06: This seems to be it, although we were not able to correctly read task 3.

Week 2/6 Composite Numerical Integration: Trapezoid and Simpson Rules.ipynb

@@ -75,7 +75,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 11,
+   "execution_count": 1,
    "metadata": {
     "deletable": false,
     "nbgrader": {
@@ -93,7 +93,10 @@
    "outputs": [],
    "source": [
     "import numpy as np\n",
-    "import scipy.integrate"
+    "import scipy.integrate\n",
+    "\n",
+    "# And for printing the lambdas:\n",
+    "import inspect"
    ]
   },
   {
@@ -124,7 +127,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 1,
+   "execution_count": 2,
    "metadata": {
     "deletable": false,
     "nbgrader": {
@@ -158,7 +161,31 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 13,
+   "execution_count": 3,
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "def compare_integration(f, a, b, n):\n",
+    "    # Let's check whether f is callable.\n",
+    "    # TODO: Improve checks on f, or not.\n",
+    "    assert callable(f)\n",
+    "    \n",
+    "    h    = (b - a)/n\n",
+    "    xk   = np.linspace(a, b, n + 1)\n",
+    "    yk   = f(xk)\n",
+    "    \n",
+    "    print(\"For function\", inspect.getsource(f))\n",
+    "    print(\"for boundaries a =\", a, \", b =\", b, \"and steps n =\", n, \"the algorithms say:\")\n",
+    "    print(\"trapezoid:\\t\\t\", trapz(yk, h))\n",
+    "    print(\"Simpson:\\t\\t\", simps(yk, h))\n",
+    "    print(\"scipy.integrate.trapz:\\t\", scipy.integrate.trapz(yk, xk))\n",
+    "    print(\"scipy.integrate.simps:\\t\", scipy.integrate.simps(yk, xk))\n",
+    "    print()"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 4,
    "metadata": {
     "deletable": false,
     "nbgrader": {
@@ -179,37 +206,40 @@
     "f = lambda x: x**2\n",
     "\n",
     "n    = 100001\n",
-    "a, b = 0, 1\n",
+    "a, b = 0, 1"
-    "h    = (b - a)/n\n",
-    "xk   = np.linspace(a, b, n + 1)\n",
-    "yk   = f(xk)"
    ]
   },
   {
    "cell_type": "code",
-   "execution_count": 14,
+   "execution_count": 5,
    "metadata": {},
    "outputs": [
     {
      "name": "stdout",
      "output_type": "stream",
      "text": [
-      "For function f(x) = x^2\n",
+      "For function f = lambda x: x**2\n",
+      "\n",
       "for boundaries a = 0 , b = 1 and steps n = 100001 the algorithms say:\n",
       "trapezoid:\t\t 0.33333333334999976\n",
       "Simpson:\t\t 0.3333300000666658\n",
       "scipy.integrate.trapz:\t 0.33333333334999965\n",
-      "scipy.integrate.simps:\t 0.3333333333333335\n"
+      "scipy.integrate.simps:\t 0.3333333333333335\n",
+      "\n",
+      "For function f = lambda x: x**2\n",
+      "\n",
+      "for boundaries a = 0 , b = 1 and steps n = 100002 the algorithms say:\n",
+      "trapezoid:\t\t 0.3333333333499994\n",
+      "Simpson:\t\t 0.33333333333333337\n",
+      "scipy.integrate.trapz:\t 0.3333333333499993\n",
+      "scipy.integrate.simps:\t 0.3333333333333333\n",
+      "\n"
      ]
     }
    ],
    "source": [
-    "print(\"For function f(x) = x^2\")\n",
+    "compare_integration(f, a, b, n)\n",
-    "print(\"for boundaries a =\", a, \", b =\", b, \"and steps n =\", n, \"the algorithms say:\")\n",
+    "compare_integration(f, a, b, n + 1)"
-    "print(\"trapezoid:\\t\\t\", trapz(yk, h))\n",
-    "print(\"Simpson:\\t\\t\", simps(yk, h))\n",
-    "print(\"scipy.integrate.trapz:\\t\", scipy.integrate.trapz(yk, xk))\n",
-    "print(\"scipy.integrate.simps:\\t\", scipy.integrate.simps(yk, xk))"
    ]
   },
   {
@@ -236,7 +266,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": null,
+   "execution_count": 6,
    "metadata": {
     "deletable": false,
     "nbgrader": {
@@ -251,15 +281,67 @@
      "task": false
     }
    },
-   "outputs": [],
+   "outputs": [
+    {
+     "name": "stdout",
+     "output_type": "stream",
+     "text": [
+      "For function f(x) = x^3 + 6x\n",
+      "for boundaries a = 2 , b = 16 and steps n = 82198 the algorithms say:\n",
+      "trapezoid:\t\t 1362.6666667343538\n",
+      "scipy.integrate.trapz:\t 1362.6666667343543\n",
+      "with difference trapz(yk, h) - scipy.integrate.trapz(yk, xk) = -4.547473508864641e-13\n",
+      "\n",
+      "For function f(x) = -x^3 + 6x\n",
+      "for boundaries a = 2 , b = 17 and steps n = 82228 the algorithms say:\n",
+      "Simpson:\t\t 1635.0\n",
+      "scipy.integrate.simps:\t 1635.0000000000002\n",
+      "with difference simps(yk, h) - scipy.integrate.simps(yk, xk) = -2.2737367544323206e-13\n",
+      "\n"
+     ]
+    }
+   ],
    "source": [
+    "# In the comparison of n even and n odd, and the testing of the integrations,\n",
+    "# we have already tested the functions, but as it is asked, here we go again.\n",
+    "\n",
     "def test_trapz():\n",
-    "    # YOUR CODE HERE\n",
+    "    fun  = lambda x: x**3 + 6*x\n",
-    "    raise NotImplementedError()\n",
+    "    a, b = 2, 16\n",
+    "    n    = 82198\n",
+    "    \n",
+    "    h    = (b - a)/n\n",
+    "    xk   = np.linspace(a, b, n + 1)\n",
+    "    yk   = f(xk)\n",
+    "\n",
+    "    trapz_our   = trapz(yk, h)\n",
+    "    trapz_scipy = scipy.integrate.trapz(yk, xk)\n",
+    "    \n",
+    "    print(\"For function f(x) = x^3 + 6x\")\n",
+    "    print(\"for boundaries a =\", a, \", b =\", b, \"and steps n =\", n, \"the algorithms say:\")\n",
+    "    print(\"trapezoid:\\t\\t\", trapz_our)\n",
+    "    print(\"scipy.integrate.trapz:\\t\", trapz_scipy)\n",
+    "    print(\"with difference trapz(yk, h) - scipy.integrate.trapz(yk, xk) =\", trapz_our - trapz_scipy)\n",
+    "    print()\n",
     "    \n",
     "def test_simps():\n",
-    "    # YOUR CODE HERE\n",
+    "    fun  = lambda x: -x**3 + 6*x\n",
-    "    raise NotImplementedError()\n",
+    "    a, b = 2, 17\n",
+    "    n    = 82228\n",
+    "    \n",
+    "    h    = (b - a)/n\n",
+    "    xk   = np.linspace(a, b, n + 1)\n",
+    "    yk   = f(xk)\n",
+    "\n",
+    "    simps_our   = simps(yk, h)\n",
+    "    simps_scipy = scipy.integrate.simps(yk, xk)\n",
+    "    \n",
+    "    print(\"For function f(x) = -x^3 + 6x\")\n",
+    "    print(\"for boundaries a =\", a, \", b =\", b, \"and steps n =\", n, \"the algorithms say:\")\n",
+    "    print(\"Simpson:\\t\\t\", simps_our)\n",
+    "    print(\"scipy.integrate.simps:\\t\", simps_scipy)\n",
+    "    print(\"with difference simps(yk, h) - scipy.integrate.simps(yk, xk) =\", simps_our - simps_scipy)\n",
+    "    print()\n",
     "    \n",
     "test_trapz()\n",
     "test_simps()"
@@ -295,7 +377,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": null,
+   "execution_count": 7,
    "metadata": {
     "deletable": false,
     "nbgrader": {
@@ -310,10 +392,60 @@
      "task": false
     }
    },
-   "outputs": [],
+   "outputs": [
+    {
+     "name": "stdout",
+     "output_type": "stream",
+     "text": [
+      "As we are calculus geniuses, we know that the first integral gives 1/2.\n",
+      "For function f1 = lambda x: x\n",
+      "\n",
+      "for boundaries a = 0 , b = 1 and steps n = 100000 the algorithms say:\n",
+      "trapezoid:\t\t 0.5000000000000001\n",
+      "Simpson:\t\t 0.5000000000000001\n",
+      "scipy.integrate.trapz:\t 0.5000000000000001\n",
+      "scipy.integrate.simps:\t 0.5\n",
+      "\n",
+      "As we are calculus geniuses, we know that the second integral gives 1/3.\n",
+      "For function f2 = lambda x: x**2\n",
+      "\n",
+      "for boundaries a = 0 , b = 1 and steps n = 100000 the algorithms say:\n",
+      "trapezoid:\t\t 0.33333333335000004\n",
+      "Simpson:\t\t 0.3333333333333335\n",
+      "scipy.integrate.trapz:\t 0.33333333335\n",
+      "scipy.integrate.simps:\t 0.3333333333333333\n",
+      "\n",
+      "As we are calculus geniuses, we know that the third integral gives 2/3.\n",
+      "For function f3 = lambda x: x**(1/2)\n",
+      "\n",
+      "for boundaries a = 0 , b = 1 and steps n = 100000 the algorithms say:\n",
+      "trapezoid:\t\t 0.6666666600968939\n",
+      "Simpson:\t\t 0.6666666640993837\n",
+      "scipy.integrate.trapz:\t 0.6666666600968938\n",
+      "scipy.integrate.simps:\t 0.6666666640993836\n",
+      "\n",
+      "For all three cases, the results are very close, and the functions work quickly.\n"
+     ]
+    }
+   ],
    "source": [
-    "# YOUR CODE HERE\n",
+    "f1 = lambda x: x\n",
-    "raise NotImplementedError()"
+    "f2 = lambda x: x**2\n",
+    "f3 = lambda x: x**(1/2)\n",
+    "\n",
+    "n    = 100000\n",
+    "a, b = 0, 1\n",
+    "\n",
+    "print(\"As we are calculus geniuses, we know that the first integral gives 1/2.\")\n",
+    "compare_integration(f1, a, b, n)\n",
+    "\n",
+    "print(\"As we are calculus geniuses, we know that the second integral gives 1/3.\")\n",
+    "compare_integration(f2, a, b, n)\n",
+    "\n",
+    "print(\"As we are calculus geniuses, we know that the third integral gives 2/3.\")\n",
+    "compare_integration(f3, a, b, n)\n",
+    "\n",
+    "print(\"For all three cases, the results are very close, and the functions work quickly.\")"
    ]
   },
   {