small fixes

src/algo2_testing.ipynb

@@ -2,7 +2,7 @@
  "cells": [
   {
    "cell_type": "code",
-   "execution_count": 2,
+   "execution_count": null,
    "metadata": {},
    "outputs": [],
    "source": [
@@ -29,7 +29,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 3,
+   "execution_count": null,
    "metadata": {},
    "outputs": [],
    "source": [
@@ -71,18 +71,20 @@
     "#     return V, H, v, beta, j"
    ]
   },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "Defined as Algorithm 2 in the paper. It's needed since it's called by Algorithm 4"
+   ]
+  },
   {
    "cell_type": "code",
-   "execution_count": 18,
+   "execution_count": null,
    "metadata": {},
    "outputs": [],
    "source": [
-    "#  review the restarted GMRES method (hereafter referred to as GMRES in short), which is a nonsymmetric Krylov subspace solver based on the Arnoldi decomposition procedure\n",
+    "def Arnoldi(A,v0,m):\n",
-    "\n",
-    "# return the matrix V, the matrix Hp, the vector v, the scalar beta and the number of iterations j. Where V is a matrix of dimension n x m, Hp is an upper Hessenberg matrix of dimension (m+1) x m. Return also the m+1 vector of the basis V. Return h as the m+1,m element of Hp and the vector v as the m+1 element of the basis V.\n",
-    "\n",
-    "\n",
-    "def ArnoldiGMRES(A,v0,m):\n",
     "    v = v0\n",
     "    beta = norm(v)\n",
     "    v = v/beta\n",
@@ -129,52 +131,26 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 14,
+   "execution_count": null,
    "metadata": {},
    "outputs": [],
    "source": [
     "m = 100\n",
     "n = 110\n",
     "A = sp.sparse.rand(n,n, density=0.1, format='lil')\n",
-    "v = sp.sparse.rand(n,1, density=0.1, format='lil')"
+    "# generate a probability vector, with all the entries as 1/n\n",
+    "v = sp.sparse.lil_matrix((n,1))\n",
+    "for i in range(n):\n",
+    "    v[i] = 1/n"
    ]
   },
   {
    "cell_type": "code",
-   "execution_count": 19,
+   "execution_count": null,
    "metadata": {},
-   "outputs": [
+   "outputs": [],
-    {
-     "name": "stdout",
-     "output_type": "stream",
-     "text": [
-      "99  iterations completed\n",
-      "V =  (110, 101)\n",
-      "H =  (101, 100)\n",
-      "v =  (110, 1)\n",
-      "beta =  2.1103247239370497\n"
-     ]
-    },
-    {
-     "data": {
-      "text/plain": [
-       "(<110x101 sparse matrix of type '<class 'numpy.float64'>'\n",
-       " \twith 10863 stored elements in List of Lists format>,\n",
-       " <101x100 sparse matrix of type '<class 'numpy.float64'>'\n",
-       " \twith 4923 stored elements in List of Lists format>,\n",
-       " <110x1 sparse matrix of type '<class 'numpy.float64'>'\n",
-       " \twith 110 stored elements in Compressed Sparse Row format>,\n",
-       " 2.1103247239370497,\n",
-       " 99)"
-      ]
-     },
-     "execution_count": 19,
-     "metadata": {},
-     "output_type": "execute_result"
-    }
-   ],
    "source": [
-    "ArnoldiGMRES(A,v,m)"
+    "Arnoldi(A,v,m)"
    ]
   }
  ],