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151 lines
3.3 KiB
Plaintext
151 lines
3.3 KiB
Plaintext
{
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"cells": [
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{
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"cell_type": "code",
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"execution_count": 1,
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"metadata": {},
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"outputs": [],
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"source": [
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"import numpy as np\n",
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"import scipy as sp"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"# Arnoldi"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"Defined as Algorithm 2 in the paper."
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]
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},
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{
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"cell_type": "code",
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"execution_count": 2,
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"metadata": {},
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"outputs": [],
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"source": [
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"def Arnoldi(A,v0,m):\n",
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" v = v0\n",
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" beta = sp.linalg.norm(v)\n",
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" v = v/beta\n",
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" H = sp.sparse.lil_matrix((m+1,m)) \n",
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" V = sp.sparse.lil_matrix((A.shape[0],m+1))\n",
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" V[:,0] = v # each column of V is a vector v\n",
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"\n",
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" for j in range(m):\n",
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" w = A @ v \n",
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" for i in range(j):\n",
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" H[i,j] = v.T @ w # tmp is a 1x1 matrix, so it's O(1) in memory\n",
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" w = w - H[i,j]*v \n",
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" \n",
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" H[j+1,j] = np.linalg.norm(w)\n",
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"\n",
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" if H[j+1,j] == 0:\n",
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" print(\"Arnoldi breakdown\")\n",
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" m = j\n",
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" v = 0\n",
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" break\n",
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" else:\n",
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" if j < m-1:\n",
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" v = w/H[j+1,j]\n",
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" V[:,j+1] = v\n",
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"\n",
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" return V, H, beta, j "
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"## Small test case\n",
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"\n",
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"The final implementation will be using all sparse arrays and matrices, no numpy arrays"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 3,
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"metadata": {},
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"outputs": [],
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"source": [
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"n = 110\n",
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"\n",
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"# start with a random sparse matrix\n",
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"A = sp.sparse.rand(n,n, density=0.1, format='lil')\n",
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"\n",
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"# Starting vector\n",
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"v = np.repeat(1/n,n)"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"Now we can run the Arnoldi Process"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 4,
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"metadata": {},
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"outputs": [
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{
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"name": "stdout",
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"output_type": "stream",
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"text": [
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"The number of iterations is: 109\n",
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"The matrix H is: (111, 110)\n",
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"The matrix V is: (110, 111)\n"
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]
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}
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],
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"source": [
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"V, H, beta, j = Arnoldi(A,v,110)\n",
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"print(\"The number of iterations is: \", j)\n",
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"print(\"The matrix H is: \", H.shape)\n",
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"print(\"The matrix V is: \", V.shape)"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"As we can see, it returns $H_{m+1}$ that is an upper-hassemberg matrix, with one extra row."
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]
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}
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],
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"metadata": {
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"kernelspec": {
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"display_name": "Python 3.10.8 64-bit",
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"language": "python",
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"name": "python3"
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},
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"language_info": {
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"codemirror_mode": {
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"name": "ipython",
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"version": 3
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},
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"file_extension": ".py",
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"mimetype": "text/x-python",
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"name": "python",
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"nbconvert_exporter": "python",
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"pygments_lexer": "ipython3",
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"version": "3.10.8"
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},
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"orig_nbformat": 4,
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"vscode": {
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"interpreter": {
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"hash": "e7370f93d1d0cde622a1f8e1c04877d8463912d04d973331ad4851f04de6915a"
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}
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}
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},
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"nbformat": 4,
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"nbformat_minor": 2
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}
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