gibson:teaching:fall-2016:math753:quadrature
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gibson:teaching:fall-2016:math753:quadrature [2016/11/17 06:59] gibson |
gibson:teaching:fall-2016:math753:quadrature [2016/11/17 07:04] gibson |
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| interpolation between triplets of datapoints. | interpolation between triplets of datapoints. | ||
| \begin{equation*} | \begin{equation*} | ||
| - | \int_a^b f(x) \, dx = \frac{h}{3} \left( y_0 + 4 \sum_{i=1, i \text{odd}}^{N-1} y_i + 2\sum_{i=2, i \text{even}}^{N-2} y_i + y_N\right) - \frac{b-a}{180} h^4 f^{(iv)}(c) | + | \int_a^b f(x) \, dx = \frac{h}{3} \left( y_0 + 4 \sum_{i=1, i \text{odd}}^{N-1} y_i + 2\sum_{i=2, i \text{even}}^{N-2} y_i + y_N\right) - \frac{b-a}{180} \, h^4 \, f^{(iv)}(c) |
| \end{equation*} | \end{equation*} | ||
| for some $c$ where $a<c<b$, and where $f^{(iv)}$ is the fourth derivative of $f$. Simpson's rule requires that $N$ is even, so that the total number of gridpoints, $N+1$, is odd. | for some $c$ where $a<c<b$, and where $f^{(iv)}$ is the fourth derivative of $f$. Simpson's rule requires that $N$ is even, so that the total number of gridpoints, $N+1$, is odd. | ||
| - | For example, for $N=6$ and 7 gridpoints, we have | + | For example, for $N=6$ and $N+1=7$ gridpoints, we have |
| \begin{equation*} | \begin{equation*} | ||
| \int_a^b f(x) \, dx = \frac{h}{3} \left(y_0 + 4y_1 + 2y_2 + 4y_3 + 2y_4 + 4y_5 + y_6 \right) + O(h^4) | \int_a^b f(x) \, dx = \frac{h}{3} \left(y_0 + 4y_1 + 2y_2 + 4y_3 + 2y_4 + 4y_5 + y_6 \right) + O(h^4) | ||
| \end{equation*} | \end{equation*} | ||
| + | |||
| + | |||
| + | === Further reading === | ||
| + | |||
| + | * [[https://en.wikipedia.org/wiki/Trapezoidal_rule|Trapezoidal rule]] (wikipedia) | ||
| + | * [[http://mathworld.wolfram.com/TrapezoidalRule.html|Trapezoidal rule]] (math world) | ||
| + | * [[https://en.wikipedia.org/wiki/Simpson%27s_rule|Simpson's rule]] (wikipedia) | ||
| + | * [[http://mathworld.wolfram.com/SimpsonsRule.html|Simpson's rule]] (math world) | ||
| + | * [[https://en.wikipedia.org/wiki/Newton%E2%80%93Cotes_formulas|Newton-Cotes formulae]] (wikipedia) | ||
| + | * [[http://mathworld.wolfram.com/Newton-CotesFormulas.html|Newton-Cotes formulae]] (math world) | ||
| + | * [[http://mathworld.wolfram.com/TrapezoidalRule.html|Trapezoidal rule]] (math world) | ||
gibson/teaching/fall-2016/math753/quadrature.txt ยท Last modified: 2016/11/17 07:04 by gibson
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