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

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Kees van Kempen
2022-02-15 15:56:25 +01:00
parent 84ccfe59bf
commit d09e0c685e
1 changed files with 159 additions and 27 deletions
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186
Week 2/6 Composite Numerical Integration: Trapezoid and Simpson Rules.ipynb
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@ -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",
"h = (b - a)/n\n",
"xk = np.linspace(a, b, n + 1)\n",
"yk = f(xk)"
"a, b = 0, 1"
]
},
{
"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",
"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))"
"compare_integration(f, a, b, n)\n",
"compare_integration(f, a, b, n + 1)"
]
},
{
@ -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",
" raise NotImplementedError()\n",
" fun = lambda x: x**3 + 6*x\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",
" raise NotImplementedError()\n",
" fun = lambda x: -x**3 + 6*x\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",
"raise NotImplementedError()"
"f1 = lambda x: x\n",
"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.\")"
]
},
{