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869 lines
31 KiB
Plaintext
869 lines
31 KiB
Plaintext
2 years ago
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{
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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 networkx as nx\n",
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"import time\n",
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"import math\n",
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"import pandas as pd\n",
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"import scipy as sp\n",
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"import plotly.express as px\n",
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"import plotly.graph_objs as go\n",
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"from scipy.sparse import *\n",
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"from scipy import linalg\n",
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"from scipy.sparse.linalg import norm\n",
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"from scipy.optimize import least_squares"
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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 \n",
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"\n",
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"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",
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"\n",
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"Everything will be reorganized in the main.py file once everything is working."
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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 = np.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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"attachments": {},
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"# Algorithm 4 testing\n",
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"\n",
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"This algorithm is based on the \"Algorithm 4\" of the paper, the pseudocode provided by the authors is the following \n",
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"\n",
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""
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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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"def compute_gamma(res, a, k): # function to compute gamma\n",
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" gamma = np.ones(len(a))\n",
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" for i in range(len(a)):\n",
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" if i != k:\n",
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" gamma[i] = (res[i]*a[k])/(res[k]*a[i])\n",
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" else:\n",
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" gamma[i] = 0\n",
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" return gamma"
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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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"Basic test case with random numbers to test the algorithm."
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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": 66,
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"metadata": {},
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"outputs": [],
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"source": [
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"def compute_ptilde(G: nx.Graph):\n",
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" \"\"\"\n",
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" Compute the ptilde matrix and the probability vector v\n",
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" :param G: the graph\n",
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" :return: the ptilde matrix and the probability vector v\n",
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"\n",
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" \"\"\"\n",
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"\n",
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" # given the graph G, return it's sparse matrix representation\n",
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" A = nx.to_scipy_sparse_array(G, format='lil')\n",
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"\n",
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" # create a vector d (sparse), where d[i] = 1 if the i-th row of A not null, else 0\n",
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" d = sp.sparse.lil_matrix((1, A.shape[0]))\n",
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" for i in range(A.shape[0]):\n",
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" # s is the sum of the i-th row of A\n",
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" s = sp.sparse.lil_matrix.sum(A[[i],:])\n",
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" if s == 0:\n",
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" d[0,[i]] = 0\n",
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"\n",
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" # probability vector v = 1/n\n",
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" v = np.repeat(1/A.shape[0], A.shape[0])\n",
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"\n",
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" # initialize the ptilde matrix\n",
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" P = sp.sparse.lil_matrix((A.shape[0], A.shape[1]))\n",
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"\n",
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" # P(i,j) = 1/(number of non null entries in column j) if A(i,j) != 0, else 0\n",
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" for j in range(A.shape[1]):\n",
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" for i in range(A.shape[0]):\n",
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" if A[i,j] != 0:\n",
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" P[i,j] = 1/sp.sparse.lil_matrix.sum(A[:,[j]] != 0)\n",
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"\n",
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" Pt = P + v @ d.T\n",
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"\n",
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" return Pt, v "
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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": 85,
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"metadata": {},
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"outputs": [],
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"source": [
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"def Algo4(Pt, v, m, a: list, tau, maxit: int, x):\n",
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"\n",
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" mv, iter = 0, 1 # mv is the number of matrix-vector products, iter is the number of iterations\n",
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" \n",
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" # initialize x as a random sparse matrix. Each col is the pagerank vector for a different alpha\n",
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" x = sp.sparse.lil_matrix((Pt.shape[0], len(a)))\n",
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"\n",
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" # initialize the identity matrix of size Pt.shape\n",
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" I = sp.sparse.eye(Pt.shape[0], Pt.shape[1], format='lil')\n",
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"\n",
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" # compute the residual vector, it is a matrix of size (n, len(a)). Each col is the residual vector for a different alpha. \n",
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" r = sp.sparse.lil_matrix((Pt.shape[0], len(a)))\n",
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" res = np.zeros(len(a))\n",
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"\n",
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" # compute the residual vector and the norm of each col in the vector res\n",
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" for i in range(len(a)):\n",
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" r[:,[i]] = sp.sparse.linalg.spsolve(I - a[i]*Pt, v)\n",
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" col = r[:,[i]].toarray()\n",
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" res[i] = np.linalg.norm(col)\n",
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"\n",
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" # this is a while loop in the paper\n",
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" for _ in range(maxit):\n",
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" # check if we have converged\n",
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" err = np.absolute(np.max(res))\n",
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" if err < tau:\n",
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" print(\"Computation ended successfully in \", iter, \" iterations and \", mv, \" matrix-vector products.\")\n",
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" return x, iter, mv\n",
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"\n",
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" print(\"\\niter = \", iter)\n",
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" print(\"res: \", res)\n",
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" print(\"err = \", err)\n",
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"\n",
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" # find k as the index of the largest residual\n",
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" k = int(np.argmax(res))\n",
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" print(\"k = \", k)\n",
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"\n",
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" # compute gamma as defined in the paper\n",
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" gamma = compute_gamma(res, a, k)\n",
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" \n",
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" # Run Arnoldi\n",
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" r_k = r[:,[k]].toarray() # r_k is the residual vector for alpha_k\n",
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" A_arnoldi = (1/a[k])*I - Pt # A_arnoldi is the matrix used in Arnoldi\n",
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" V, H, beta, j = Arnoldi((1/a[k])*I - Pt, r_k, m) # V is the matrix of vectors v, H is the Hessenberg matrix, beta is the norm of the last vector v, j is the number of iterations of Arnoldi\n",
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" H = H[:-1,:] # remove the last row of H\n",
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" V = V[:,:-1] # remove the last col of V\n",
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" mv = mv + j # update the number of matrix-vector products\n",
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"\n",
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" H_e1 = np.zeros(H.shape[0]) \n",
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" H_e1[0] = 1 # canonical vector e1 of size H.shape[0]\n",
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"\n",
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" # compute y as the minimizer of || beta*e1 - Hy ||_2 using the least squares method\n",
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" y = sp.sparse.lil_matrix((H.shape[1],len(a))) # y is the matrix of vectors y, each col is the vector y for a different alpha\n",
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"\n",
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" # we only need the k-th col of y in this iteration\n",
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" y[:,[k]] = sp.sparse.linalg.lsqr(H, beta*H_e1)[0]\n",
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" y_k = y[:,[k]].toarray()\n",
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"\n",
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" # # Update x\n",
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" x_new = x\n",
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" x_new[:,[k]] = x[:,[k]] + V @ y_k\n",
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"\n",
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" V_e1 = np.zeros(V.shape[0])\n",
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" V_e1[0] = 1 # canonical vector e1 of size V.shape[0]\n",
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"\n",
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" # Update res[k]\n",
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" norm_k =np.linalg.norm(beta*V_e1 - V @ y_k) # this returns a scalar\n",
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" res[k] = a[k]*norm_k\n",
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"\n",
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" # multi shift\n",
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" for i in range(len(a)):\n",
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" if i != k and res[i] >= tau:\n",
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" if res[i] >= tau:\n",
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" \n",
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" H = H + ((1-a[i])/a[i] - (1-a[k])/a[k])*sp.sparse.eye(H.shape[0], H.shape[1], format='lil')\n",
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"\n",
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" # Compute z as described in the paper\n",
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" z1 = H_e1*beta\n",
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" z1 = z1.reshape(z1.shape[0],1)\n",
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" z2 = H @ y[:,[1]]\n",
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" z2 = z2.reshape(z2.shape[0],1)\n",
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" z = z1 - z2\n",
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"\n",
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" # Solve the linear system for A and b\n",
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" A = sp.sparse.hstack([H, z])\n",
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" b = (beta*H_e1)\n",
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"\n",
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" # use the least squares method to solve the linear system\n",
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" to_split = sp.sparse.linalg.lsqr(A, b.reshape(b.shape[0],1))[0]\n",
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" \n",
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" # the last element of to_split is the last element of gamma[i], the other elements are the elements of y[:[i]]\n",
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" y[:,[i]] = to_split[:-1]\n",
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" gamma[i] = to_split[-1]\n",
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"\n",
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" # update x\n",
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" x_new[:,i] = x[:,i] + V @ y[:,[i]]\n",
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"\n",
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" # update the residual vector\n",
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" # print(\"\\tupdating res[\", i, \"]\")\n",
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" # print(\"\\tgamma[\", i, \"] = \", gamma[i])\n",
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" # print(\"\\tres[\", k, \"] = \", res[k])\n",
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" # print(\"\\ta[\", i, \"] = \", a[i])\n",
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" # print(\"\\ta[\", k, \"] = \", a[k])\n",
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" res[i] = (a[i]/a[k])*gamma[i]*res[k]\n",
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" # print(\"\\tupdated res[\", i, \"] = \", res[i])\n",
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" # print()\n",
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"\n",
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" else:\n",
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" if res[i] < tau:\n",
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" print(\"res[\", i, \"] is smaller than tau = \", tau, \" at iteration \", iter)\n",
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"\n",
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" iter = iter + 1\n",
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" x = x_new\n",
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"\n",
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" raise Exception('Maximum number of iterations reached')"
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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": 86,
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||
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"metadata": {},
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"outputs": [],
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"source": [
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"G = nx.watts_strogatz_graph(1000, 4, 0.1)\n",
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"Pt, v = compute_ptilde(G)\n",
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"# a = [0.85, 0.86, 0.87, 0.88, 0.89, 0.90, 0.91, 0.92, 0.93, 0.94, 0.95, 0.96, 0.97, 0.98, 0.99]\n",
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"a = [0.85, 0.86, 0.87, 0.88]\n",
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"tau = 1e-6\n",
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"maxit = 100\n",
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"n = len(G.nodes)\n",
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"x = sp.sparse.random(n, len(a), density=0.1, format='lil')"
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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": 87,
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||
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"metadata": {},
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"outputs": [
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{
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"name": "stderr",
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"output_type": "stream",
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"text": [
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"/usr/lib/python3.10/site-packages/scipy/sparse/linalg/_dsolve/linsolve.py:168: SparseEfficiencyWarning: spsolve requires A be CSC or CSR matrix format\n",
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" warn('spsolve requires A be CSC or CSR matrix format',\n"
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]
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},
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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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"\n",
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"\n",
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"iter = 1\n",
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"res: [0.21235414 0.22756312 0.24511308 0.26558937]\n",
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"err = 0.26558937251088227\n",
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"k = 3\n",
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"\n",
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"\n",
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"iter = 2\n",
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"res: [ 0.81703891 0.8829876 0.10190349 10.69433448]\n",
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"err = 10.69433447757171\n",
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"k = 3\n",
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"\n",
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"\n",
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"iter = 3\n",
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"res: [ 0.81703891 0.8829876 0.10190349 10.69433448]\n",
|
||
|
|
"err = 10.69433447757171\n",
|
||
|
|
"k = 3\n",
|
||
|
|
"\n",
|
||
|
|
"\n",
|
||
|
|
"iter = 4\n",
|
||
|
|
"res: [ 0.81703891 0.8829876 0.10190349 10.69433448]\n",
|
||
|
|
"err = 10.69433447757171\n",
|
||
|
|
"k = 3\n",
|
||
|
|
"\n",
|
||
|
|
"\n",
|
||
|
|
"iter = 5\n",
|
||
|
|
"res: [ 0.81703891 0.8829876 0.10190349 10.69433448]\n",
|
||
|
|
"err = 10.69433447757171\n",
|
||
|
|
"k = 3\n",
|
||
|
|
"\n",
|
||
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|
"\n",
|
||
|
|
"iter = 6\n",
|
||
|
|
"res: [ 0.81703891 0.8829876 0.10190349 10.69433448]\n",
|
||
|
|
"err = 10.69433447757171\n",
|
||
|
|
"k = 3\n",
|
||
|
|
"\n",
|
||
|
|
"\n",
|
||
|
|
"iter = 7\n",
|
||
|
|
"res: [ 0.81703891 0.8829876 0.10190349 10.69433448]\n",
|
||
|
|
"err = 10.69433447757171\n",
|
||
|
|
"k = 3\n",
|
||
|
|
"\n",
|
||
|
|
"\n",
|
||
|
|
"iter = 8\n",
|
||
|
|
"res: [ 0.81703891 0.8829876 0.10190349 10.69433448]\n",
|
||
|
|
"err = 10.69433447757171\n",
|
||
|
|
"k = 3\n",
|
||
|
|
"\n",
|
||
|
|
"\n",
|
||
|
|
"iter = 9\n",
|
||
|
|
"res: [ 0.81703891 0.8829876 0.10190349 10.69433448]\n",
|
||
|
|
"err = 10.69433447757171\n",
|
||
|
|
"k = 3\n",
|
||
|
|
"\n",
|
||
|
|
"\n",
|
||
|
|
"iter = 10\n",
|
||
|
|
"res: [ 0.81703891 0.8829876 0.10190349 10.69433448]\n",
|
||
|
|
"err = 10.69433447757171\n",
|
||
|
|
"k = 3\n",
|
||
|
|
"\n",
|
||
|
|
"\n",
|
||
|
|
"iter = 11\n",
|
||
|
|
"res: [ 0.81703891 0.8829876 0.10190349 10.69433448]\n",
|
||
|
|
"err = 10.69433447757171\n",
|
||
|
|
"k = 3\n",
|
||
|
|
"\n",
|
||
|
|
"\n",
|
||
|
|
"iter = 12\n",
|
||
|
|
"res: [ 0.81703891 0.8829876 0.10190349 10.69433448]\n",
|
||
|
|
"err = 10.69433447757171\n",
|
||
|
|
"k = 3\n",
|
||
|
|
"\n",
|
||
|
|
"\n",
|
||
|
|
"iter = 13\n",
|
||
|
|
"res: [ 0.81703891 0.8829876 0.10190349 10.69433448]\n",
|
||
|
|
"err = 10.69433447757171\n",
|
||
|
|
"k = 3\n",
|
||
|
|
"\n",
|
||
|
|
"\n",
|
||
|
|
"iter = 14\n",
|
||
|
|
"res: [ 0.81703891 0.8829876 0.10190349 10.69433448]\n",
|
||
|
|
"err = 10.69433447757171\n",
|
||
|
|
"k = 3\n",
|
||
|
|
"\n",
|
||
|
|
"\n",
|
||
|
|
"iter = 15\n",
|
||
|
|
"res: [ 0.81703891 0.8829876 0.10190349 10.69433448]\n",
|
||
|
|
"err = 10.69433447757171\n",
|
||
|
|
"k = 3\n",
|
||
|
|
"\n",
|
||
|
|
"\n",
|
||
|
|
"iter = 16\n",
|
||
|
|
"res: [ 0.81703891 0.8829876 0.10190349 10.69433448]\n",
|
||
|
|
"err = 10.69433447757171\n",
|
||
|
|
"k = 3\n",
|
||
|
|
"\n",
|
||
|
|
"\n",
|
||
|
|
"iter = 17\n",
|
||
|
|
"res: [ 0.81703891 0.8829876 0.10190349 10.69433448]\n",
|
||
|
|
"err = 10.69433447757171\n",
|
||
|
|
"k = 3\n",
|
||
|
|
"\n",
|
||
|
|
"\n",
|
||
|
|
"iter = 18\n",
|
||
|
|
"res: [ 0.81703891 0.8829876 0.10190349 10.69433448]\n",
|
||
|
|
"err = 10.69433447757171\n",
|
||
|
|
"k = 3\n",
|
||
|
|
"\n",
|
||
|
|
"\n",
|
||
|
|
"iter = 19\n",
|
||
|
|
"res: [ 0.81703891 0.8829876 0.10190349 10.69433448]\n",
|
||
|
|
"err = 10.69433447757171\n",
|
||
|
|
"k = 3\n",
|
||
|
|
"\n",
|
||
|
|
"\n",
|
||
|
|
"iter = 20\n",
|
||
|
|
"res: [ 0.81703891 0.8829876 0.10190349 10.69433448]\n",
|
||
|
|
"err = 10.69433447757171\n",
|
||
|
|
"k = 3\n",
|
||
|
|
"\n",
|
||
|
|
"\n",
|
||
|
|
"iter = 21\n",
|
||
|
|
"res: [ 0.81703891 0.8829876 0.10190349 10.69433448]\n",
|
||
|
|
"err = 10.69433447757171\n",
|
||
|
|
"k = 3\n",
|
||
|
|
"\n",
|
||
|
|
"\n",
|
||
|
|
"iter = 22\n",
|
||
|
|
"res: [ 0.81703891 0.8829876 0.10190349 10.69433448]\n",
|
||
|
|
"err = 10.69433447757171\n",
|
||
|
|
"k = 3\n",
|
||
|
|
"\n",
|
||
|
|
"\n",
|
||
|
|
"iter = 23\n",
|
||
|
|
"res: [ 0.81703891 0.8829876 0.10190349 10.69433448]\n",
|
||
|
|
"err = 10.69433447757171\n",
|
||
|
|
"k = 3\n",
|
||
|
|
"\n",
|
||
|
|
"\n",
|
||
|
|
"iter = 24\n",
|
||
|
|
"res: [ 0.81703891 0.8829876 0.10190349 10.69433448]\n",
|
||
|
|
"err = 10.69433447757171\n",
|
||
|
|
"k = 3\n",
|
||
|
|
"\n",
|
||
|
|
"\n",
|
||
|
|
"iter = 25\n",
|
||
|
|
"res: [ 0.81703891 0.8829876 0.10190349 10.69433448]\n",
|
||
|
|
"err = 10.69433447757171\n",
|
||
|
|
"k = 3\n",
|
||
|
|
"\n",
|
||
|
|
"\n",
|
||
|
|
"iter = 26\n",
|
||
|
|
"res: [ 0.81703891 0.8829876 0.10190349 10.69433448]\n",
|
||
|
|
"err = 10.69433447757171\n",
|
||
|
|
"k = 3\n",
|
||
|
|
"\n",
|
||
|
|
"\n",
|
||
|
|
"iter = 27\n",
|
||
|
|
"res: [ 0.81703891 0.8829876 0.10190349 10.69433448]\n",
|
||
|
|
"err = 10.69433447757171\n",
|
||
|
|
"k = 3\n",
|
||
|
|
"\n",
|
||
|
|
"\n",
|
||
|
|
"iter = 28\n",
|
||
|
|
"res: [ 0.81703891 0.8829876 0.10190349 10.69433448]\n",
|
||
|
|
"err = 10.69433447757171\n",
|
||
|
|
"k = 3\n",
|
||
|
|
"\n",
|
||
|
|
"\n",
|
||
|
|
"iter = 29\n",
|
||
|
|
"res: [ 0.81703891 0.8829876 0.10190349 10.69433448]\n",
|
||
|
|
"err = 10.69433447757171\n",
|
||
|
|
"k = 3\n",
|
||
|
|
"\n",
|
||
|
|
"\n",
|
||
|
|
"iter = 30\n",
|
||
|
|
"res: [ 0.81703891 0.8829876 0.10190349 10.69433448]\n",
|
||
|
|
"err = 10.69433447757171\n",
|
||
|
|
"k = 3\n",
|
||
|
|
"\n",
|
||
|
|
"\n",
|
||
|
|
"iter = 31\n",
|
||
|
|
"res: [ 0.81703891 0.8829876 0.10190349 10.69433448]\n",
|
||
|
|
"err = 10.69433447757171\n",
|
||
|
|
"k = 3\n",
|
||
|
|
"\n",
|
||
|
|
"\n",
|
||
|
|
"iter = 32\n",
|
||
|
|
"res: [ 0.81703891 0.8829876 0.10190349 10.69433448]\n",
|
||
|
|
"err = 10.69433447757171\n",
|
||
|
|
"k = 3\n",
|
||
|
|
"\n",
|
||
|
|
"\n",
|
||
|
|
"iter = 33\n",
|
||
|
|
"res: [ 0.81703891 0.8829876 0.10190349 10.69433448]\n",
|
||
|
|
"err = 10.69433447757171\n",
|
||
|
|
"k = 3\n",
|
||
|
|
"\n",
|
||
|
|
"\n",
|
||
|
|
"iter = 34\n",
|
||
|
|
"res: [ 0.81703891 0.8829876 0.10190349 10.69433448]\n",
|
||
|
|
"err = 10.69433447757171\n",
|
||
|
|
"k = 3\n",
|
||
|
|
"\n",
|
||
|
|
"\n",
|
||
|
|
"iter = 35\n",
|
||
|
|
"res: [ 0.81703891 0.8829876 0.10190349 10.69433448]\n",
|
||
|
|
"err = 10.69433447757171\n",
|
||
|
|
"k = 3\n",
|
||
|
|
"\n",
|
||
|
|
"\n",
|
||
|
|
"iter = 36\n",
|
||
|
|
"res: [ 0.81703891 0.8829876 0.10190349 10.69433448]\n",
|
||
|
|
"err = 10.69433447757171\n",
|
||
|
|
"k = 3\n",
|
||
|
|
"\n",
|
||
|
|
"\n",
|
||
|
|
"iter = 37\n",
|
||
|
|
"res: [ 0.81703891 0.8829876 0.10190349 10.69433448]\n",
|
||
|
|
"err = 10.69433447757171\n",
|
||
|
|
"k = 3\n",
|
||
|
|
"\n",
|
||
|
|
"\n",
|
||
|
|
"iter = 38\n",
|
||
|
|
"res: [ 0.81703891 0.8829876 0.10190349 10.69433448]\n",
|
||
|
|
"err = 10.69433447757171\n",
|
||
|
|
"k = 3\n",
|
||
|
|
"\n",
|
||
|
|
"\n",
|
||
|
|
"iter = 39\n",
|
||
|
|
"res: [ 0.81703891 0.8829876 0.10190349 10.69433448]\n",
|
||
|
|
"err = 10.69433447757171\n",
|
||
|
|
"k = 3\n",
|
||
|
|
"\n",
|
||
|
|
"\n",
|
||
|
|
"iter = 40\n",
|
||
|
|
"res: [ 0.81703891 0.8829876 0.10190349 10.69433448]\n",
|
||
|
|
"err = 10.69433447757171\n",
|
||
|
|
"k = 3\n",
|
||
|
|
"\n",
|
||
|
|
"\n",
|
||
|
|
"iter = 41\n",
|
||
|
|
"res: [ 0.81703891 0.8829876 0.10190349 10.69433448]\n",
|
||
|
|
"err = 10.69433447757171\n",
|
||
|
|
"k = 3\n",
|
||
|
|
"\n",
|
||
|
|
"\n",
|
||
|
|
"iter = 42\n",
|
||
|
|
"res: [ 0.81703891 0.8829876 0.10190349 10.69433448]\n",
|
||
|
|
"err = 10.69433447757171\n",
|
||
|
|
"k = 3\n",
|
||
|
|
"\n",
|
||
|
|
"\n",
|
||
|
|
"iter = 43\n",
|
||
|
|
"res: [ 0.81703891 0.8829876 0.10190349 10.69433448]\n",
|
||
|
|
"err = 10.69433447757171\n",
|
||
|
|
"k = 3\n",
|
||
|
|
"\n",
|
||
|
|
"\n",
|
||
|
|
"iter = 44\n",
|
||
|
|
"res: [ 0.81703891 0.8829876 0.10190349 10.69433448]\n",
|
||
|
|
"err = 10.69433447757171\n",
|
||
|
|
"k = 3\n",
|
||
|
|
"\n",
|
||
|
|
"\n",
|
||
|
|
"iter = 45\n",
|
||
|
|
"res: [ 0.81703891 0.8829876 0.10190349 10.69433448]\n",
|
||
|
|
"err = 10.69433447757171\n",
|
||
|
|
"k = 3\n",
|
||
|
|
"\n",
|
||
|
|
"\n",
|
||
|
|
"iter = 46\n",
|
||
|
|
"res: [ 0.81703891 0.8829876 0.10190349 10.69433448]\n",
|
||
|
|
"err = 10.69433447757171\n",
|
||
|
|
"k = 3\n",
|
||
|
|
"\n",
|
||
|
|
"\n",
|
||
|
|
"iter = 47\n",
|
||
|
|
"res: [ 0.81703891 0.8829876 0.10190349 10.69433448]\n",
|
||
|
|
"err = 10.69433447757171\n",
|
||
|
|
"k = 3\n",
|
||
|
|
"\n",
|
||
|
|
"\n",
|
||
|
|
"iter = 48\n",
|
||
|
|
"res: [ 0.81703891 0.8829876 0.10190349 10.69433448]\n",
|
||
|
|
"err = 10.69433447757171\n",
|
||
|
|
"k = 3\n",
|
||
|
|
"\n",
|
||
|
|
"\n",
|
||
|
|
"iter = 49\n",
|
||
|
|
"res: [ 0.81703891 0.8829876 0.10190349 10.69433448]\n",
|
||
|
|
"err = 10.69433447757171\n",
|
||
|
|
"k = 3\n",
|
||
|
|
"\n",
|
||
|
|
"\n",
|
||
|
|
"iter = 50\n",
|
||
|
|
"res: [ 0.81703891 0.8829876 0.10190349 10.69433448]\n",
|
||
|
|
"err = 10.69433447757171\n",
|
||
|
|
"k = 3\n",
|
||
|
|
"\n",
|
||
|
|
"\n",
|
||
|
|
"iter = 51\n",
|
||
|
|
"res: [ 0.81703891 0.8829876 0.10190349 10.69433448]\n",
|
||
|
|
"err = 10.69433447757171\n",
|
||
|
|
"k = 3\n",
|
||
|
|
"\n",
|
||
|
|
"\n",
|
||
|
|
"iter = 52\n",
|
||
|
|
"res: [ 0.81703891 0.8829876 0.10190349 10.69433448]\n",
|
||
|
|
"err = 10.69433447757171\n",
|
||
|
|
"k = 3\n",
|
||
|
|
"\n",
|
||
|
|
"\n",
|
||
|
|
"iter = 53\n",
|
||
|
|
"res: [ 0.81703891 0.8829876 0.10190349 10.69433448]\n",
|
||
|
|
"err = 10.69433447757171\n",
|
||
|
|
"k = 3\n",
|
||
|
|
"\n",
|
||
|
|
"\n",
|
||
|
|
"iter = 54\n",
|
||
|
|
"res: [ 0.81703891 0.8829876 0.10190349 10.69433448]\n",
|
||
|
|
"err = 10.69433447757171\n",
|
||
|
|
"k = 3\n",
|
||
|
|
"\n",
|
||
|
|
"\n",
|
||
|
|
"iter = 55\n",
|
||
|
|
"res: [ 0.81703891 0.8829876 0.10190349 10.69433448]\n",
|
||
|
|
"err = 10.69433447757171\n",
|
||
|
|
"k = 3\n",
|
||
|
|
"\n",
|
||
|
|
"\n",
|
||
|
|
"iter = 56\n",
|
||
|
|
"res: [ 0.81703891 0.8829876 0.10190349 10.69433448]\n",
|
||
|
|
"err = 10.69433447757171\n",
|
||
|
|
"k = 3\n",
|
||
|
|
"\n",
|
||
|
|
"\n",
|
||
|
|
"iter = 57\n",
|
||
|
|
"res: [ 0.81703891 0.8829876 0.10190349 10.69433448]\n",
|
||
|
|
"err = 10.69433447757171\n",
|
||
|
|
"k = 3\n",
|
||
|
|
"\n",
|
||
|
|
"\n",
|
||
|
|
"iter = 58\n",
|
||
|
|
"res: [ 0.81703891 0.8829876 0.10190349 10.69433448]\n",
|
||
|
|
"err = 10.69433447757171\n",
|
||
|
|
"k = 3\n",
|
||
|
|
"\n",
|
||
|
|
"\n",
|
||
|
|
"iter = 59\n",
|
||
|
|
"res: [ 0.81703891 0.8829876 0.10190349 10.69433448]\n",
|
||
|
|
"err = 10.69433447757171\n",
|
||
|
|
"k = 3\n",
|
||
|
|
"\n",
|
||
|
|
"\n",
|
||
|
|
"iter = 60\n",
|
||
|
|
"res: [ 0.81703891 0.8829876 0.10190349 10.69433448]\n",
|
||
|
|
"err = 10.69433447757171\n",
|
||
|
|
"k = 3\n",
|
||
|
|
"\n",
|
||
|
|
"\n",
|
||
|
|
"iter = 61\n",
|
||
|
|
"res: [ 0.81703891 0.8829876 0.10190349 10.69433448]\n",
|
||
|
|
"err = 10.69433447757171\n",
|
||
|
|
"k = 3\n",
|
||
|
|
"\n",
|
||
|
|
"\n",
|
||
|
|
"iter = 62\n",
|
||
|
|
"res: [ 0.81703891 0.8829876 0.10190349 10.69433448]\n",
|
||
|
|
"err = 10.69433447757171\n",
|
||
|
|
"k = 3\n",
|
||
|
|
"\n",
|
||
|
|
"\n",
|
||
|
|
"iter = 63\n",
|
||
|
|
"res: [ 0.81703891 0.8829876 0.10190349 10.69433448]\n",
|
||
|
|
"err = 10.69433447757171\n",
|
||
|
|
"k = 3\n",
|
||
|
|
"\n",
|
||
|
|
"\n",
|
||
|
|
"iter = 64\n",
|
||
|
|
"res: [ 0.81703891 0.8829876 0.10190349 10.69433448]\n",
|
||
|
|
"err = 10.69433447757171\n",
|
||
|
|
"k = 3\n",
|
||
|
|
"\n",
|
||
|
|
"\n",
|
||
|
|
"iter = 65\n",
|
||
|
|
"res: [ 0.81703891 0.8829876 0.10190349 10.69433448]\n",
|
||
|
|
"err = 10.69433447757171\n",
|
||
|
|
"k = 3\n",
|
||
|
|
"\n",
|
||
|
|
"\n",
|
||
|
|
"iter = 66\n",
|
||
|
|
"res: [ 0.81703891 0.8829876 0.10190349 10.69433448]\n",
|
||
|
|
"err = 10.69433447757171\n",
|
||
|
|
"k = 3\n",
|
||
|
|
"\n",
|
||
|
|
"\n",
|
||
|
|
"iter = 67\n",
|
||
|
|
"res: [ 0.81703891 0.8829876 0.10190349 10.69433448]\n",
|
||
|
|
"err = 10.69433447757171\n",
|
||
|
|
"k = 3\n",
|
||
|
|
"\n",
|
||
|
|
"\n",
|
||
|
|
"iter = 68\n",
|
||
|
|
"res: [ 0.81703891 0.8829876 0.10190349 10.69433448]\n",
|
||
|
|
"err = 10.69433447757171\n",
|
||
|
|
"k = 3\n",
|
||
|
|
"\n",
|
||
|
|
"\n",
|
||
|
|
"iter = 69\n",
|
||
|
|
"res: [ 0.81703891 0.8829876 0.10190349 10.69433448]\n",
|
||
|
|
"err = 10.69433447757171\n",
|
||
|
|
"k = 3\n",
|
||
|
|
"\n",
|
||
|
|
"\n",
|
||
|
|
"iter = 70\n",
|
||
|
|
"res: [ 0.81703891 0.8829876 0.10190349 10.69433448]\n",
|
||
|
|
"err = 10.69433447757171\n",
|
||
|
|
"k = 3\n",
|
||
|
|
"\n",
|
||
|
|
"\n",
|
||
|
|
"iter = 71\n",
|
||
|
|
"res: [ 0.81703891 0.8829876 0.10190349 10.69433448]\n",
|
||
|
|
"err = 10.69433447757171\n",
|
||
|
|
"k = 3\n",
|
||
|
|
"\n",
|
||
|
|
"\n",
|
||
|
|
"iter = 72\n",
|
||
|
|
"res: [ 0.81703891 0.8829876 0.10190349 10.69433448]\n",
|
||
|
|
"err = 10.69433447757171\n",
|
||
|
|
"k = 3\n",
|
||
|
|
"\n",
|
||
|
|
"\n",
|
||
|
|
"iter = 73\n",
|
||
|
|
"res: [ 0.81703891 0.8829876 0.10190349 10.69433448]\n",
|
||
|
|
"err = 10.69433447757171\n",
|
||
|
|
"k = 3\n",
|
||
|
|
"\n",
|
||
|
|
"\n",
|
||
|
|
"iter = 74\n",
|
||
|
|
"res: [ 0.81703891 0.8829876 0.10190349 10.69433448]\n",
|
||
|
|
"err = 10.69433447757171\n",
|
||
|
|
"k = 3\n",
|
||
|
|
"\n",
|
||
|
|
"\n",
|
||
|
|
"iter = 75\n",
|
||
|
|
"res: [ 0.81703891 0.8829876 0.10190349 10.69433448]\n",
|
||
|
|
"err = 10.69433447757171\n",
|
||
|
|
"k = 3\n",
|
||
|
|
"\n",
|
||
|
|
"\n",
|
||
|
|
"iter = 76\n",
|
||
|
|
"res: [ 0.81703891 0.8829876 0.10190349 10.69433448]\n",
|
||
|
|
"err = 10.69433447757171\n",
|
||
|
|
"k = 3\n",
|
||
|
|
"\n",
|
||
|
|
"\n",
|
||
|
|
"iter = 77\n",
|
||
|
|
"res: [ 0.81703891 0.8829876 0.10190349 10.69433448]\n",
|
||
|
|
"err = 10.69433447757171\n",
|
||
|
|
"k = 3\n",
|
||
|
|
"\n",
|
||
|
|
"\n",
|
||
|
|
"iter = 78\n",
|
||
|
|
"res: [ 0.81703891 0.8829876 0.10190349 10.69433448]\n",
|
||
|
|
"err = 10.69433447757171\n",
|
||
|
|
"k = 3\n",
|
||
|
|
"\n",
|
||
|
|
"\n",
|
||
|
|
"iter = 79\n",
|
||
|
|
"res: [ 0.81703891 0.8829876 0.10190349 10.69433448]\n",
|
||
|
|
"err = 10.69433447757171\n",
|
||
|
|
"k = 3\n",
|
||
|
|
"\n",
|
||
|
|
"\n",
|
||
|
|
"iter = 80\n",
|
||
|
|
"res: [ 0.81703891 0.8829876 0.10190349 10.69433448]\n",
|
||
|
|
"err = 10.69433447757171\n",
|
||
|
|
"k = 3\n",
|
||
|
|
"\n",
|
||
|
|
"\n",
|
||
|
|
"iter = 81\n",
|
||
|
|
"res: [ 0.81703891 0.8829876 0.10190349 10.69433448]\n",
|
||
|
|
"err = 10.69433447757171\n",
|
||
|
|
"k = 3\n",
|
||
|
|
"\n",
|
||
|
|
"\n",
|
||
|
|
"iter = 82\n",
|
||
|
|
"res: [ 0.81703891 0.8829876 0.10190349 10.69433448]\n",
|
||
|
|
"err = 10.69433447757171\n",
|
||
|
|
"k = 3\n",
|
||
|
|
"\n",
|
||
|
|
"\n",
|
||
|
|
"iter = 83\n",
|
||
|
|
"res: [ 0.81703891 0.8829876 0.10190349 10.69433448]\n",
|
||
|
|
"err = 10.69433447757171\n",
|
||
|
|
"k = 3\n",
|
||
|
|
"\n",
|
||
|
|
"\n",
|
||
|
|
"iter = 84\n",
|
||
|
|
"res: [ 0.81703891 0.8829876 0.10190349 10.69433448]\n",
|
||
|
|
"err = 10.69433447757171\n",
|
||
|
|
"k = 3\n",
|
||
|
|
"\n",
|
||
|
|
"\n",
|
||
|
|
"iter = 85\n",
|
||
|
|
"res: [ 0.81703891 0.8829876 0.10190349 10.69433448]\n",
|
||
|
|
"err = 10.69433447757171\n",
|
||
|
|
"k = 3\n",
|
||
|
|
"\n",
|
||
|
|
"\n",
|
||
|
|
"iter = 86\n",
|
||
|
|
"res: [ 0.81703891 0.8829876 0.10190349 10.69433448]\n",
|
||
|
|
"err = 10.69433447757171\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)",
|
||
|
|
"Cell \u001b[0;32mIn[87], line 1\u001b[0m\n\u001b[0;32m----> 1\u001b[0m x, \u001b[38;5;28miter\u001b[39m, mv \u001b[38;5;241m=\u001b[39m \u001b[43mAlgo4\u001b[49m\u001b[43m(\u001b[49m\u001b[43mPt\u001b[49m\u001b[43m,\u001b[49m\u001b[43m \u001b[49m\u001b[43mv\u001b[49m\u001b[43m,\u001b[49m\u001b[43m \u001b[49m\u001b[38;5;241;43m100\u001b[39;49m\u001b[43m,\u001b[49m\u001b[43m \u001b[49m\u001b[43ma\u001b[49m\u001b[43m,\u001b[49m\u001b[43m \u001b[49m\u001b[43mtau\u001b[49m\u001b[43m,\u001b[49m\u001b[43m \u001b[49m\u001b[43mmaxit\u001b[49m\u001b[43m,\u001b[49m\u001b[43m \u001b[49m\u001b[43mx\u001b[49m\u001b[43m)\u001b[49m\n",
|
||
|
|
"Cell \u001b[0;32mIn[85], line 46\u001b[0m, in \u001b[0;36mAlgo4\u001b[0;34m(Pt, v, m, a, tau, maxit, x)\u001b[0m\n\u001b[1;32m 44\u001b[0m V, H, beta, j \u001b[38;5;241m=\u001b[39m Arnoldi((\u001b[38;5;241m1\u001b[39m\u001b[38;5;241m/\u001b[39ma[k])\u001b[38;5;241m*\u001b[39mI \u001b[38;5;241m-\u001b[39m Pt, r_k, m) \u001b[38;5;66;03m# V is the matrix of vectors v, H is the Hessenberg matrix, beta is the norm of the last vector v, j is the number of iterations of Arnoldi\u001b[39;00m\n\u001b[1;32m 45\u001b[0m H \u001b[38;5;241m=\u001b[39m H[:\u001b[38;5;241m-\u001b[39m\u001b[38;5;241m1\u001b[39m,:] \u001b[38;5;66;03m# remove the last row of H\u001b[39;00m\n\u001b[0;32m---> 46\u001b[0m V \u001b[38;5;241m=\u001b[39m \u001b[43mV\u001b[49m\u001b[43m[\u001b[49m\u001b[43m:\u001b[49m\u001b[43m,\u001b[49m\u001b[43m:\u001b[49m\u001b[38;5;241;43m-\u001b[39;49m\u001b[38;5;241;43m1\u001b[39;49m\u001b[43m]\u001b[49m \u001b[38;5;66;03m# remove the last col of V\u001b[39;00m\n\u001b[1;32m 47\u001b[0m mv \u001b[38;5;241m=\u001b[39m mv \u001b[38;5;241m+\u001b[39m j \u001b[38;5;66;03m# update the number of matrix-vector products\u001b[39;00m\n\u001b[1;32m 49\u001b[0m H_e1 \u001b[38;5;241m=\u001b[39m np\u001b[38;5;241m.\u001b[39mzeros(H\u001b[38;5;241m.\u001b[39mshape[\u001b[38;5;241m0\u001b[39m]) \n",
|
||
|
|
"File \u001b[0;32m/usr/lib/python3.10/site-packages/scipy/sparse/_lil.py:211\u001b[0m, in \u001b[0;36mlil_matrix.__getitem__\u001b[0;34m(self, key)\u001b[0m\n\u001b[1;32m 209\u001b[0m \u001b[39mreturn\u001b[39;00m \u001b[39mself\u001b[39m\u001b[39m.\u001b[39m_get_intXint(\u001b[39m*\u001b[39mkey)\n\u001b[1;32m 210\u001b[0m \u001b[39m# Everything else takes the normal path.\u001b[39;00m\n\u001b[0;32m--> 211\u001b[0m \u001b[39mreturn\u001b[39;00m IndexMixin\u001b[39m.\u001b[39;49m\u001b[39m__getitem__\u001b[39;49m(\u001b[39mself\u001b[39;49m, key)\n",
|
||
|
|
"File \u001b[0;32m/usr/lib/python3.10/site-packages/scipy/sparse/_index.py:69\u001b[0m, in \u001b[0;36mIndexMixin.__getitem__\u001b[0;34m(self, key)\u001b[0m\n\u001b[1;32m 67\u001b[0m \u001b[39mif\u001b[39;00m row \u001b[39m==\u001b[39m \u001b[39mslice\u001b[39m(\u001b[39mNone\u001b[39;00m) \u001b[39mand\u001b[39;00m row \u001b[39m==\u001b[39m col:\n\u001b[1;32m 68\u001b[0m \u001b[39mreturn\u001b[39;00m \u001b[39mself\u001b[39m\u001b[39m.\u001b[39mcopy()\n\u001b[0;32m---> 69\u001b[0m \u001b[39mreturn\u001b[39;00m \u001b[39mself\u001b[39;49m\u001b[39m.\u001b[39;49m_get_sliceXslice(row, col)\n\u001b[1;32m 70\u001b[0m \u001b[39melif\u001b[39;00m col\u001b[39m.\u001b[39mndim \u001b[39m==\u001b[39m \u001b[39m1\u001b[39m:\n\u001b[1;32m 71\u001b[0m \u001b[39mreturn\u001b[39;00m \u001b[39mself\u001b[39m\u001b[39m.\u001b[39m_get_sliceXarray(row, col)\n",
|
||
|
|
"File \u001b[0;32m/usr/lib/python3.10/site-packages/scipy/sparse/_lil.py:241\u001b[0m, in \u001b[0;36mlil_matrix._get_sliceXslice\u001b[0;34m(self, row, col)\u001b[0m\n\u001b[1;32m 239\u001b[0m \u001b[39mdef\u001b[39;00m \u001b[39m_get_sliceXslice\u001b[39m(\u001b[39mself\u001b[39m, row, col):\n\u001b[1;32m 240\u001b[0m row \u001b[39m=\u001b[39m \u001b[39mrange\u001b[39m(\u001b[39m*\u001b[39mrow\u001b[39m.\u001b[39mindices(\u001b[39mself\u001b[39m\u001b[39m.\u001b[39mshape[\u001b[39m0\u001b[39m]))\n\u001b[0;32m--> 241\u001b[0m \u001b[39mreturn\u001b[39;00m \u001b[39mself\u001b[39;49m\u001b[39m.\u001b[39;49m_get_row_ranges(row, col)\n",
|
||
|
|
"File \u001b[0;32m/usr/lib/python3.10/site-packages/scipy/sparse/_lil.py:290\u001b[0m, in \u001b[0;36mlil_matrix._get_row_ranges\u001b[0;34m(self, rows, col_slice)\u001b[0m\n\u001b[1;32m 287\u001b[0m nj \u001b[39m=\u001b[39m \u001b[39mlen\u001b[39m(col_range)\n\u001b[1;32m 288\u001b[0m new \u001b[39m=\u001b[39m \u001b[39mself\u001b[39m\u001b[39m.\u001b[39m_lil_container((\u001b[39mlen\u001b[39m(rows), nj), dtype\u001b[39m=\u001b[39m\u001b[39mself\u001b[39m\u001b[39m.\u001b[39mdtype)\n\u001b[0;32m--> 290\u001b[0m _csparsetools\u001b[39m.\u001b[39;49mlil_get_row_ranges(\u001b[39mself\u001b[39;49m\u001b[39m.\u001b[39;49mshape[\u001b[39m0\u001b[39;49m], \u001b[39mself\u001b[39;49m\u001b[39m.\u001b[39;49mshape[\u001b[39m1\u001b[39;49m],\n\u001b[1;32m 291\u001b[0m \u001b[39mself\u001b[39;49m\u001b[39m.\u001b[39;49mrows, \u001b[39mself\u001b[39;49m\u001b[39m.\u001b[39;49mdata,\n\u001b[1;32m 292\u001b[0m new\u001b[39m.\u001b[39;49mrows, new\u001b[39m.\u001b[39;49mdata,\n\u001b[1;32m 293\u001b[0m rows,\n\u001b[1;32m 294\u001b[0m j_start, j_stop, j_stride, nj)\n\u001b[1;32m 296\u001b[0m \u001b[39mreturn\u001b[39;00m new\n",
|
||
|
|
"\u001b[0;31mKeyboardInterrupt\u001b[0m: "
|
||
|
|
]
|
||
|
|
}
|
||
|
|
],
|
||
|
|
"source": [
|
||
|
|
"x, iter, mv = Algo4(Pt, v, 100, a, tau, maxit, x)"
|
||
|
|
]
|
||
|
|
}
|
||
|
|
],
|
||
|
|
"metadata": {
|
||
|
|
"kernelspec": {
|
||
|
|
"display_name": "Python 3.10.8 64-bit",
|
||
|
|
"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": "3.10.8"
|
||
|
|
},
|
||
|
|
"orig_nbformat": 4,
|
||
|
|
"vscode": {
|
||
|
|
"interpreter": {
|
||
|
|
"hash": "e7370f93d1d0cde622a1f8e1c04877d8463912d04d973331ad4851f04de6915a"
|
||
|
|
}
|
||
|
|
}
|
||
|
|
},
|
||
|
|
"nbformat": 4,
|
||
|
|
"nbformat_minor": 2
|
||
|
|
}
|