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added final project
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rcc02007 committed Nov 20, 2017
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241 changes: 88 additions & 153 deletions 17_integrals_and_derivatives/17_integrals.ipynb
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},
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"source": [
"# Numerical Integrals"
]
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"source": [
"# Numerical Integrals\n",
"\n",
"# Integrals in practice\n",
"\n",
"### Example: Compare toughness of Stainless steel to Structural steel\n",
"\n",
"![Stress-strain plot of steel](steel_psi.jpg)\n",
"\n",
"### Step 1 - G3Data to get points \n",
"### Step 1 - G3Data to get points\n",
"\n",
"Use the plot shown to determine the toughness of stainless steel and the toughness of structural steel.\n",
"\n"
"Use the plot shown to determine the toughness of stainless steel and the\n",
"toughness of structural steel."
]
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},
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"source": [
"# Gauss Quadrature (for functions)\n",
"\n",
"Evaluating an integral, we assumed a polynomial form for each Newton-Cotes approximation.\n",
"Evaluating an integral, we assumed a polynomial form for each Newton-Cotes\n",
"approximation.\n",
"\n",
"If we can evaluate the function at any point, it makes more sense to choose points more wisely rather than just using endpoints\n",
"If we can evaluate the function at any point, it makes more sense to choose\n",
"points more wisely rather than just using endpoints\n",
"\n",
"![trapezoidal example](trap_example.png)"
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"Set up two unknown constants, $c_{0}$ and $x_{0}$ and determine a *wise* place to evaluate f(x) such that \n",
"![trapezoidal example](trap_example.png)\n",
"\n",
"Set up two unknown constants, $c_{0}$ and $x_{0}$ and determine a *wise* place\n",
"to evaluate f(x) such that\n",
"\n",
"$I=c_{0}f(x_{0})$\n",
"\n",
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"This process is called **Gauss Quadrature**. Usually, the bounds are fixed at -1 and 1 instead of a and b\n",
"This process is called **Gauss Quadrature**. Usually, the bounds are fixed at -1\n",
"and 1 instead of a and b\n",
"\n",
"$I=c_{0}f(x_{0})$\n",
"\n",
Expand All @@ -366,17 +345,8 @@
"\n",
"so $c_{0}=2$ and $x_{0}=0$\n",
"\n",
"$I=\\int_{-1}^{1}f(x)dx \\approx 2f\\left(0\\right)$"
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"source": [
"$I=\\int_{-1}^{1}f(x)dx \\approx 2f\\left(0\\right)$\n",
"\n",
"Now, integrals can be performed with a change of variable\n",
"\n",
"a=2\n",
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"\n",
"at $x_{d}=-1$, x=a\n",
"\n",
"at $x_{d}=1$, x=b"
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"so \n",
"at $x_{d}=1$, x=b\n",
"\n",
"so\n",
"\n",
"$x=\\frac{(b+a) +(b-a)x_{d}}{2}$\n",
"\n",
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"\n",
"$\\int_{2}^{3}x+1dx=\\int_{-1}^{1}\\left(\\frac{5 +x_{d}}{2}\n",
"+1\\right)\n",
"\\frac{3-2}{2}dx_{d}$"
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"$\\int_{2}^{3}x+1dx=\\int_{-1}^{1}\\left(\\frac{7}{4}+\\frac{1}{4}x_{d}\\right)dx_{d}=2\\frac{7}{4}=3.5$"
"\\frac{3-2}{2}dx_{d}$\n",
"\n",
"$\\int_{2}^{3}x+1dx=\\int_{-1}^{1}\\left(\\frac{7}{4}+\\frac{1}{4}x_{d}\\right)dx_{d}=\n",
"2\\frac{7}{4}=3.5$"
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},
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"## General Gauss weights and points\n",
"\n",
"![Gauss quadrature table](gauss_weights.png)"
"![Gauss quadrature table](gauss_weights.png)\n",
"\n",
"### If you need to evaluate an integral, to increase accuracy, increase number\n",
"of Gauss points\n",
"\n",
"### Adaptive Quadrature\n",
"\n",
"Matlab/Octave built-in functions use two types of adaptive quadrature to\n",
"increase accuracy of integrals of functions.\n",
"\n",
"1. `quad`: Simpson quadrature good for nonsmooth functions\n",
"\n",
"2. `quadl`: Lobatto quadrature good for smooth functions"
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"### If you need to evaluate an integral, to increase accuracy, increase number of Gauss points\n",
"\n",
"### Adaptive Quadrature\n",
"\n",
"Matlab/Octave built-in functions use two types of adaptive quadrature to increase accuracy of integrals of functions. \n",
"\n",
"1. `quad`: Simpson quadrature good for nonsmooth functions\n",
"\n",
"2. `quadl`: Lobatto quadrature good for smooth functions\n",
"\n",
"```matlab\n",
"q = quad(fun, a, b, tol, trace, p1, p2, …)\n",
"fun : function to be integrates\n",
"a, b: integration bounds\n",
"tol: desired absolute tolerance (default: 10-6)\n",
"trace: flag to display details or not\n",
"p1, p2, …: extra parameters for fun\n",
"quadl has the same arguments\n",
"```"
"quadl has the same arguments"
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"source": [
"# Thanks"
]
}
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"text": "MetaKernel Magics",
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