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844 lines
29 KiB
Plaintext
844 lines
29 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 time\n",
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"import numpy as np\n",
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"import scipy as sp\n",
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"import pandas as pd\n",
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"import networkx as nx"
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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 `arnoldi.ipynb`. It's an implementation of the Algorithm 2 from the paper. It's needed during the computation of the shifted GMRES algorithm that we are trying to implement here."
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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": "markdown",
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"metadata": {},
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"source": [
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"Auxiliary function"
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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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"\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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"Auxiliary function to compute the $\\tilde P$ matrix described in the reference 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": 4,
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"metadata": {},
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"outputs": [],
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"source": [
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"def google_matrix_sparse(G, alpha=0.85, personalization=None, nodelist=None, weight=\"weight\", dangling=None) -> np.matrix:\n",
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"\n",
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" if nodelist is None:\n",
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" nodelist = list(G)\n",
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"\n",
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" A = nx.to_scipy_sparse_array(G, nodelist=nodelist, weight=weight, format=\"lil\", dtype=int)\n",
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"\n",
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" N = len(G)\n",
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" if N == 0:\n",
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" return np.asmatrix(A)\n",
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"\n",
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" # Personalization vector\n",
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" if personalization is None:\n",
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" p = np.repeat(1.0 / N, N)\n",
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" p = sp.sparse.lil_array(p)\n",
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" else:\n",
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" p = np.array([personalization.get(n, 0) for n in nodelist], dtype=float)\n",
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" if p.sum() == 0:\n",
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" raise ZeroDivisionError\n",
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" p /= p.sum()\n",
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"\n",
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" # Dangling nodes\n",
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" if dangling is None:\n",
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" dangling_weights = np.ones(N, dtype=int)\n",
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" dangling_weights = sp.sparse.lil_array(dangling_weights, dtype=int)\n",
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" else:\n",
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" # Convert the dangling dictionary into an array in nodelist order\n",
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" dangling_weights = np.array([dangling.get(n, 0) for n in nodelist], dtype=float)\n",
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" dangling_weights /= dangling_weights.sum()\n",
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"\n",
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" # replace rows with all zeros with dangling_weights\n",
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" A[[A.sum(axis=1)==0],:] = dangling_weights\n",
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"\n",
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" # Normalize rows\n",
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" row_sums = A.sum(axis=1) # row sums\n",
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" r_inv = np.power(row_sums.astype(float), -1).flatten() # inverse of row sums\n",
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" r_inv[np.isinf(r_inv)] = 0.0 # replace inf with 0\n",
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" R = sp.sparse.diags(r_inv) # create diagonal matrix\n",
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" A = R.dot(A) \n",
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"\n",
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" return A, p.T"
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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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"`Algorithm 4` described in the reference 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": 5,
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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]] = ((1 - a[i])/a[i])*v - ((1/a[i])*I - Pt).dot(x[:,[i]])\n",
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" res[i] = sp.sparse.linalg.norm(r[:,[i]], ord='fro') # frobenius norm since l2 norm is not defined for sparse matrices in scipy\n",
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"\n",
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" for _ in range(maxit):\n",
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" err = np.absolute(np.max(res)) # check if we have converged\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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" V, H, beta, j = Arnoldi((1/a[k])*I - Pt, r_k, m) # run 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))) \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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"\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 \n",
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" V_e1 = sp.sparse.lil_matrix(V_e1) # 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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" res[k] = a[k]* (sp.sparse.linalg.norm(beta*V_e1.T - V @ y[:,[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.T*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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" 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 res[i]\n",
|
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" res[i] = (a[i]/a[k])*gamma[i]*res[k]\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 += 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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|
},
|
|
{
|
|
"cell_type": "markdown",
|
|
"metadata": {},
|
|
"source": [
|
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"Random Numerical Test Case on a small-world random graph"
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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": 6,
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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 = google_matrix_sparse(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, 0.90, 0.95, 0.99]\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": 7,
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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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"\n",
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"iter = 1\n",
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"res: [0.00558049 0.00351364 0.00166436 0.00031942]\n",
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"err = 0.005580489988532435\n",
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"k = 0\n",
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"\n",
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"iter = 2\n",
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"res: [ 6.30826211e-03 1.27202498e-07 -1.07175457e-08 -1.83681405e-08]\n",
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"err = 0.006308262114154177\n",
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"k = 0\n",
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"\n",
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"iter = 3\n",
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"res: [ 6.30826211e-03 1.27202498e-07 -1.07175457e-08 -1.83681405e-08]\n",
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"err = 0.006308262114154177\n",
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"k = 0\n",
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"\n",
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"iter = 4\n",
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"res: [ 6.30826211e-03 1.27202498e-07 -1.07175457e-08 -1.83681405e-08]\n",
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"err = 0.006308262114154177\n",
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"k = 0\n",
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"\n",
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"iter = 5\n",
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"res: [ 6.30826211e-03 1.27202498e-07 -1.07175457e-08 -1.83681405e-08]\n",
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"err = 0.006308262114154177\n",
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"k = 0\n",
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"\n",
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"iter = 6\n",
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"res: [ 6.30826211e-03 1.27202498e-07 -1.07175457e-08 -1.83681405e-08]\n",
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"err = 0.006308262114154177\n",
|
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"k = 0\n",
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"\n",
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"iter = 7\n",
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"res: [ 6.30826211e-03 1.27202498e-07 -1.07175457e-08 -1.83681405e-08]\n",
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"err = 0.006308262114154177\n",
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"k = 0\n",
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"\n",
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"iter = 8\n",
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"res: [ 6.30826211e-03 1.27202498e-07 -1.07175457e-08 -1.83681405e-08]\n",
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"err = 0.006308262114154177\n",
|
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"k = 0\n",
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"\n",
|
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"iter = 9\n",
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"res: [ 6.30826211e-03 1.27202498e-07 -1.07175457e-08 -1.83681405e-08]\n",
|
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"err = 0.006308262114154177\n",
|
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"k = 0\n",
|
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"\n",
|
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"iter = 10\n",
|
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"res: [ 6.30826211e-03 1.27202498e-07 -1.07175457e-08 -1.83681405e-08]\n",
|
|
"err = 0.006308262114154177\n",
|
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"k = 0\n",
|
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"\n",
|
|
"iter = 11\n",
|
|
"res: [ 6.30826211e-03 1.27202498e-07 -1.07175457e-08 -1.83681405e-08]\n",
|
|
"err = 0.006308262114154177\n",
|
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"k = 0\n",
|
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"\n",
|
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"iter = 12\n",
|
|
"res: [ 6.30826211e-03 1.27202498e-07 -1.07175457e-08 -1.83681405e-08]\n",
|
|
"err = 0.006308262114154177\n",
|
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"k = 0\n",
|
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"\n",
|
|
"iter = 13\n",
|
|
"res: [ 6.30826211e-03 1.27202498e-07 -1.07175457e-08 -1.83681405e-08]\n",
|
|
"err = 0.006308262114154177\n",
|
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"k = 0\n",
|
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"\n",
|
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"iter = 14\n",
|
|
"res: [ 6.30826211e-03 1.27202498e-07 -1.07175457e-08 -1.83681405e-08]\n",
|
|
"err = 0.006308262114154177\n",
|
|
"k = 0\n",
|
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"\n",
|
|
"iter = 15\n",
|
|
"res: [ 6.30826211e-03 1.27202498e-07 -1.07175457e-08 -1.83681405e-08]\n",
|
|
"err = 0.006308262114154177\n",
|
|
"k = 0\n",
|
|
"\n",
|
|
"iter = 16\n",
|
|
"res: [ 6.30826211e-03 1.27202498e-07 -1.07175457e-08 -1.83681405e-08]\n",
|
|
"err = 0.006308262114154177\n",
|
|
"k = 0\n",
|
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"\n",
|
|
"iter = 17\n",
|
|
"res: [ 6.30826211e-03 1.27202498e-07 -1.07175457e-08 -1.83681405e-08]\n",
|
|
"err = 0.006308262114154177\n",
|
|
"k = 0\n",
|
|
"\n",
|
|
"iter = 18\n",
|
|
"res: [ 6.30826211e-03 1.27202498e-07 -1.07175457e-08 -1.83681405e-08]\n",
|
|
"err = 0.006308262114154177\n",
|
|
"k = 0\n",
|
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"\n",
|
|
"iter = 19\n",
|
|
"res: [ 6.30826211e-03 1.27202498e-07 -1.07175457e-08 -1.83681405e-08]\n",
|
|
"err = 0.006308262114154177\n",
|
|
"k = 0\n",
|
|
"\n",
|
|
"iter = 20\n",
|
|
"res: [ 6.30826211e-03 1.27202498e-07 -1.07175457e-08 -1.83681405e-08]\n",
|
|
"err = 0.006308262114154177\n",
|
|
"k = 0\n",
|
|
"\n",
|
|
"iter = 21\n",
|
|
"res: [ 6.30826211e-03 1.27202498e-07 -1.07175457e-08 -1.83681405e-08]\n",
|
|
"err = 0.006308262114154177\n",
|
|
"k = 0\n",
|
|
"\n",
|
|
"iter = 22\n",
|
|
"res: [ 6.30826211e-03 1.27202498e-07 -1.07175457e-08 -1.83681405e-08]\n",
|
|
"err = 0.006308262114154177\n",
|
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"k = 0\n",
|
|
"\n",
|
|
"iter = 23\n",
|
|
"res: [ 6.30826211e-03 1.27202498e-07 -1.07175457e-08 -1.83681405e-08]\n",
|
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"err = 0.006308262114154177\n",
|
|
"k = 0\n",
|
|
"\n",
|
|
"iter = 24\n",
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"res: [ 6.30826211e-03 1.27202498e-07 -1.07175457e-08 -1.83681405e-08]\n",
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"err = 0.006308262114154177\n",
|
|
"k = 0\n",
|
|
"\n",
|
|
"iter = 25\n",
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"res: [ 6.30826211e-03 1.27202498e-07 -1.07175457e-08 -1.83681405e-08]\n",
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"err = 0.006308262114154177\n",
|
|
"k = 0\n",
|
|
"\n",
|
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"iter = 26\n",
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"res: [ 6.30826211e-03 1.27202498e-07 -1.07175457e-08 -1.83681405e-08]\n",
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"err = 0.006308262114154177\n",
|
|
"k = 0\n",
|
|
"\n",
|
|
"iter = 27\n",
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"res: [ 6.30826211e-03 1.27202498e-07 -1.07175457e-08 -1.83681405e-08]\n",
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"err = 0.006308262114154177\n",
|
|
"k = 0\n",
|
|
"\n",
|
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"iter = 28\n",
|
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"res: [ 6.30826211e-03 1.27202498e-07 -1.07175457e-08 -1.83681405e-08]\n",
|
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"err = 0.006308262114154177\n",
|
|
"k = 0\n",
|
|
"\n",
|
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"iter = 29\n",
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"res: [ 6.30826211e-03 1.27202498e-07 -1.07175457e-08 -1.83681405e-08]\n",
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"err = 0.006308262114154177\n",
|
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"k = 0\n",
|
|
"\n",
|
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"iter = 30\n",
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"res: [ 6.30826211e-03 1.27202498e-07 -1.07175457e-08 -1.83681405e-08]\n",
|
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"err = 0.006308262114154177\n",
|
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"k = 0\n",
|
|
"\n",
|
|
"iter = 31\n",
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"res: [ 6.30826211e-03 1.27202498e-07 -1.07175457e-08 -1.83681405e-08]\n",
|
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"err = 0.006308262114154177\n",
|
|
"k = 0\n",
|
|
"\n",
|
|
"iter = 32\n",
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"res: [ 6.30826211e-03 1.27202498e-07 -1.07175457e-08 -1.83681405e-08]\n",
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"err = 0.006308262114154177\n",
|
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"k = 0\n",
|
|
"\n",
|
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"iter = 33\n",
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"res: [ 6.30826211e-03 1.27202498e-07 -1.07175457e-08 -1.83681405e-08]\n",
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"err = 0.006308262114154177\n",
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"k = 0\n",
|
|
"\n",
|
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"iter = 34\n",
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"res: [ 6.30826211e-03 1.27202498e-07 -1.07175457e-08 -1.83681405e-08]\n",
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"err = 0.006308262114154177\n",
|
|
"k = 0\n",
|
|
"\n",
|
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"iter = 35\n",
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"res: [ 6.30826211e-03 1.27202498e-07 -1.07175457e-08 -1.83681405e-08]\n",
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"err = 0.006308262114154177\n",
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"k = 0\n",
|
|
"\n",
|
|
"iter = 36\n",
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|
"res: [ 6.30826211e-03 1.27202498e-07 -1.07175457e-08 -1.83681405e-08]\n",
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"err = 0.006308262114154177\n",
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"k = 0\n",
|
|
"\n",
|
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"iter = 37\n",
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"res: [ 6.30826211e-03 1.27202498e-07 -1.07175457e-08 -1.83681405e-08]\n",
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"err = 0.006308262114154177\n",
|
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"k = 0\n",
|
|
"\n",
|
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"iter = 38\n",
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|
"res: [ 6.30826211e-03 1.27202498e-07 -1.07175457e-08 -1.83681405e-08]\n",
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"err = 0.006308262114154177\n",
|
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"k = 0\n",
|
|
"\n",
|
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"iter = 39\n",
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|
"res: [ 6.30826211e-03 1.27202498e-07 -1.07175457e-08 -1.83681405e-08]\n",
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"err = 0.006308262114154177\n",
|
|
"k = 0\n",
|
|
"\n",
|
|
"iter = 40\n",
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"res: [ 6.30826211e-03 1.27202498e-07 -1.07175457e-08 -1.83681405e-08]\n",
|
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"err = 0.006308262114154177\n",
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"k = 0\n",
|
|
"\n",
|
|
"iter = 41\n",
|
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"res: [ 6.30826211e-03 1.27202498e-07 -1.07175457e-08 -1.83681405e-08]\n",
|
|
"err = 0.006308262114154177\n",
|
|
"k = 0\n",
|
|
"\n",
|
|
"iter = 42\n",
|
|
"res: [ 6.30826211e-03 1.27202498e-07 -1.07175457e-08 -1.83681405e-08]\n",
|
|
"err = 0.006308262114154177\n",
|
|
"k = 0\n",
|
|
"\n",
|
|
"iter = 43\n",
|
|
"res: [ 6.30826211e-03 1.27202498e-07 -1.07175457e-08 -1.83681405e-08]\n",
|
|
"err = 0.006308262114154177\n",
|
|
"k = 0\n",
|
|
"\n",
|
|
"iter = 44\n",
|
|
"res: [ 6.30826211e-03 1.27202498e-07 -1.07175457e-08 -1.83681405e-08]\n",
|
|
"err = 0.006308262114154177\n",
|
|
"k = 0\n",
|
|
"\n",
|
|
"iter = 45\n",
|
|
"res: [ 6.30826211e-03 1.27202498e-07 -1.07175457e-08 -1.83681405e-08]\n",
|
|
"err = 0.006308262114154177\n",
|
|
"k = 0\n",
|
|
"\n",
|
|
"iter = 46\n",
|
|
"res: [ 6.30826211e-03 1.27202498e-07 -1.07175457e-08 -1.83681405e-08]\n",
|
|
"err = 0.006308262114154177\n",
|
|
"k = 0\n",
|
|
"\n",
|
|
"iter = 47\n",
|
|
"res: [ 6.30826211e-03 1.27202498e-07 -1.07175457e-08 -1.83681405e-08]\n",
|
|
"err = 0.006308262114154177\n",
|
|
"k = 0\n",
|
|
"\n",
|
|
"iter = 48\n",
|
|
"res: [ 6.30826211e-03 1.27202498e-07 -1.07175457e-08 -1.83681405e-08]\n",
|
|
"err = 0.006308262114154177\n",
|
|
"k = 0\n",
|
|
"\n",
|
|
"iter = 49\n",
|
|
"res: [ 6.30826211e-03 1.27202498e-07 -1.07175457e-08 -1.83681405e-08]\n",
|
|
"err = 0.006308262114154177\n",
|
|
"k = 0\n",
|
|
"\n",
|
|
"iter = 50\n",
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|
"res: [ 6.30826211e-03 1.27202498e-07 -1.07175457e-08 -1.83681405e-08]\n",
|
|
"err = 0.006308262114154177\n",
|
|
"k = 0\n",
|
|
"\n",
|
|
"iter = 51\n",
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|
"res: [ 6.30826211e-03 1.27202498e-07 -1.07175457e-08 -1.83681405e-08]\n",
|
|
"err = 0.006308262114154177\n",
|
|
"k = 0\n",
|
|
"\n",
|
|
"iter = 52\n",
|
|
"res: [ 6.30826211e-03 1.27202498e-07 -1.07175457e-08 -1.83681405e-08]\n",
|
|
"err = 0.006308262114154177\n",
|
|
"k = 0\n",
|
|
"\n",
|
|
"iter = 53\n",
|
|
"res: [ 6.30826211e-03 1.27202498e-07 -1.07175457e-08 -1.83681405e-08]\n",
|
|
"err = 0.006308262114154177\n",
|
|
"k = 0\n",
|
|
"\n",
|
|
"iter = 54\n",
|
|
"res: [ 6.30826211e-03 1.27202498e-07 -1.07175457e-08 -1.83681405e-08]\n",
|
|
"err = 0.006308262114154177\n",
|
|
"k = 0\n",
|
|
"\n",
|
|
"iter = 55\n",
|
|
"res: [ 6.30826211e-03 1.27202498e-07 -1.07175457e-08 -1.83681405e-08]\n",
|
|
"err = 0.006308262114154177\n",
|
|
"k = 0\n",
|
|
"\n",
|
|
"iter = 56\n",
|
|
"res: [ 6.30826211e-03 1.27202498e-07 -1.07175457e-08 -1.83681405e-08]\n",
|
|
"err = 0.006308262114154177\n",
|
|
"k = 0\n",
|
|
"\n",
|
|
"iter = 57\n",
|
|
"res: [ 6.30826211e-03 1.27202498e-07 -1.07175457e-08 -1.83681405e-08]\n",
|
|
"err = 0.006308262114154177\n",
|
|
"k = 0\n",
|
|
"\n",
|
|
"iter = 58\n",
|
|
"res: [ 6.30826211e-03 1.27202498e-07 -1.07175457e-08 -1.83681405e-08]\n",
|
|
"err = 0.006308262114154177\n",
|
|
"k = 0\n",
|
|
"\n",
|
|
"iter = 59\n",
|
|
"res: [ 6.30826211e-03 1.27202498e-07 -1.07175457e-08 -1.83681405e-08]\n",
|
|
"err = 0.006308262114154177\n",
|
|
"k = 0\n",
|
|
"\n",
|
|
"iter = 60\n",
|
|
"res: [ 6.30826211e-03 1.27202498e-07 -1.07175457e-08 -1.83681405e-08]\n",
|
|
"err = 0.006308262114154177\n",
|
|
"k = 0\n",
|
|
"\n",
|
|
"iter = 61\n",
|
|
"res: [ 6.30826211e-03 1.27202498e-07 -1.07175457e-08 -1.83681405e-08]\n",
|
|
"err = 0.006308262114154177\n",
|
|
"k = 0\n",
|
|
"\n",
|
|
"iter = 62\n",
|
|
"res: [ 6.30826211e-03 1.27202498e-07 -1.07175457e-08 -1.83681405e-08]\n",
|
|
"err = 0.006308262114154177\n",
|
|
"k = 0\n",
|
|
"\n",
|
|
"iter = 63\n",
|
|
"res: [ 6.30826211e-03 1.27202498e-07 -1.07175457e-08 -1.83681405e-08]\n",
|
|
"err = 0.006308262114154177\n",
|
|
"k = 0\n",
|
|
"\n",
|
|
"iter = 64\n",
|
|
"res: [ 6.30826211e-03 1.27202498e-07 -1.07175457e-08 -1.83681405e-08]\n",
|
|
"err = 0.006308262114154177\n",
|
|
"k = 0\n",
|
|
"\n",
|
|
"iter = 65\n",
|
|
"res: [ 6.30826211e-03 1.27202498e-07 -1.07175457e-08 -1.83681405e-08]\n",
|
|
"err = 0.006308262114154177\n",
|
|
"k = 0\n",
|
|
"\n",
|
|
"iter = 66\n",
|
|
"res: [ 6.30826211e-03 1.27202498e-07 -1.07175457e-08 -1.83681405e-08]\n",
|
|
"err = 0.006308262114154177\n",
|
|
"k = 0\n",
|
|
"\n",
|
|
"iter = 67\n",
|
|
"res: [ 6.30826211e-03 1.27202498e-07 -1.07175457e-08 -1.83681405e-08]\n",
|
|
"err = 0.006308262114154177\n",
|
|
"k = 0\n",
|
|
"\n",
|
|
"iter = 68\n",
|
|
"res: [ 6.30826211e-03 1.27202498e-07 -1.07175457e-08 -1.83681405e-08]\n",
|
|
"err = 0.006308262114154177\n",
|
|
"k = 0\n",
|
|
"\n",
|
|
"iter = 69\n",
|
|
"res: [ 6.30826211e-03 1.27202498e-07 -1.07175457e-08 -1.83681405e-08]\n",
|
|
"err = 0.006308262114154177\n",
|
|
"k = 0\n",
|
|
"\n",
|
|
"iter = 70\n",
|
|
"res: [ 6.30826211e-03 1.27202498e-07 -1.07175457e-08 -1.83681405e-08]\n",
|
|
"err = 0.006308262114154177\n",
|
|
"k = 0\n",
|
|
"\n",
|
|
"iter = 71\n",
|
|
"res: [ 6.30826211e-03 1.27202498e-07 -1.07175457e-08 -1.83681405e-08]\n",
|
|
"err = 0.006308262114154177\n",
|
|
"k = 0\n",
|
|
"\n",
|
|
"iter = 72\n",
|
|
"res: [ 6.30826211e-03 1.27202498e-07 -1.07175457e-08 -1.83681405e-08]\n",
|
|
"err = 0.006308262114154177\n",
|
|
"k = 0\n",
|
|
"\n",
|
|
"iter = 73\n",
|
|
"res: [ 6.30826211e-03 1.27202498e-07 -1.07175457e-08 -1.83681405e-08]\n",
|
|
"err = 0.006308262114154177\n",
|
|
"k = 0\n",
|
|
"\n",
|
|
"iter = 74\n",
|
|
"res: [ 6.30826211e-03 1.27202498e-07 -1.07175457e-08 -1.83681405e-08]\n",
|
|
"err = 0.006308262114154177\n",
|
|
"k = 0\n",
|
|
"\n",
|
|
"iter = 75\n",
|
|
"res: [ 6.30826211e-03 1.27202498e-07 -1.07175457e-08 -1.83681405e-08]\n",
|
|
"err = 0.006308262114154177\n",
|
|
"k = 0\n",
|
|
"\n",
|
|
"iter = 76\n",
|
|
"res: [ 6.30826211e-03 1.27202498e-07 -1.07175457e-08 -1.83681405e-08]\n",
|
|
"err = 0.006308262114154177\n",
|
|
"k = 0\n",
|
|
"\n",
|
|
"iter = 77\n",
|
|
"res: [ 6.30826211e-03 1.27202498e-07 -1.07175457e-08 -1.83681405e-08]\n",
|
|
"err = 0.006308262114154177\n",
|
|
"k = 0\n",
|
|
"\n",
|
|
"iter = 78\n",
|
|
"res: [ 6.30826211e-03 1.27202498e-07 -1.07175457e-08 -1.83681405e-08]\n",
|
|
"err = 0.006308262114154177\n",
|
|
"k = 0\n",
|
|
"\n",
|
|
"iter = 79\n",
|
|
"res: [ 6.30826211e-03 1.27202498e-07 -1.07175457e-08 -1.83681405e-08]\n",
|
|
"err = 0.006308262114154177\n",
|
|
"k = 0\n",
|
|
"\n",
|
|
"iter = 80\n",
|
|
"res: [ 6.30826211e-03 1.27202498e-07 -1.07175457e-08 -1.83681405e-08]\n",
|
|
"err = 0.006308262114154177\n",
|
|
"k = 0\n",
|
|
"\n",
|
|
"iter = 81\n",
|
|
"res: [ 6.30826211e-03 1.27202498e-07 -1.07175457e-08 -1.83681405e-08]\n",
|
|
"err = 0.006308262114154177\n",
|
|
"k = 0\n",
|
|
"\n",
|
|
"iter = 82\n",
|
|
"res: [ 6.30826211e-03 1.27202498e-07 -1.07175457e-08 -1.83681405e-08]\n",
|
|
"err = 0.006308262114154177\n",
|
|
"k = 0\n",
|
|
"\n",
|
|
"iter = 83\n",
|
|
"res: [ 6.30826211e-03 1.27202498e-07 -1.07175457e-08 -1.83681405e-08]\n",
|
|
"err = 0.006308262114154177\n",
|
|
"k = 0\n",
|
|
"\n",
|
|
"iter = 84\n",
|
|
"res: [ 6.30826211e-03 1.27202498e-07 -1.07175457e-08 -1.83681405e-08]\n",
|
|
"err = 0.006308262114154177\n",
|
|
"k = 0\n",
|
|
"\n",
|
|
"iter = 85\n",
|
|
"res: [ 6.30826211e-03 1.27202498e-07 -1.07175457e-08 -1.83681405e-08]\n",
|
|
"err = 0.006308262114154177\n",
|
|
"k = 0\n",
|
|
"\n",
|
|
"iter = 86\n",
|
|
"res: [ 6.30826211e-03 1.27202498e-07 -1.07175457e-08 -1.83681405e-08]\n",
|
|
"err = 0.006308262114154177\n",
|
|
"k = 0\n",
|
|
"\n",
|
|
"iter = 87\n",
|
|
"res: [ 6.30826211e-03 1.27202498e-07 -1.07175457e-08 -1.83681405e-08]\n",
|
|
"err = 0.006308262114154177\n",
|
|
"k = 0\n",
|
|
"\n",
|
|
"iter = 88\n",
|
|
"res: [ 6.30826211e-03 1.27202498e-07 -1.07175457e-08 -1.83681405e-08]\n",
|
|
"err = 0.006308262114154177\n",
|
|
"k = 0\n",
|
|
"\n",
|
|
"iter = 89\n",
|
|
"res: [ 6.30826211e-03 1.27202498e-07 -1.07175457e-08 -1.83681405e-08]\n",
|
|
"err = 0.006308262114154177\n",
|
|
"k = 0\n",
|
|
"\n",
|
|
"iter = 90\n",
|
|
"res: [ 6.30826211e-03 1.27202498e-07 -1.07175457e-08 -1.83681405e-08]\n",
|
|
"err = 0.006308262114154177\n",
|
|
"k = 0\n",
|
|
"\n",
|
|
"iter = 91\n",
|
|
"res: [ 6.30826211e-03 1.27202498e-07 -1.07175457e-08 -1.83681405e-08]\n",
|
|
"err = 0.006308262114154177\n",
|
|
"k = 0\n",
|
|
"\n",
|
|
"iter = 92\n",
|
|
"res: [ 6.30826211e-03 1.27202498e-07 -1.07175457e-08 -1.83681405e-08]\n",
|
|
"err = 0.006308262114154177\n",
|
|
"k = 0\n",
|
|
"\n",
|
|
"iter = 93\n",
|
|
"res: [ 6.30826211e-03 1.27202498e-07 -1.07175457e-08 -1.83681405e-08]\n",
|
|
"err = 0.006308262114154177\n",
|
|
"k = 0\n",
|
|
"\n",
|
|
"iter = 94\n",
|
|
"res: [ 6.30826211e-03 1.27202498e-07 -1.07175457e-08 -1.83681405e-08]\n",
|
|
"err = 0.006308262114154177\n",
|
|
"k = 0\n",
|
|
"\n",
|
|
"iter = 95\n",
|
|
"res: [ 6.30826211e-03 1.27202498e-07 -1.07175457e-08 -1.83681405e-08]\n",
|
|
"err = 0.006308262114154177\n",
|
|
"k = 0\n",
|
|
"\n",
|
|
"iter = 96\n",
|
|
"res: [ 6.30826211e-03 1.27202498e-07 -1.07175457e-08 -1.83681405e-08]\n",
|
|
"err = 0.006308262114154177\n",
|
|
"k = 0\n",
|
|
"\n",
|
|
"iter = 97\n",
|
|
"res: [ 6.30826211e-03 1.27202498e-07 -1.07175457e-08 -1.83681405e-08]\n",
|
|
"err = 0.006308262114154177\n",
|
|
"k = 0\n",
|
|
"\n",
|
|
"iter = 98\n",
|
|
"res: [ 6.30826211e-03 1.27202498e-07 -1.07175457e-08 -1.83681405e-08]\n",
|
|
"err = 0.006308262114154177\n",
|
|
"k = 0\n",
|
|
"\n",
|
|
"iter = 99\n",
|
|
"res: [ 6.30826211e-03 1.27202498e-07 -1.07175457e-08 -1.83681405e-08]\n",
|
|
"err = 0.006308262114154177\n",
|
|
"k = 0\n",
|
|
"\n",
|
|
"iter = 100\n",
|
|
"res: [ 6.30826211e-03 1.27202498e-07 -1.07175457e-08 -1.83681405e-08]\n",
|
|
"err = 0.006308262114154177\n",
|
|
"k = 0\n"
|
|
]
|
|
},
|
|
{
|
|
"ename": "Exception",
|
|
"evalue": "Maximum number of iterations reached",
|
|
"output_type": "error",
|
|
"traceback": [
|
|
"\u001b[0;31m---------------------------------------------------------------------------\u001b[0m",
|
|
"\u001b[0;31mException\u001b[0m Traceback (most recent call last)",
|
|
"Cell \u001b[0;32mIn[7], 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[38;5;241;43m1e-6\u001b[39;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[43mx\u001b[49m\u001b[43m)\u001b[49m\n",
|
|
"Cell \u001b[0;32mIn[5], line 101\u001b[0m, in \u001b[0;36mAlgo4\u001b[0;34m(Pt, v, m, a, tau, maxit, x)\u001b[0m\n\u001b[1;32m 98\u001b[0m \u001b[38;5;28miter\u001b[39m \u001b[38;5;241m+\u001b[39m\u001b[38;5;241m=\u001b[39m \u001b[38;5;241m1\u001b[39m\n\u001b[1;32m 99\u001b[0m x \u001b[38;5;241m=\u001b[39m x_new\n\u001b[0;32m--> 101\u001b[0m \u001b[38;5;28;01mraise\u001b[39;00m \u001b[38;5;167;01mException\u001b[39;00m(\u001b[38;5;124m'\u001b[39m\u001b[38;5;124mMaximum number of iterations reached\u001b[39m\u001b[38;5;124m'\u001b[39m)\n",
|
|
"\u001b[0;31mException\u001b[0m: Maximum number of iterations reached"
|
|
]
|
|
}
|
|
],
|
|
"source": [
|
|
"x, iter, mv = Algo4(Pt, v, 100, a, 1e-6, 100, 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": "3.10.9"
|
|
},
|
|
"orig_nbformat": 4,
|
|
"vscode": {
|
|
"interpreter": {
|
|
"hash": "e7370f93d1d0cde622a1f8e1c04877d8463912d04d973331ad4851f04de6915a"
|
|
}
|
|
}
|
|
},
|
|
"nbformat": 4,
|
|
"nbformat_minor": 2
|
|
}
|