experiment with toeplitz mtxs

lecture_1/Lab_1_helpers.ipynb

@@ -217,44 +217,60 @@
    "source": [
     "# Don't ever solve a linear system with an inverse matrix\n",
     "* Seriously, you promised\n",
-    "\n"
+    "\n",
+    "The code below solves a linear system in two different ways, first by computing the matrix inverse then multiplying.  Then using a smart way `np.linalg.solve`) "
    ]
   },
   {
    "cell_type": "code",
-   "execution_count": 8,
+   "execution_count": 13,
    "metadata": {},
    "outputs": [
     {
      "name": "stdout",
      "output_type": "stream",
      "text": [
-      "it took 0.876535177230835 ms the slow way\n",
-      "it took 0.30843091011047363 ms the fast way\n"
+      "it took 6.21192193031311 s the slow way\n",
+      "it took 1.9188358783721924 s the fast way\n",
+      "are the results different? : False\n"
      ]
     }
    ],
    "source": [
     "import time\n",
     "\n",
-    "# Try making it bigger than 2000, if you dare.\n",
+    "# Try making it bigger than 2000, if you dare \n",
+    "# (maybe stay under 10k)\n",
     "\n",
-    "medium_mtx = np.array( rng.rand( 2000, 2000 ) )\n",
-    "medium_vec = np.array( rng.rand( 2000, 1 ) )\n",
+    "medium_mtx = np.array( rng.rand( 4000, 4000 ) )\n",
+    "medium_vec = np.array( rng.rand( 4000, 1 ) )\n",
     "\n",
+    "# Solve by computing the matrix inverse, then multiplying\n",
+    "# See how long it takes\n",
     "t0 = time.time()\n",
     "slow_result = np.linalg.inv( medium_mtx ) @ medium_vec\n",
     "t1 = time.time()\n",
-    "print( 'it took {} ms the slow way'.format( t1-t0 ))\n",
+    "print( 'it took {} s the slow way'.format( t1-t0 ))\n",
     "\n",
+    "# Solve in a smart way\n",
+    "# See how long it takes\n",
     "t0_f = time.time()\n",
     "fast_result = np.linalg.solve( medium_mtx, medium_vec )\n",
     "t1_f = time.time()\n",
-    "print( 'it took {} ms the fast way'.format( t1_f - t0_f ))\n",
+    "print( 'it took {} s the fast way'.format( t1_f - t0_f ))\n",
     "\n",
     "# Check that the results are the same\n",
-    "# Subtract the results and see if any of the elements are bigger\n",
-    "( slow_result - fast_result ) > 0.0001"
+    "# Subtract the results and see if any of the elements are bigger than a tiny number\n",
+    "print( 'are the results different? : {}'.format( np.any( ( slow_result - fast_result ) > 1e-6 )))"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "## Bonus\n",
+    "\n",
+    "Learn about [Toeplitz](https://en.wikipedia.org/wiki/Toeplitz_matrix) matrices. Notices there's a [special way to solve them](https://docs.scipy.org/doc/scipy/reference/generated/scipy.linalg.solve_toeplitz.html#scipy.linalg.solve_toeplitz).  Try it - see how the speed compares to solving the random matrix above"
    ]
   },
   {