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06: Task 1 be done.

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Kees van Kempen
2022-02-15 15:32:51 +01:00
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Week 2/6 Composite Numerical Integration: Trapezoid and Simpson Rules.ipynb
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@ -39,7 +39,7 @@
"cell_type": "raw", "cell_type": "raw",
"metadata": {}, "metadata": {},
"source": [ "source": [
"team_members = \"\"" "team_members = \"Koen Vendrig, Kees van Kempen\""
] ]
}, },
{ {
@ -75,7 +75,7 @@
}, },
{ {
"cell_type": "code", "cell_type": "code",
"execution_count": null, "execution_count": 11,
"metadata": { "metadata": {
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"nbgrader": { "nbgrader": {
@ -92,10 +92,8 @@
}, },
"outputs": [], "outputs": [],
"source": [ "source": [
"# Import packages here ...\n", "import numpy as np\n",
"\n", "import scipy.integrate"
"# YOUR CODE HERE\n",
"raise NotImplementedError()"
] ]
}, },
{ {
@ -126,7 +124,7 @@
}, },
{ {
"cell_type": "code", "cell_type": "code",
"execution_count": null, "execution_count": 1,
"metadata": { "metadata": {
"deletable": false, "deletable": false,
"nbgrader": { "nbgrader": {
@ -144,17 +142,23 @@
"outputs": [], "outputs": [],
"source": [ "source": [
"def trapz(yk, dx):\n", "def trapz(yk, dx):\n",
" # YOUR CODE HERE\n", " a, b = yk[0], yk[-1]\n",
" raise NotImplementedError()\n", " h = dx\n",
" integral = h/2*(a + 2*np.sum(yk[1:-1]) + b)\n",
" return integral\n",
" \n", " \n",
"def simps(yk, dx):\n", "def simps(yk, dx):\n",
" # YOUR CODE HERE\n", " a, b = yk[0], yk[-1]\n",
" raise NotImplementedError()" " h = dx\n",
" # Instead of summing over something with n/2, we use step size 2,\n",
" # thus avoiding any issues with 2 ∤ n.\n",
" integral = h/3*(a + 2*np.sum(yk[2:-1:2]) + 4*np.sum(yk[1:-1:2]) + b)\n",
" return integral"
] ]
}, },
{ {
"cell_type": "code", "cell_type": "code",
"execution_count": null, "execution_count": 13,
"metadata": { "metadata": {
"deletable": false, "deletable": false,
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@ -171,10 +175,41 @@
}, },
"outputs": [], "outputs": [],
"source": [ "source": [
"# Compare your results here ...\n", "# We need a function to integrate, so here we go.\n",
"f = lambda x: x**2\n",
"\n", "\n",
"# YOUR CODE HERE\n", "n = 100001\n",
"raise NotImplementedError()" "a, b = 0, 1\n",
"h = (b - a)/n\n",
"xk = np.linspace(a, b, n + 1)\n",
"yk = f(xk)"
]
},
{
"cell_type": "code",
"execution_count": 14,
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"For function f(x) = x^2\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"
]
}
],
"source": [
"print(\"For function f(x) = x^2\")\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))"
] ]
}, },
{ {
@ -291,7 +326,7 @@
], ],
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"display_name": "Python 3 (ipykernel)", "display_name": "Python 3",
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