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solved the problem of the linear system. The res vector gets zeroed, to do

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Luca Lombardo 2 years ago
parent 55d9631ee4
commit 646b2fe624
2 changed files with 146 additions and 60 deletions
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2
algo2_testing.ipynb vendored
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@ -199,7 +199,7 @@
"name": "python", "name": "python",
"nbconvert_exporter": "python", "nbconvert_exporter": "python",
"pygments_lexer": "ipython3", "pygments_lexer": "ipython3",
"version": "3.10.6" "version": "3.10.6 (main, Nov 2 2022, 18:53:38) [GCC 11.3.0]"
}, },
"orig_nbformat": 4, "orig_nbformat": 4,
"vscode": { "vscode": {

204
algo4_testing.ipynb vendored
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"cells": [ "cells": [
{ {
"cell_type": "code", "cell_type": "code",
"execution_count": null, "execution_count": 3,
"metadata": {}, "metadata": {},
"outputs": [], "outputs": [],
"source": [ "source": [
@ -33,7 +33,7 @@
}, },
{ {
"cell_type": "code", "cell_type": "code",
"execution_count": null, "execution_count": 4,
"metadata": {}, "metadata": {},
"outputs": [], "outputs": [],
"source": [ "source": [
@ -85,7 +85,7 @@
}, },
{ {
"cell_type": "code", "cell_type": "code",
"execution_count": null, "execution_count": 5,
"metadata": {}, "metadata": {},
"outputs": [], "outputs": [],
"source": [ "source": [
@ -180,94 +180,121 @@
}, },
{ {
"cell_type": "code", "cell_type": "code",
"execution_count": null, "execution_count": 109,
"metadata": {},
"outputs": [],
"source": [
"def find_k(res): # function to find the index of the largest element in res\n",
" k = 0\n",
" for i in range(len(a)):\n",
" if res[i] == max(res):\n",
" k = i\n",
" break\n",
" return k\n",
"\n",
"def compute_gamma(res, a, k): # function to compute gamma\n",
" gamma = np.zeros(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": "code",
"execution_count": 123,
"metadata": {}, "metadata": {},
"outputs": [], "outputs": [],
"source": [ "source": [
"def Algo4(Pt, v, m, a: list, tau, maxit: int, x):\n", "def Algo4(Pt, v, m, a: list, tau, maxit: int, x):\n",
" \n",
" # I'm using a non declared variable n here , it's declared in the next cell when I call this function. This will be fixed later in the main.py file\n",
"\n", "\n",
" x = sp.sparse.lil_matrix((Pt.shape[0], len(a)))\n",
" mv = 0\n", " mv = 0\n",
"\n", "\n",
" # I'm defining 3 canonical vectors of different sizes. It's probably stupid, will be fixed once the algorithm actually works\n",
"\n",
" H_e1 = np.zeros((m+1,1)) # canonical basis vector of size H.shape[0]\n",
" H_e1[0] = 1\n",
"\n",
" V_e1 = np.zeros((n,1)) # canonical basis vector of size V.shape[0]\n",
" V_e1[0] = 1\n",
"\n",
" s_e1 = np.zeros((len(a),1)) # canonical basis vector of size s.shape[0]\n",
" s_e1[0] = 1\n",
"\n",
" def find_k(res): # function to find the index of the largest element in res\n",
" k = 0\n",
" for i in range(len(a)):\n",
" if res[i] == max(res):\n",
" k = i\n",
" break\n",
" return k\n",
"\n",
" def compute_gamma(res, a, k): # function to compute gamma\n",
" gamma = np.zeros(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\n",
"\n",
" # compute the residual vector\n", " # compute the residual vector\n",
"\n",
" I = sp.sparse.eye(Pt.shape[0], Pt.shape[1], format='lil')\n", " I = sp.sparse.eye(Pt.shape[0], Pt.shape[1], format='lil')\n",
" res = np.zeros(len(a))\n", " res = np.zeros(len(a))\n",
" for i in range(len(a)):\n", " for i in range(len(a)):\n",
" r = sp.sparse.linalg.spsolve(I - a[i]*Pt, v)\n", " r = sp.sparse.linalg.spsolve(I - a[i]*Pt, v)\n",
" res[i] = a[i]*np.linalg.norm(r)\n", " res[i] = a[i]*np.linalg.norm(r)\n",
"\n", "\n",
" iter = 0\n",
" for _ in range(maxit):\n", " for _ in range(maxit):\n",
" print(\"\\niter: \", iter)\n",
" k = find_k(res)\n", " k = find_k(res)\n",
" gamma = compute_gamma(res, a, k)\n", " gamma = compute_gamma(res, a, k)\n",
" \n", " \n",
" # Run Arnoldi\n", " # Run Arnoldi\n",
" V, H, v, beta, j = Arnoldi((1/a[k])*I - Pt, r, m)\n", " V, H, v, beta, j = Arnoldi((1/a[k])*I - Pt, r, m)\n",
"\n", "\n",
" H = H[:-1,:]\n",
" V = V[:,:-1]\n",
" mv = mv + j\n", " mv = mv + j\n",
"\n", "\n",
" # compute y as the minimizer of || beta*e1 - Hy ||_2 using the least squares method\n", " # compute y as the minimizer of || beta*e1 - Hy ||_2 using the least squares method\n",
" H_e1 = np.zeros(H.shape[0])\n",
" H_e1[0] = 1\n",
" y = sp.sparse.linalg.lsqr(H, beta*H_e1)[0]\n", " y = sp.sparse.linalg.lsqr(H, beta*H_e1)[0]\n",
" \n", " \n",
" # reshape y to be a column vector\n", " # Update res[k]\n",
" y = y.reshape(y.shape[0],1)\n", " y = y.reshape(y.shape[0],1)\n",
" V_e1 = np.zeros(V.shape[0])\n",
" V_e1[0] = 1\n",
"\n", "\n",
" # update x \n", " res[k] = a[k]*np.linalg.norm(beta*V_e1 - V @ y) \n",
" x_new = x + V[:,0:y.shape[0]] @ y\n", " print(\"res[\", k, \"] = \", res[k]) \n",
"\n",
"\n",
" res[k] = a[k]*np.linalg.norm(beta*V_e1 - V[:,0:y.shape[0]] @ y)\n",
" \n", " \n",
" for i in range(len(a)):\n", " for i in range(len(a)):\n",
" print(\"res = \", res)\n",
" if i != k and res[i] >= tau:\n", " if i != k and res[i] >= tau:\n",
" print(\"res[\", i, \"] is greater than tau = \", tau, \" at iteration \", iter, \" entering the for loop\")\n",
"\n",
" # Compute H as described in the paper\n", " # Compute H as described in the paper\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", " H_new = H + ((1-a[i])/a[i] - (1-a[k])/a[k])*sp.sparse.eye(H.shape[0], H.shape[1], format='lil')\n",
" z = beta*H_e1 - H @ y\n", "\n",
" # Compute z as described in the paper\n",
" tmp = H_e1*beta \n",
" z1, z2 = tmp.reshape(tmp.shape[0],1), H @ y\n",
" z = z1 - z2\n",
"\n",
" # Solve the linear system \n",
" A = sp.sparse.hstack([H_new, z])\n",
" b = gamma[i]*(beta*H_e1)\n",
" b = b.reshape(b.shape[0],1)\n",
" \n",
" # use the least squares method to solve the linear system\n",
" y_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\n",
" y = y_to_split[:-1].reshape(y_to_split[:-1].shape[0],1)\n",
" gamma[i] = y_to_split[-1]\n",
"\n", "\n",
" # TO DO: SOLVE THE LINEAR SYSTEM HERE\n", " x_new = x\n",
" x_new[:,[k]] = x[:,[k]] + V @ y\n",
"\n", "\n",
" # update x\n", " # update x\n",
" x_new = x + V[:,0:y.shape[0]] @ y\n", " x_new[:,i] = x[:,i] + V @ y\n",
"\n", "\n",
" # update the residual vector\n", " # update the residual vector\n",
" res[i] = (a[i]/a[k])*gamma[k]*res[k]\n", " res[i] = (a[i]/a[k])*gamma[k]*res[k]\n",
" \n", " print(\"res[\", i, \"] = \", res[i])\n",
" err = np.absolute(x - x_new).sum()\n", " \n",
" x = x_new\n", " else:\n",
" if res[i] < tau:\n",
" print(\"res[\", i, \"] = \", res[i], \" is smaller than tau = \", tau, \" at iteration \", iter)\n",
"\n", "\n",
" err = np.absolute(res).max()\n",
" print(\"err = \", err)\n",
" if err < tau:\n", " if err < tau:\n",
" raise StopIteration\n", " return x_new, iter, mv\n",
" \n",
" iter = iter + 1\n",
" x = x_new\n",
"\n", "\n",
" return x, iter, mv" " raise Exception('Maximum number of iterations reached')\n",
" \n"
] ]
}, },
{ {
@ -279,21 +306,21 @@
}, },
{ {
"cell_type": "code", "cell_type": "code",
"execution_count": null, "execution_count": 124,
"metadata": {}, "metadata": {},
"outputs": [], "outputs": [],
"source": [ "source": [
"n = 100\n", "n = 1000\n",
"m = 110\n", "m = 1100\n",
"tau = 1e-9\n", "tau = 1e-5\n",
"a = [0.85, 0.9, 0.95, 0.99]\n", "a = [0.85, 0.9, 0.95, 0.99]\n",
"\n", "\n",
"x = sp.sparse.lil_matrix((n,1))\n", "x = sp.sparse.lil_matrix((n,1))\n",
"x[0,0] = 1\n", "x[0,0] = 1\n",
"\n", "\n",
"# generate a random graph\n", "# generate a random graph\n",
"G = nx.gnp_random_graph(n, 1)\n", "G = nx.gnp_random_graph(n, 0.01)\n",
"v = np.repeat(1.0 / 100, 100) # p is the personalization vector\n", "v = np.repeat(1.0 / 1000, 1000) # p is the personalization vector\n",
"v = v.reshape(v.shape[0],1)\n", "v = v.reshape(v.shape[0],1)\n",
"\n", "\n",
"A = nx.to_scipy_sparse_array(G, dtype=float)\n", "A = nx.to_scipy_sparse_array(G, dtype=float)\n",
@ -304,11 +331,70 @@
}, },
{ {
"cell_type": "code", "cell_type": "code",
"execution_count": null, "execution_count": 125,
"metadata": {}, "metadata": {},
"outputs": [], "outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"\n",
"iter: 0\n",
"res[ 3 ] = 1.9500347533542841\n",
"res = [0.02974037 0.03169717 0.03368163 1.95003475]\n",
"res[ 0 ] is greater than tau = 1e-05 at iteration 0 entering the for loop\n",
"res[ 0 ] = 0.0\n",
"res = [0. 0.03169717 0.03368163 1.95003475]\n",
"res[ 1 ] is greater than tau = 1e-05 at iteration 0 entering the for loop\n",
"res[ 1 ] = 0.0\n",
"res = [0. 0. 0.03368163 1.95003475]\n",
"res[ 2 ] is greater than tau = 1e-05 at iteration 0 entering the for loop\n",
"res[ 2 ] = 0.0\n",
"res = [0. 0. 0. 1.95003475]\n",
"err = 1.9500347533542841\n",
"\n",
"iter: 1\n",
"res[ 3 ] = 1.9500347533542841\n",
"res = [0. 0. 0. 1.95003475]\n",
"res[ 0 ] = 0.0 is smaller than tau = 1e-05 at iteration 1\n",
"res = [0. 0. 0. 1.95003475]\n",
"res[ 1 ] = 0.0 is smaller than tau = 1e-05 at iteration 1\n",
"res = [0. 0. 0. 1.95003475]\n",
"res[ 2 ] = 0.0 is smaller than tau = 1e-05 at iteration 1\n",
"res = [0. 0. 0. 1.95003475]\n",
"err = 1.9500347533542841\n",
"\n",
"iter: 2\n",
"res[ 3 ] = 1.9500347533542841\n",
"res = [0. 0. 0. 1.95003475]\n",
"res[ 0 ] = 0.0 is smaller than tau = 1e-05 at iteration 2\n",
"res = [0. 0. 0. 1.95003475]\n",
"res[ 1 ] = 0.0 is smaller than tau = 1e-05 at iteration 2\n",
"res = [0. 0. 0. 1.95003475]\n",
"res[ 2 ] = 0.0 is smaller than tau = 1e-05 at iteration 2\n",
"res = [0. 0. 0. 1.95003475]\n",
"err = 1.9500347533542841\n",
"\n",
"iter: 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_85082/3677688099.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_85082/4220477307.py\u001b[0m in \u001b[0;36mAlgo4\u001b[0;34m(Pt, v, m, a, tau, maxit, x)\u001b[0m\n\u001b[1;32m 18\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m 19\u001b[0m \u001b[0;31m# Run Arnoldi\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m---> 20\u001b[0;31m \u001b[0mV\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mH\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mv\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\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 21\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m 22\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[0;32m/tmp/ipykernel_85082/1990871025.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 320\u001b[0m \u001b[0mself\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0m_set_arrayXarray\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mrow\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mcol\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[1;32m 321\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m--> 322\u001b[0;31m \u001b[0;32mdef\u001b[0m \u001b[0m__setitem__\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mself\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mkey\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[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m 323\u001b[0m \u001b[0;31m# Fast path for simple (int, int) indexing.\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m 324\u001b[0m if (isinstance(key, tuple) and len(key) == 2 and\n",
"\u001b[0;31mKeyboardInterrupt\u001b[0m: "
]
}
],
"source": [ "source": [
"Algo4(Q, v, m, a, tau, 100, x)" "x, iter, mv = Algo4(Q, v, m, a, tau, 100, x)"
] ]
} }
], ],

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