07: Implement the Conjugate Gradient alg

Finishes task 5, but adds a TODO!

Week 3/7 Linear Equation Systems.ipynb

@@ -574,7 +574,8 @@
     "    assert len(A.shape) == 2 and A.shape[0] == A.shape[1]\n",
     "    \n",
     "    n = len(A)\n",
-    "    x_cur = np.zeros(n)\n",
+    "    \n",
+    "    x_cur  = np.zeros(n)\n",
     "    x_prev = np.zeros(n)\n",
     "    \n",
     "    k = 0\n",
@@ -583,7 +584,7 @@
     "        x_prev = x_cur.copy()\n",
     "        \n",
     "        v     = b - A @ x_prev\n",
-    "        t     = np.dot(v,v)/np.dot(v, A @ v)\n",
+    "        t     = np.dot(v, v)/np.dot(v, A @ v)\n",
     "        x_cur = x_prev.copy() + t*v\n",
     "        \n",
     "    return x_cur, k"
@@ -705,7 +706,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": null,
+   "execution_count": 9,
    "metadata": {
     "deletable": false,
     "nbgrader": {
@@ -722,14 +723,41 @@
    },
    "outputs": [],
    "source": [
-    "def CG(A, b, eps):\n",
+    "def CG(A, b, eps, k_max = 10000):\n",
-    "    # YOUR CODE HERE\n",
+    "    \"\"\"\n",
-    "    raise NotImplementedError()"
+    "    Return the Conjugate Gradient algorithm estimate solution x to the problem\n",
+    "    Ax = b and the number of iterations k it took to decrease maximum\n",
+    "    norm error below eps or to exceed iteration maximum k_max.\n",
+    "    \"\"\"\n",
+    "    \n",
+    "    # Assert n by n matrix.\n",
+    "    assert len(A.shape) == 2 and A.shape[0] == A.shape[1]\n",
+    "    \n",
+    "    n = len(A)\n",
+    "    \n",
+    "    x_cur  = np.zeros(n)\n",
+    "    x_prev = x_cur.copy()\n",
+    "    r_cur = b - A @ x_cur\n",
+    "    v     = r_cur\n",
+    "    \n",
+    "    k = 0\n",
+    "    while diff(x_cur, x_prev) > eps and k < k_max or k == 0:\n",
+    "        k += 1\n",
+    "        x_prev = x_cur.copy()\n",
+    "        r_prev = r_cur\n",
+    "        \n",
+    "        t     = np.dot(r_prev, r_prev)/np.dot(v, A @ v)\n",
+    "        x_cur = x_prev + t*v\n",
+    "        r_cur = r_prev - t*A @ v\n",
+    "        s     = np.dot(r_cur, r_cur)/np.dot(r_prev, r_prev)\n",
+    "        v     = r_cur + s*v\n",
+    "        \n",
+    "    return x_cur, k"
    ]
   },
   {
    "cell_type": "code",
-   "execution_count": null,
+   "execution_count": 10,
    "metadata": {
     "deletable": false,
     "nbgrader": {
@@ -744,11 +772,56 @@
      "task": false
     }
    },
-   "outputs": [],
+   "outputs": [
+    {
+     "name": "stdout",
+     "output_type": "stream",
+     "text": [
+      "Starting with A =\n",
+      "[[10 -1  2  0]\n",
+      " [-1 11 -1  3]\n",
+      " [ 2 -1 10 -1]\n",
+      " [ 0  3 -1  8]]\n",
+      "and b = [  6  25 -11  15]\n",
+      "We apply the Conjugate Gradient algorithm to solve Ax = b.\n",
+      "\n",
+      "For eps = 1e-01\tafter k = 4\t iterations:\n",
+      "x =\t\t\t [ 1.  2. -1.  1.]\n",
+      "Ax =\t\t\t [  6.  25. -11.  15.]\n",
+      "diff(Ax, b) =\t\t 1.7763568394002505e-15\n",
+      "diff(x, x_exact) =\t 2.220446049250313e-16\n",
+      "\n",
+      "For eps = 1e-02\tafter k = 5\t iterations:\n",
+      "x =\t\t\t [ 1.  2. -1.  1.]\n",
+      "Ax =\t\t\t [  6.  25. -11.  15.]\n",
+      "diff(Ax, b) =\t\t 1.7763568394002505e-15\n",
+      "diff(x, x_exact) =\t 2.220446049250313e-16\n",
+      "\n",
+      "For eps = 1e-03\tafter k = 5\t iterations:\n",
+      "x =\t\t\t [ 1.  2. -1.  1.]\n",
+      "Ax =\t\t\t [  6.  25. -11.  15.]\n",
+      "diff(Ax, b) =\t\t 1.7763568394002505e-15\n",
+      "diff(x, x_exact) =\t 2.220446049250313e-16\n",
+      "\n",
+      "For eps = 1e-04\tafter k = 5\t iterations:\n",
+      "x =\t\t\t [ 1.  2. -1.  1.]\n",
+      "Ax =\t\t\t [  6.  25. -11.  15.]\n",
+      "diff(Ax, b) =\t\t 1.7763568394002505e-15\n",
+      "diff(x, x_exact) =\t 2.220446049250313e-16\n",
+      "\n"
+     ]
+    }
+   ],
    "source": [
     "def test_CG():\n",
-    "    # YOUR CODE HERE\n",
+    "    \"\"\"\n",
-    "    raise NotImplementedError()\n",
+    "    Check that CG returns solutions for the example system Ax = b\n",
+    "    within the error defined by the same eps as used for the iteration.\n",
+    "    \"\"\"\n",
+    "    \n",
+    "    # TODO: Do we need to plot the error over epsilon like in task 4?\n",
+    "    \n",
+    "    return test_alg(CG, \"Conjugate Gradient\")\n",
     "    \n",
     "test_CG()"
    ]