{
 "cells": [
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "import time\n",
    "import numpy as np\n",
    "import scipy as sp\n",
    "import pandas as pd\n",
    "import networkx as nx"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Arnoldi \n",
    "\n",
    "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."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [],
   "source": [
    "def Arnoldi(A,v0,m):\n",
    "    v = v0\n",
    "    beta = np.linalg.norm(v)\n",
    "    v = v/beta\n",
    "    H = sp.sparse.lil_matrix((m+1,m)) \n",
    "    V = sp.sparse.lil_matrix((A.shape[0],m+1))\n",
    "    V[:,0] = v # each column of V is a vector v\n",
    "\n",
    "    for j in range(m):\n",
    "        w = A @ v  \n",
    "        for i in range(j):\n",
    "            H[i,j] = v.T @ w # tmp is a 1x1 matrix, so it's O(1) in memory\n",
    "            w = w - H[i,j]*v \n",
    "            \n",
    "        H[j+1,j] = np.linalg.norm(w)\n",
    "\n",
    "        if H[j+1,j] == 0:\n",
    "            # print(\"Arnoldi breakdown\")\n",
    "            m = j\n",
    "            v = 0\n",
    "            break\n",
    "        else:\n",
    "            if j < m-1:\n",
    "                v = w/H[j+1,j]\n",
    "                V[:,j+1] = v\n",
    "\n",
    "    return V, H, beta, j  "
   ]
  },
  {
   "attachments": {},
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Algorithm 4 testing\n",
    "\n",
    "This algorithm is based on the \"Algorithm 4\" of the paper, the pseudocode provided by the authors is the following \n",
    "\n",
    "![](https://i.imgur.com/H92fru7.png)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Auxiliary function"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [],
   "source": [
    "def compute_gamma(res, a, k): # function to compute gamma\n",
    "    gamma = np.ones(len(a))\n",
    "    for i in range(len(a)):\n",
    "        if i != k:\n",
    "            gamma[i] = (res[i]*a[k])/(res[k]*a[i])\n",
    "\n",
    "    return gamma"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Auxiliary function to compute the $\\tilde P$ matrix described in the reference paper"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [],
   "source": [
    "def google_matrix_sparse(G, alpha=0.85, personalization=None, nodelist=None, weight=\"weight\", dangling=None) -> np.matrix:\n",
    "\n",
    "  if nodelist is None:\n",
    "      nodelist = list(G)\n",
    "\n",
    "  A = nx.to_scipy_sparse_array(G, nodelist=nodelist, weight=weight, format=\"lil\", dtype=int)\n",
    "\n",
    "  N = len(G)\n",
    "  if N == 0:\n",
    "      return np.asmatrix(A)\n",
    "\n",
    "  # Personalization vector\n",
    "  if personalization is None:\n",
    "      p = np.repeat(1.0 / N, N)\n",
    "      p = sp.sparse.lil_array(p)\n",
    "  else:\n",
    "      p = np.array([personalization.get(n, 0) for n in nodelist], dtype=float)\n",
    "      if p.sum() == 0:\n",
    "          raise ZeroDivisionError\n",
    "      p /= p.sum()\n",
    "\n",
    "  # Dangling nodes\n",
    "  if dangling is None:\n",
    "      dangling_weights = np.ones(N, dtype=int)\n",
    "      dangling_weights = sp.sparse.lil_array(dangling_weights, dtype=int)\n",
    "  else:\n",
    "      # Convert the dangling dictionary into an array in nodelist order\n",
    "      dangling_weights = np.array([dangling.get(n, 0) for n in nodelist], dtype=float)\n",
    "      dangling_weights /= dangling_weights.sum()\n",
    "\n",
    "  # replace rows with all zeros with dangling_weights\n",
    "  A[[A.sum(axis=1)==0],:] = dangling_weights\n",
    "\n",
    "  # Normalize rows\n",
    "  row_sums = A.sum(axis=1) # row sums\n",
    "  r_inv = np.power(row_sums.astype(float), -1).flatten() # inverse of row sums\n",
    "  r_inv[np.isinf(r_inv)] = 0.0 # replace inf with 0\n",
    "  R = sp.sparse.diags(r_inv) # create diagonal matrix\n",
    "  A = R.dot(A) \n",
    "\n",
    "  return A, p.T"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "`Algorithm 4` described in the reference paper "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [],
   "source": [
    "def Algo4(Pt, v, m, a: list, tau, maxit: int, x):\n",
    "\n",
    "    mv, iter = 0, 1 # mv is the number of matrix-vector products, iter is the number of iterations\n",
    "    \n",
    "    # initialize x as a random sparse matrix. Each col is the pagerank vector for a different alpha\n",
    "    x = sp.sparse.lil_matrix((Pt.shape[0], len(a)))\n",
    "\n",
    "    # initialize the identity matrix of size Pt.shape\n",
    "    I = sp.sparse.eye(Pt.shape[0], Pt.shape[1], format='lil')\n",
    "\n",
    "    # compute the residual vector, it is a matrix of size (n, len(a)). Each col is the residual vector for a different alpha. \n",
    "    r = sp.sparse.lil_matrix((Pt.shape[0], len(a)))\n",
    "    res = np.zeros(len(a))\n",
    "\n",
    "    # compute the residual vector and the norm of each col in the vector res\n",
    "    for i in range(len(a)):\n",
    "        r[:,[i]] = ((1 - a[i])/a[i])*v - ((1/a[i])*I - Pt).dot(x[:,[i]])\n",
    "        res[i] = sp.sparse.linalg.norm(r[:,[i]], ord='fro') # frobenius norm since l2 norm is not defined for sparse matrices in scipy\n",
    "\n",
    "    for _ in range(maxit):\n",
    "        err = np.absolute(np.max(res)) # check if we have converged\n",
    "        if err < tau:\n",
    "            print(\"Computation ended successfully in \", iter, \" iterations and \", mv, \" matrix-vector products.\")\n",
    "            return x, iter, mv\n",
    "\n",
    "        print(\"\\niter = \", iter)\n",
    "        print(\"res: \", res)\n",
    "        print(\"err = \", err)\n",
    "\n",
    "        # find k as the index of the largest residual\n",
    "        k = int(np.argmax(res))\n",
    "        print(\"k = \", k)\n",
    "\n",
    "        # compute gamma as defined in the paper\n",
    "        gamma = compute_gamma(res, a, k)\n",
    "       \n",
    "        # Run Arnoldi\n",
    "        r_k = r[:,[k]].toarray() # r_k is the residual vector for alpha_k\n",
    "        V, H, beta, j = Arnoldi((1/a[k])*I - Pt, r_k, m) # run Arnoldi\n",
    "        H = H[:-1,:] # remove the last row of H\n",
    "        V = V[:,:-1] # remove the last col of V\n",
    "        mv = mv + j # update the number of matrix-vector products\n",
    "\n",
    "        H_e1 = np.zeros(H.shape[0]) \n",
    "        H_e1[0] = 1 # canonical vector e1 of size H.shape[0]\n",
    "\n",
    "        # compute y as the minimizer of || beta*e1 - Hy ||_2 using the least squares method\n",
    "        y = sp.sparse.lil_matrix((H.shape[1],len(a))) \n",
    "\n",
    "        # we only need the k-th col of y in this iteration\n",
    "        y[:,[k]] = sp.sparse.linalg.lsqr(H, beta*H_e1)[0]\n",
    "\n",
    "        # # Update x\n",
    "        x_new = x\n",
    "        x_new[:,[k]] = x[:,[k]] + V @ y[:,[k]]\n",
    "\n",
    "        V_e1 = np.zeros(V.shape[0])\n",
    "        V_e1[0] = 1 \n",
    "        V_e1 = sp.sparse.lil_matrix(V_e1) # canonical vector e1 of size V.shape[0]\n",
    "\n",
    "        # Update res[k]\n",
    "        res[k] = a[k]* (sp.sparse.linalg.norm(beta*V_e1.T - V @ y[:,[k]]))\n",
    "\n",
    "        # multi shift\n",
    "        for i in range(len(a)):\n",
    "            if i != k and res[i] >= tau:\n",
    "                if res[i] >= tau:\n",
    "                    \n",
    "                    H = H + ((1-a[i])/a[i] - (1-a[k])/a[k])*sp.sparse.eye(H.shape[0], H.shape[1], format='lil')\n",
    "\n",
    "                    # Compute z as described in the paper\n",
    "                    z1 = H_e1.T*beta\n",
    "                    z1 = z1.reshape(z1.shape[0],1)\n",
    "                    z2 = H @ y[:,[1]]\n",
    "                    z2 = z2.reshape(z2.shape[0],1)\n",
    "                    z = z1 - z2\n",
    "\n",
    "                    # Solve the linear system for A and b\n",
    "                    A = sp.sparse.hstack([H, z])\n",
    "                    b = (beta*H_e1)\n",
    "\n",
    "                    to_split = sp.sparse.linalg.lsqr(A, b.reshape(b.shape[0],1))[0]\n",
    "                    \n",
    "                    # the last element of to_split is the last element of gamma[i], the other elements are the elements of y[:[i]]\n",
    "                    y[:,[i]] = to_split[:-1]\n",
    "                    gamma[i] = to_split[-1]\n",
    "\n",
    "                    # update x\n",
    "                    x_new[:,i] = x[:,i] + V @ y[:,[i]]\n",
    "\n",
    "                    # update res[i]\n",
    "                    res[i] = (a[i]/a[k])*gamma[i]*res[k]\n",
    "\n",
    "                else:\n",
    "                    if res[i] < tau:\n",
    "                        print(\"res[\", i, \"] is smaller than tau = \", tau, \" at iteration \", iter)\n",
    "\n",
    "        iter += 1\n",
    "        x = x_new\n",
    "\n",
    "    raise Exception('Maximum number of iterations reached')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Random Numerical Test Case on a small-world random graph"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [],
   "source": [
    "G = nx.watts_strogatz_graph(1000, 4, 0.1)\n",
    "Pt, v = google_matrix_sparse(G)\n",
    "# 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",
    "a = [0.85, 0.90, 0.95, 0.99]\n",
    "n = len(G.nodes)\n",
    "x = sp.sparse.random(n, len(a), density=0.1, format='lil')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "\n",
      "iter =  1\n",
      "res:  [0.00558049 0.00351364 0.00166436 0.00031942]\n",
      "err =  0.005580489988532435\n",
      "k =  0\n",
      "\n",
      "iter =  2\n",
      "res:  [ 6.30826211e-03  1.27202498e-07 -1.07175457e-08 -1.83681405e-08]\n",
      "err =  0.006308262114154177\n",
      "k =  0\n",
      "\n",
      "iter =  3\n",
      "res:  [ 6.30826211e-03  1.27202498e-07 -1.07175457e-08 -1.83681405e-08]\n",
      "err =  0.006308262114154177\n",
      "k =  0\n",
      "\n",
      "iter =  4\n",
      "res:  [ 6.30826211e-03  1.27202498e-07 -1.07175457e-08 -1.83681405e-08]\n",
      "err =  0.006308262114154177\n",
      "k =  0\n",
      "\n",
      "iter =  5\n",
      "res:  [ 6.30826211e-03  1.27202498e-07 -1.07175457e-08 -1.83681405e-08]\n",
      "err =  0.006308262114154177\n",
      "k =  0\n",
      "\n",
      "iter =  6\n",
      "res:  [ 6.30826211e-03  1.27202498e-07 -1.07175457e-08 -1.83681405e-08]\n",
      "err =  0.006308262114154177\n",
      "k =  0\n",
      "\n",
      "iter =  7\n",
      "res:  [ 6.30826211e-03  1.27202498e-07 -1.07175457e-08 -1.83681405e-08]\n",
      "err =  0.006308262114154177\n",
      "k =  0\n",
      "\n",
      "iter =  8\n",
      "res:  [ 6.30826211e-03  1.27202498e-07 -1.07175457e-08 -1.83681405e-08]\n",
      "err =  0.006308262114154177\n",
      "k =  0\n",
      "\n",
      "iter =  9\n",
      "res:  [ 6.30826211e-03  1.27202498e-07 -1.07175457e-08 -1.83681405e-08]\n",
      "err =  0.006308262114154177\n",
      "k =  0\n",
      "\n",
      "iter =  10\n",
      "res:  [ 6.30826211e-03  1.27202498e-07 -1.07175457e-08 -1.83681405e-08]\n",
      "err =  0.006308262114154177\n",
      "k =  0\n",
      "\n",
      "iter =  11\n",
      "res:  [ 6.30826211e-03  1.27202498e-07 -1.07175457e-08 -1.83681405e-08]\n",
      "err =  0.006308262114154177\n",
      "k =  0\n",
      "\n",
      "iter =  12\n",
      "res:  [ 6.30826211e-03  1.27202498e-07 -1.07175457e-08 -1.83681405e-08]\n",
      "err =  0.006308262114154177\n",
      "k =  0\n",
      "\n",
      "iter =  13\n",
      "res:  [ 6.30826211e-03  1.27202498e-07 -1.07175457e-08 -1.83681405e-08]\n",
      "err =  0.006308262114154177\n",
      "k =  0\n",
      "\n",
      "iter =  14\n",
      "res:  [ 6.30826211e-03  1.27202498e-07 -1.07175457e-08 -1.83681405e-08]\n",
      "err =  0.006308262114154177\n",
      "k =  0\n",
      "\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",
      "\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",
      "\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",
      "k =  0\n",
      "\n",
      "iter =  23\n",
      "res:  [ 6.30826211e-03  1.27202498e-07 -1.07175457e-08 -1.83681405e-08]\n",
      "err =  0.006308262114154177\n",
      "k =  0\n",
      "\n",
      "iter =  24\n",
      "res:  [ 6.30826211e-03  1.27202498e-07 -1.07175457e-08 -1.83681405e-08]\n",
      "err =  0.006308262114154177\n",
      "k =  0\n",
      "\n",
      "iter =  25\n",
      "res:  [ 6.30826211e-03  1.27202498e-07 -1.07175457e-08 -1.83681405e-08]\n",
      "err =  0.006308262114154177\n",
      "k =  0\n",
      "\n",
      "iter =  26\n",
      "res:  [ 6.30826211e-03  1.27202498e-07 -1.07175457e-08 -1.83681405e-08]\n",
      "err =  0.006308262114154177\n",
      "k =  0\n",
      "\n",
      "iter =  27\n",
      "res:  [ 6.30826211e-03  1.27202498e-07 -1.07175457e-08 -1.83681405e-08]\n",
      "err =  0.006308262114154177\n",
      "k =  0\n",
      "\n",
      "iter =  28\n",
      "res:  [ 6.30826211e-03  1.27202498e-07 -1.07175457e-08 -1.83681405e-08]\n",
      "err =  0.006308262114154177\n",
      "k =  0\n",
      "\n",
      "iter =  29\n",
      "res:  [ 6.30826211e-03  1.27202498e-07 -1.07175457e-08 -1.83681405e-08]\n",
      "err =  0.006308262114154177\n",
      "k =  0\n",
      "\n",
      "iter =  30\n",
      "res:  [ 6.30826211e-03  1.27202498e-07 -1.07175457e-08 -1.83681405e-08]\n",
      "err =  0.006308262114154177\n",
      "k =  0\n",
      "\n",
      "iter =  31\n",
      "res:  [ 6.30826211e-03  1.27202498e-07 -1.07175457e-08 -1.83681405e-08]\n",
      "err =  0.006308262114154177\n",
      "k =  0\n",
      "\n",
      "iter =  32\n",
      "res:  [ 6.30826211e-03  1.27202498e-07 -1.07175457e-08 -1.83681405e-08]\n",
      "err =  0.006308262114154177\n",
      "k =  0\n",
      "\n",
      "iter =  33\n",
      "res:  [ 6.30826211e-03  1.27202498e-07 -1.07175457e-08 -1.83681405e-08]\n",
      "err =  0.006308262114154177\n",
      "k =  0\n",
      "\n",
      "iter =  34\n",
      "res:  [ 6.30826211e-03  1.27202498e-07 -1.07175457e-08 -1.83681405e-08]\n",
      "err =  0.006308262114154177\n",
      "k =  0\n",
      "\n",
      "iter =  35\n",
      "res:  [ 6.30826211e-03  1.27202498e-07 -1.07175457e-08 -1.83681405e-08]\n",
      "err =  0.006308262114154177\n",
      "k =  0\n",
      "\n",
      "iter =  36\n",
      "res:  [ 6.30826211e-03  1.27202498e-07 -1.07175457e-08 -1.83681405e-08]\n",
      "err =  0.006308262114154177\n",
      "k =  0\n",
      "\n",
      "iter =  37\n",
      "res:  [ 6.30826211e-03  1.27202498e-07 -1.07175457e-08 -1.83681405e-08]\n",
      "err =  0.006308262114154177\n",
      "k =  0\n",
      "\n",
      "iter =  38\n",
      "res:  [ 6.30826211e-03  1.27202498e-07 -1.07175457e-08 -1.83681405e-08]\n",
      "err =  0.006308262114154177\n",
      "k =  0\n",
      "\n",
      "iter =  39\n",
      "res:  [ 6.30826211e-03  1.27202498e-07 -1.07175457e-08 -1.83681405e-08]\n",
      "err =  0.006308262114154177\n",
      "k =  0\n",
      "\n",
      "iter =  40\n",
      "res:  [ 6.30826211e-03  1.27202498e-07 -1.07175457e-08 -1.83681405e-08]\n",
      "err =  0.006308262114154177\n",
      "k =  0\n",
      "\n",
      "iter =  41\n",
      "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",
      "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",
      "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)"
   ]
  }
 ],
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