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{
"metadata": {
"kernelspec": {
"display_name": "Python 3 (ipykernel)",
"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": "3.10.12"
}
},
"nbformat": 4,
"nbformat_minor": 4,
"cells": [
{
"cell_type": "markdown",
"metadata": {
"lang": "en"
},
"source": [
"## The gradient method to solve A x = b\n",
"\n",
"The goal of this lab is to let you move forward on your own. Look at previous course and program the gradient method\n",
"to solve the $A {\\bf x} = {\\bf b}$ matrix system with A symmetric and diagonally dominant\n",
"($a_{ii} > \\sum_{k \\ne i} |a_{ik}|$).\n",
"\n",
"* Start in 2D with a 2x2 matrix, check that the result is good and plot the convergence curve\n",
"* Switch to nxn (we will show that it works with a 9x9 matrix)\n",
"\n",
"It may be interesting to normalize the matrix A to prevent the calculations from exploding."
]
},
{
"cell_type": "code",
"execution_count": 33,
"metadata": {},
"outputs": [],
"source": [
"def solve_with_gradiant(A, b):\n",
" mu = 0.01\n"
]
},
{
"cell_type": "code",
"execution_count": 34,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"array([[49, 13],\n",
" [13, 13]])"
]
},
"execution_count": 34,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"import numpy as np\n",
"A = np.random.randint(10, size=(2,2))\n",
"A = A + A.T\n",
"A += np.diag(A.sum(axis=1))\n",
"A"
]
},
{
"cell_type": "code",
"execution_count": 35,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"array([62, 26])"
]
},
"execution_count": 35,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"b = A.sum(axis=1)\n",
"b"
]
},
{
"cell_type": "code",
"execution_count": null,
"metadata": {},
"outputs": [],
"source": [
"solve_with_gradiant(A,b)"
]
},
{
"cell_type": "markdown",
"metadata": {
"lang": "en"
},
"source": [
"## Introduce inertia\n",
"\n",
"Introduce inertia into the gradient method. What do we notice?"
]
},
{
"cell_type": "code",
"execution_count": null,
"metadata": {},
"outputs": [],
"source": []
},
{
"cell_type": "markdown",
"metadata": {
"lang": "en"
},
"source": [
"## Optimal value of µ\n",
"\n",
"Note that two successive slope directions are orthogonal if the next point is the minimum in\n",
"given direction ($\\nabla J ({\\bf x}^k$)).\n",
"\n",
"$$\\nabla J ({\\bf x}^{k+1})^T \\; \\nabla J ({\\bf x}^k) = 0$$\n",
"\n",
"#### Demonstration\n",
"\n",
"\n",
"<table><tr>\n",
" <th>\n",
"We want to adjust µ to arrive at the minimum of J when we advance in the direction $- \\nabla J({\\bf x}^{k})$.\n",
"This means that the partial derivative of $J({\\bf x}^{k+1})$ with respect to µ must be\n",
"equal to 0 or by making µ appear in the equation:\n",
"\n",
"$$\n",
"\\frac{\\partial J ({\\bf x}^k - µ \\; \\nabla J ({\\bf x}^k))}{\\partial µ} = 0\n",
"$$\n",
"\n",
"By expanding we have\n",
"\n",
"$$\n",
"\\begin{align}\n",
"\\frac{\\partial J ({\\bf x}^{k+1})}{\\partial {\\bf x}^{k+1}} \\; \n",
"\\frac{\\partial {\\bf x}^{k+1}}{\\partial µ} & = 0 \\\\\n",
"J'({\\bf x}^{k+1}) \\, . \\, (- \\nabla J ({\\bf x}^k)) & = 0 \\\\\n",
"(A\\, {\\bf x}^{k+1} - b) \\, . \\, \\nabla J ({\\bf x}^k) & = 0 \\quad \\textrm{since A is symetric}\\\\\n",
"\\nabla J ({\\bf x}^{k+1}) \\, . \\, \\nabla J ({\\bf x}^k) & = 0 \\quad \\textrm{QED}\n",
"\\end{align}\n",
"$$\n",
"\n",
"\n",
"</th><th>\n",
"<img src=\"images/gradient.png\" width = \"400px\"/>\n",
"</th></tr></table>\n",
"\n",
"Using this property, evaluate the optimal value of µ to reach the minimum in the direction of\n",
"$\\nabla J ({\\bf x}^k)$.\n",
"\n",
"Write the gradient method with the calculation of the optimal µ at each iteration to solve $A {\\bf x} = {\\bf b}$."
]
},
{
"cell_type": "code",
"execution_count": null,
"metadata": {},
"outputs": [],
"source": []
}
]
}
|
