEQVRR5SI9p4j1WYVscAf1xr1FFJc7F7uEhRiSk6I5t2sivVrH0qdtHH3bxtEo7EfuubhvLax6B9ZMg9I8iEqEM253WiEte9fBmYkEBwV3A1dcXsXGvYWs3/PDLDo9p4jKamepfZOIUHq0imX0wDb0ah1Lr9ZxpLSI/vEh/b2SPFg7w5ld71sLIeHQbST0nwDJw3XBjMhpUHA3IDlFZe4Z9A8z6e3uVR4A8VER9GwdyzldO7tDOpYO8VGn15s+maoKSJ/vvNCY/gm4qqD1AOfqxj5jdLGMyI+k4A5A3/ejN+w9uh+dW1R+eEy75o3p1SqOawa2pVcbZyadFNPIe5vgWgv71jhhvXYGlOZDdAs4807odx206OmdzysShBTc9dz3/egjZ9Ib9hYe7keHhhhSkqI5OyWBXq3j6NU6lh6tYolr7KMWRHGOs3xv9TuQvQ5CI6DbpU4rpMsFWnMt4gX163/VlvnQvBM07wIhp7liIUBVuyy79peSnlNMek4RGdnFbM4uIj27mIpqFwCNw0Pp0SqGUQOO7kdHhtdRP9pTVeWw5WPnhcb0+WCroc0guOxp6DVarRARL6s/wV1dCdNvgKoyiGzqBEHbVGg72HncQMKgstrFjvwS0rOL3SFdTHp2EVvzSqioch0e1youkuSkaCae1fHwTLqjN/rRnrLWueT8+1bIoQMQ3RKG3eOsuda9r0V8pv4EtwmFSQthdxrsXgZ7ljt3gLPuMGve2R3iqU6gt+hdry/OKK+qZlveDwGdkePMnrfllVDl+uHmiW2bNSYlKZpzuiaSnBRNSlI0yUnRxETWg9UWVeWQuxm2LnRaITkbILQRdL/MaYV0Pk+tEBE/MNbW/R1YU1NTbVpa2o//QOXFkLUS9qS5Az0Nivc5x0IbQat+Tpi3HeQEetP24K0X307gUEU1mbnFZLhbHOnZzuPt+SV8n88hBjrERx0O5pQW0aQkxdA5MYomEfUg+KyFgzudYM5e77zlbIC8dKcNAs6/c//roNcoaNzMv/WKNEDGmOXW2lSPxtbr4D6WtVC4x5mR705zZuVZK532CkBUkjMbbzPIPTsfCI1i6uRTF5dXOeGcXURGzvdBXcyuA6V8/08YFmLomBDlhHNSNMktYkhJiqZTQpTv+9Ancujg8QGdsxHKj7jFetP2zm80ST2d1SCtBzi/8YiI1zTc4K5JdaUTQN+3V3anQX66+6CBpB5H98sTu0PIiUO0oLSSjNyio3rQGdlFZBWUHR4TERpC58TvZ9Ax7hl0NB3io4gIqycvqlZVOP8O2Rsgxx3S2RugcPcPYyLjjg7oFr2df5/IWP/VLRKkgiu4a3LogDvEl7sDPc15DiAiGtu6PyWJA9jZuCdrSGZtQWMycorJzC0mr7ji8IeJDA85HM5H9p/bN29C2Onep6OuWQuFWe7Z8xEBnbfFuQcIOFcrJnSFFr2cgE7q5TyObe3z1pKI1EzB7VZV7WLH/lIysovI3bkJs3sZcftX0/HQBrqxg3Dj9G+zSGBbo+4caNaX6tapxHVOpUvrRNo0bVz7W5N6U1mh09bIcYfz92FdVvDDmNi2xwd0fHK9fiFXRGoX3J7sgPMqcDmQY62tl3cCKq2oYmtuyeHec0ZOMRm5xezILzl8Tw6AlrGDSE4615k9x4fTN3QHHcvW0yp3Na33pMG+b2AfsDIEwiKdmWpomPvPcOeWo6Hhp3i+Lsa536+ugtxN7p70OucFxO9FxDih3Psad6ujt9MWatzU918AEfEpT5Y0vA78C3jTu6Wc2v6SiuPCOTOn+PD2WOBcSdiheRO6JEUzomcLkhOj6ZIUTZfEqBqW2HUFRvzwbnGO0yPfuxoqip17bFRXOi0HV/UPj6srjz5WXeW8QFrT84fHH/M+Hv6mY0IhIcXpzw+8yT2b7gVx7dTmEAlSnuyA85UxpqP3S3G4XJasgkOHwzkzt5jMnBIycovZX3J0/7lLYjSpHZsxLrEdXdz95w7xTU7/znbRSdD9UufN2071g8BVCRjnStKwRt6vR0QCRj1YROyoqnYx+qVFpGcXc6iy+vDzzZqEk5wUzcW9WtDFPXtOToyuf/3n2goJda9uifR3JSISYOosuI0xk4BJAO3bt699IaHODHpQh2Yku8M5OSma+GjNNkVEjuTRqhJ3q2Sepy9O1pdVJSIigaI2q0rqyWJkERHx1CmD2xjzDrAY6GaM2W2MudX7ZYmIyIl4sqpkvC8KERERz6hVIiISYBTcIiIBRsEtIhJgFNwiIgFGwS0iEmC8cltXY0wusOM0/3oCkFeH5QQCnXPDF2znCzrn2upgrU30ZKBXgvvHMMakeXr1UEOhc274gu18QefsTWqViIgEGAW3iEiAqY/BPdnfBfiBzrnhC7bzBZ2z19S7HreIiJxcfZxxi4jISfgtuI0xlxhjNhtjMowxD9ZwvJExZpr7+FJfbp/mDR6c7y+NMRuMMWuMMZ8bYzr4o866dKpzPmLcNcYYa4wJ+BUInpyzMeZa99d6vTFmqq9rrGsefG+3N8YsMMasdH9/+2BvQO8xxrxqjMkxxqw7wXFjjHne/e+xxhgzsM6LsNb6/A0IBTKBzkAEsBroecyYu4CX3Y/HAdP8UasPz/d8oIn78Z2BfL6enrN7XAzwFbAESPV33T74OqcAK4Fm7veT/F23D855MnCn+3FPYLu/6/6R53wOMBBYd4LjlwIfAQY4E1ha1zX4a8Y9BMiw1m611lYA7wJXHTPmKuAN9+OZwHBjAnZb81Oer7V2gbW21P3uEqCtj2usa558jQH+DDwBlPmyOC/x5Jx/BrxgrT0AYK3N8XGNdc2Tc7ZArPtxHJDlw/rqnLX2K2D/SYZcBbxpHUuApsaYVnVZg7+Cuw2w64j3d7ufq3GMtbYKKADifVJd3fPkfI90K85P7EB2ynN2/wrZzlr7P18W5kWefJ27Al2NMd8aY5YYYy7xWXXe4ck5PwJcb4zZDXwI3OOb0vymtv/fa63e7PIuDmPM9UAqcK6/a/EmY0wI8Axws59L8bUwnHbJeTi/VX1ljOljrT3o16q8azzwurX2aWPMUGCKMaa3tdbl78IClb9m3HuAdke839b9XI1jjDFhOL9i5fukurrnyflijLkQ+B1wpbW23Ee1ecupzjkG6A0sNMZsx+kFzg3wFyg9+TrvBuZaayuttduALThBHqg8OedbgekA1trFQCTOPT0aKo/+v/8Y/gruZUCKMaaTMSYC58XHuceMmQvc5H48BvjCujv/AeiU52uMGQD8Gye0A73vCac4Z2ttgbU2wVrb0VrbEaevf6W1Ns0/5dYJT76v5+DMtjHGJOC0Trb6ssg65sk57wSGAxhjeuAEd65Pq/StucCN7tUlZwIF1tq9dfoZ/PjK7KU4s41M4Hfu5x7F+c8Lzhd3BpABfAd09veryV4+38+AbGCV+22uv2v29jkfM3YhAb6qxMOvs8FpEW0A1gLj/F2zD865J/AtzoqTVcBF/q75R57vO8BeoBLnN6hbgTuAO474Gr/g/vdY643va105KSISYHTlpIhIgFFwi4gEGAW3iEiAUXCLiAQYBbeISIBRcIuIBBgFt4hIgFFwi4gEmP8He+zSTgLW4CEAAAAASUVORK5CYII=\n",
"text/plain": "<Figure size 432x288 with 1 Axes>"
},
"metadata": {},
"output_type": "display_data"
}
]
},
{
"metadata": {
"trusted": false
},
"cell_type": "code",
"source": "# function interpolation by solving exponential basis vandermonde\n# 1 x x^2 x^3 ...\nx = np.linspace(0,1,10)\ny = 1.5*x**3 + 1 - 0.5*x**2 + 2*x\nN = 4\nV = np.array([[np.exp(x_i)**i for i in range(0,4)] for x_i in x]) # change made here\ny = np.transpose(np.matrix(y))\n# lets get the coefficients V b = y \nb = la.solve(np.matmul(np.transpose(V),V), np.transpose(V)*y)\nf = lambda x: b[0,0] + b[1,0]*np.exp(x) + b[2,0]*np.exp(x)**2 + b[3,0]*np.exp(x)**3 # change made here\ny_hat = np.array([f(x_i) for x_i in x])\nplt.plot(x,y)\nplt.plot(x,y_hat)",
"execution_count": 77,
"outputs": [
{
"data": {
"text/plain": "[<matplotlib.lines.Line2D at 0x1142dbc50>]"
},
"execution_count": 77,
"metadata": {},
"output_type": "execute_result"
},
{
"data": {
"image/png": 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\n",
"text/plain": "<Figure size 432x288 with 1 Axes>"
},
"metadata": {},
"output_type": "display_data"
}
]
},
{
"metadata": {
"trusted": false
},
"cell_type": "code",
"source": "# numerical stability point\nx = np.linspace(0,1,4)\ny = 1.5*x**3 + 1 - 0.5*x**2 + 2*x\nV_n = np.array([[np.exp(x_i)**i for i in range(0,4)] for x_i in x]) # change made here\nV_e = np.array([[x_i**i for i in range(0,4)] for x_i in x])\n# look at the condition number\nprint(la.norm(la.inv(V_n),2)*la.norm(V_n,2))\nprint(la.norm(la.inv(V_e),2)*la.norm(V_e,2))",
"execution_count": 87,
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": "1315.566624409681\n98.86773850722757\n"
}
]
}
],
"metadata": {
"kernelspec": {
"name": "python3",
"display_name": "Python 3",
"language": "python"
},
"language_info": {
"mimetype": "text/x-python",
"nbconvert_exporter": "python",
"name": "python",
"pygments_lexer": "ipython3",
"version": "3.5.4",
"file_extension": ".py",
"codemirror_mode": {
"version": 3,
"name": "ipython"
}
}
},
"nbformat": 4,
"nbformat_minor": 2
}
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