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ShfitedPowGMRES/algo4_testing.ipynb

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{
"cells": [
{
"cell_type": "code",
"execution_count": 2,
"metadata": {},
"outputs": [],
"source": [
"import numpy as np\n",
"import networkx as nx\n",
"import time\n",
"import math\n",
"import pandas as pd\n",
"import scipy as sp\n",
"import plotly.express as px\n",
"import plotly.graph_objs as go\n",
"from scipy.sparse import *\n",
"from scipy import linalg\n",
"from scipy.sparse.linalg import norm\n",
"from scipy.optimize import least_squares"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## Arnoldi \n",
"\n",
"This is a copy of the algorithm defined and tested in the notebook `algo2_testing`. It's an implementation of the Algorithm 2 from the paper. It's needed in this notebook since this function is called by the `algo4` function. It's implemented to return exactly what's needed in the `algo4` function.\n",
"\n",
"Everything will be reorganized in the main.py file once everything is working."
]
},
{
"cell_type": "code",
"execution_count": 28,
"metadata": {},
"outputs": [],
"source": [
"def Arnoldi(A,v0,m):\n",
" v = v0\n",
" beta = np.linalg.norm(v)\n",
" v = v/beta\n",
" H = sp.sparse.lil_matrix((m+1,m)) \n",
" V = sp.sparse.lil_matrix((A.shape[0],m+1))\n",
" V[:,0] = v # each column of V is a vector v\n",
"\n",
" for j in range(m):\n",
" w = A @ v \n",
" for i in range(j):\n",
" H[i,j] = v.T @ w # tmp is a 1x1 matrix, so it's O(1) in memory\n",
" w = w - H[i,j]*v \n",
" \n",
" H[j+1,j] = np.linalg.norm(w)\n",
"\n",
" if H[j+1,j] == 0:\n",
" # print(\"Arnoldi breakdown\")\n",
" m = j\n",
" v = 0\n",
" break\n",
" else:\n",
" if j < m-1:\n",
" v = w/H[j+1,j]\n",
" V[:,j+1] = v\n",
"\n",
" return V, H, beta, j "
]
},
{
"attachments": {},
"cell_type": "markdown",
"metadata": {},
"source": [
"# Algorithm 4 testing\n",
"\n",
"This algorithm is based on the \"Algorithm 4\" of the paper, the pseudocode provided by the authors is the following \n",
"\n",
"![](https://i.imgur.com/H92fru7.png)"
]
},
{
"cell_type": "code",
"execution_count": 1,
"metadata": {},
"outputs": [],
"source": [
"def compute_gamma(res, a, k): # function to compute gamma\n",
" gamma = np.ones(len(a))\n",
" for i in range(len(a)):\n",
" if i != k:\n",
" gamma[i] = (res[i]*a[k])/(res[k]*a[i])\n",
" else:\n",
" gamma[i] = 0\n",
" return gamma"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Basic test case with random numbers to test the algorithm."
]
},
{
"cell_type": "code",
"execution_count": 35,
"metadata": {},
"outputs": [],
"source": [
"n = 1000\n",
"m = 1100\n",
"tau = 1e-6\n",
"a = [0.85, 0.88, 0.9, 0.95]\n",
"\n",
"x = sp.sparse.lil_matrix((n,1))\n",
"x[0,0] = 1\n",
"\n",
"# generate a random graph\n",
"G = nx.gnp_random_graph(n, 0.1)\n",
"v = np.repeat(1.0 / 1000, 1000) # p is the personalization vector\n",
"v = v.reshape(v.shape[0],1)\n",
"\n",
"A = nx.to_scipy_sparse_array(G, dtype=float)\n",
"S = A.sum(axis=1) # S[i] is the sum of the weights of edges going out of node i\n",
"S[S != 0] = 1.0 / S[S != 0] # S[i] is now the sum of the weights of edges going into node i\n",
"Q = sp.sparse.csr_array(sp.sparse.spdiags(S.T, 0, *A.shape)) # Q is the matrix of edge weights"
]
},
{
"cell_type": "code",
"execution_count": 83,
"metadata": {},
"outputs": [],
"source": [
"def Algo4(Pt, v, m, a: list, tau, maxit: int, x):\n",
"\n",
" mv, iter = 0, 1 # mv is the number of matrix-vector products, iter is the number of iterations\n",
" \n",
" # initialize x as a random sparse matrix. Each col is the pagerank vector for a different alpha\n",
" x = sp.sparse.lil_matrix((Pt.shape[0], len(a)))\n",
"\n",
"\n",
" # compute the residual vector, it is a matrix of size (n, len(a)). Each col is the residual vector for a different alpha. \n",
" I = sp.sparse.eye(Pt.shape[0], Pt.shape[1], format='lil')\n",
" r = sp.sparse.lil_matrix((Pt.shape[0], len(a)))\n",
" res = np.zeros(len(a))\n",
"\n",
" for i in range(len(a)):\n",
" r[:,[i]] = sp.sparse.linalg.spsolve(I - a[i]*Pt, v)\n",
" col = r[:,[i]].toarray()\n",
" res[i] = np.linalg.norm(col)\n",
"\n",
" for _ in range(maxit):\n",
" # check if we have converged\n",
" err = np.absolute(np.amax(res))\n",
" if err < tau:\n",
" print(\"Computation ended successfully in \", iter, \" iterations and \", mv, \" matrix-vector products.\")\n",
" return x, iter, mv\n",
"\n",
" print(\"\\niter = \", iter)\n",
" print(\"res: \", res)\n",
" print(\"err = \", err)\n",
"\n",
"\n",
" k = int(np.argmax(res))\n",
" print(\"k = \", k)\n",
" gamma = compute_gamma(res, a, k)\n",
" \n",
" # Run Arnoldi\n",
" r_k = r[:,[k]].toarray()\n",
" A_arnoldi = (1/a[k])*I - Pt\n",
" V, H, beta, j = Arnoldi((1/a[k])*I - Pt, r_k, m)\n",
" H = H[:-1,:]\n",
" V = V[:,:-1]\n",
" mv = mv + j\n",
"\n",
" H_e1 = np.zeros(H.shape[0])\n",
" H_e1[0] = 1\n",
"\n",
" # compute y as the minimizer of || beta*e1 - Hy ||_2 using the least squares method\n",
" y = sp.sparse.lil_matrix((H.shape[1],len(a)))\n",
" y[:,[k]] = sp.sparse.linalg.lsqr(H, beta*H_e1)[0]\n",
" y_k = y[:,[k]].toarray()\n",
"\n",
" # # Update x\n",
" x_new = x\n",
" x_new[:,[k]] = x[:,[k]] + V @ y_k\n",
"\n",
" # Update res[k]\n",
" V_e1 = np.zeros(V.shape[0])\n",
" V_e1[0] = 1\n",
"\n",
" norm_k =np.linalg.norm(beta*V_e1 - V @ y_k)\n",
" res[k] = a[k]*norm_k\n",
"\n",
" # multi shift\n",
" for i in range(len(a)):\n",
" if res[i] >= tau:\n",
" # print(\"res[\", i, \"] is larger than tau = \", tau)\n",
"\n",
" # # Compute H as described in the paper\n",
" # H_k = H[:,[k]].toarray()\n",
" # H_i = H_k + ((1-a[i])/a[i] - (1-a[k])/a[k])\n",
" # H[:,[i]] = H_i\n",
" H = H + ((1-a[i])/a[i] - (1-a[k])/a[k])*sp.sparse.eye(H.shape[0], H.shape[1], format='lil')\n",
"\n",
" # Compute z as described in the paper\n",
" z1 = H_e1*beta\n",
" z1 = z1.reshape(z1.shape[0],1)\n",
" z2 = H @ y[:,[1]]\n",
" z2 = z2.reshape(z2.shape[0],1)\n",
" z = z1 - z2\n",
"\n",
" # Solve the linear system \n",
" A = sp.sparse.hstack([H, z])\n",
" b = (beta*H_e1)\n",
" b = b.reshape(b.shape[0],1)\n",
" # use the least squares method to solve the linear system\n",
" to_split = sp.sparse.linalg.lsqr(A, b)[0]\n",
" \n",
" # the last element of y_to_split is the last element of gamma[i], the other elements are the elements of y[:[i]]\n",
" y[:,[i]] = to_split[:-1]\n",
" gamma[i] = to_split[-1]\n",
"\n",
" # update x\n",
" x_new[:,i] = x[:,i] + V @ y[:,[i]]\n",
"\n",
" # update the residual vector\n",
" res[i] = (a[i]/a[k])*pow(gamma[i], i)*res[k]\n",
"\n",
" else:\n",
" if res[i] < tau:\n",
" print(\"res[\", i, \"] is smaller than tau = \", tau, \" at iteration \", iter)\n",
"\n",
" iter = iter + 1\n",
" x = x_new\n",
"\n",
" raise Exception('Maximum number of iterations reached')"
]
},
{
"cell_type": "code",
"execution_count": 84,
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"\n",
"iter = 1\n",
"res: [0.03189738 0.03190716 0.03191369 0.03193001]\n",
"err = 0.031930006625941795\n",
"k = 0\n",
"\n",
"iter = 2\n",
"res: [1.11728737e+00 8.26005227e-04 5.55288870e-10 4.81520495e-13]\n",
"err = 1.1172873666904701\n",
"k = 3\n",
"res[ 2 ] is smaller than tau = 1e-06 at iteration 2\n",
"\n",
"iter = 3\n",
"res: [1.17714008e+00 1.29941354e-03 5.55288870e-10 1.93969263e-18]\n",
"err = 1.1771400826095457\n",
"k = 3\n",
"res[ 2 ] is smaller than tau = 1e-06 at iteration 3\n",
"\n",
"iter = 4\n",
"res: [1.17714008e+00 1.29941354e-03 5.55288870e-10 1.93969263e-18]\n",
"err = 1.1771400826095457\n",
"k = 3\n",
"res[ 2 ] is smaller than tau = 1e-06 at iteration 4\n",
"\n",
"iter = 5\n",
"res: [1.17714008e+00 1.29941354e-03 5.55288870e-10 1.93969263e-18]\n",
"err = 1.1771400826095457\n",
"k = 3\n",
"res[ 2 ] is smaller than tau = 1e-06 at iteration 5\n",
"\n",
"iter = 6\n",
"res: [1.17714008e+00 1.29941354e-03 5.55288870e-10 1.93969263e-18]\n",
"err = 1.1771400826095457\n",
"k = 3\n"
]
},
{
"ename": "KeyboardInterrupt",
"evalue": "",
"output_type": "error",
"traceback": [
"\u001b[0;31m---------------------------------------------------------------------------\u001b[0m",
"\u001b[0;31mKeyboardInterrupt\u001b[0m Traceback (most recent call last)",
"\u001b[0;32m/tmp/ipykernel_13660/3677688099.py\u001b[0m in \u001b[0;36m<module>\u001b[0;34m\u001b[0m\n\u001b[0;32m----> 1\u001b[0;31m \u001b[0mx\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0miter\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mmv\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mAlgo4\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mQ\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mv\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mm\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0ma\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mtau\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0;36m100\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mx\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m",
"\u001b[0;32m/tmp/ipykernel_13660/2503933778.py\u001b[0m in \u001b[0;36mAlgo4\u001b[0;34m(Pt, v, m, a, tau, maxit, x)\u001b[0m\n\u001b[1;32m 36\u001b[0m \u001b[0mr_k\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mr\u001b[0m\u001b[0;34m[\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m,\u001b[0m\u001b[0;34m[\u001b[0m\u001b[0mk\u001b[0m\u001b[0;34m]\u001b[0m\u001b[0;34m]\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mtoarray\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 37\u001b[0m \u001b[0mA_arnoldi\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0;34m(\u001b[0m\u001b[0;36m1\u001b[0m\u001b[0;34m/\u001b[0m\u001b[0ma\u001b[0m\u001b[0;34m[\u001b[0m\u001b[0mk\u001b[0m\u001b[0;34m]\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m*\u001b[0m\u001b[0mI\u001b[0m \u001b[0;34m-\u001b[0m \u001b[0mPt\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m---> 38\u001b[0;31m \u001b[0mV\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mH\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mbeta\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mj\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mArnoldi\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;36m1\u001b[0m\u001b[0;34m/\u001b[0m\u001b[0ma\u001b[0m\u001b[0;34m[\u001b[0m\u001b[0mk\u001b[0m\u001b[0;34m]\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m*\u001b[0m\u001b[0mI\u001b[0m \u001b[0;34m-\u001b[0m \u001b[0mPt\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mr_k\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mm\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 39\u001b[0m \u001b[0mH\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mH\u001b[0m\u001b[0;34m[\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m-\u001b[0m\u001b[0;36m1\u001b[0m\u001b[0;34m,\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 40\u001b[0m \u001b[0mV\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mV\u001b[0m\u001b[0;34m[\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m,\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m-\u001b[0m\u001b[0;36m1\u001b[0m\u001b[0;34m]\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
"\u001b[0;32m/tmp/ipykernel_13660/113321894.py\u001b[0m in \u001b[0;36mArnoldi\u001b[0;34m(A, v0, m)\u001b[0m\n\u001b[1;32m 10\u001b[0m \u001b[0mw\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mA\u001b[0m \u001b[0;34m@\u001b[0m \u001b[0mv\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m 11\u001b[0m \u001b[0;32mfor\u001b[0m \u001b[0mi\u001b[0m \u001b[0;32min\u001b[0m \u001b[0mrange\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mj\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[0;32m---> 12\u001b[0;31m \u001b[0mH\u001b[0m\u001b[0;34m[\u001b[0m\u001b[0mi\u001b[0m\u001b[0;34m,\u001b[0m\u001b[0mj\u001b[0m\u001b[0;34m]\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mv\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mT\u001b[0m \u001b[0;34m@\u001b[0m \u001b[0mw\u001b[0m \u001b[0;31m# tmp is a 1x1 matrix, so it's O(1) in memory\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m 13\u001b[0m \u001b[0mw\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mw\u001b[0m \u001b[0;34m-\u001b[0m \u001b[0mH\u001b[0m\u001b[0;34m[\u001b[0m\u001b[0mi\u001b[0m\u001b[0;34m,\u001b[0m\u001b[0mj\u001b[0m\u001b[0;34m]\u001b[0m\u001b[0;34m*\u001b[0m\u001b[0mv\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m 14\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n",
"\u001b[0;32m~/.local/lib/python3.10/site-packages/scipy/sparse/_lil.py\u001b[0m in \u001b[0;36m__setitem__\u001b[0;34m(self, key, x)\u001b[0m\n\u001b[1;32m 326\u001b[0m isinstance(key[1], INT_TYPES)):\n\u001b[1;32m 327\u001b[0m \u001b[0mx\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mself\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mdtype\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mtype\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mx\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m--> 328\u001b[0;31m \u001b[0;32mif\u001b[0m \u001b[0mx\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0msize\u001b[0m \u001b[0;34m>\u001b[0m \u001b[0;36m1\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 329\u001b[0m \u001b[0;32mraise\u001b[0m \u001b[0mValueError\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m\"Trying to assign a sequence to an item\"\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m 330\u001b[0m \u001b[0;32mreturn\u001b[0m \u001b[0mself\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0m_set_intXint\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mkey\u001b[0m\u001b[0;34m[\u001b[0m\u001b[0;36m0\u001b[0m\u001b[0;34m]\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mkey\u001b[0m\u001b[0;34m[\u001b[0m\u001b[0;36m1\u001b[0m\u001b[0;34m]\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mx\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
"\u001b[0;31mKeyboardInterrupt\u001b[0m: "
]
}
],
"source": [
"x, iter, mv = Algo4(Q, v, m, a, tau, 100, x)"
]
}
],
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"vscode": {
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}
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