gibson:teaching:fall-2016:math753:lagrangepoly
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gibson:teaching:fall-2016:math753:lagrangepoly [2016/11/11 12:29] gibson |
gibson:teaching:fall-2016:math753:lagrangepoly [2016/11/11 12:38] gibson |
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| \begin{equation*} | \begin{equation*} | ||
| - | P(x) = y_1 \frac{(x-x_2)(x-x_3)}{(x_1-x_2)(x_1-x_3)} + y_1 \frac{(x-x_1)(x-x_3)}{(x_2-x_1)(x_2-x_3)} + y_3 \frac{(x-x_1)(x-x_2)}{(x_3-x_1)(x_3-x_1)} | + | P(x) = y_1 \frac{(x-x_2)(x-x_3)}{(x_1-x_2)(x_1-x_3)} + y_1 \frac{(x-x_1)(x-x_3)}{(x_2-x_1)(x_2-x_3)} + y_3 \frac{(x-x_1)(x-x_2)}{(x_3-x_1)(x_3-x_2)} |
| \end{equation*} | \end{equation*} | ||
| It's easy to see the polynomial goes through each data point. If you plug in $x=x_1$, the second and third terms vanish, and the fraction in the first term is $1$, so that $P(x_1) = y_1$. Similar simplifications occur when plugging in $x=x_2$, to get $P(x_2) = y_2$, and similar for $P(x_3) = y_3$. Also, since everything on the right-hand-side except $x$ is a constant, it's clear that the $P(x)$ is a polynomial in $x$ of order 2. The generalization to higher-order polynomials is straightforward. | It's easy to see the polynomial goes through each data point. If you plug in $x=x_1$, the second and third terms vanish, and the fraction in the first term is $1$, so that $P(x_1) = y_1$. Similar simplifications occur when plugging in $x=x_2$, to get $P(x_2) = y_2$, and similar for $P(x_3) = y_3$. Also, since everything on the right-hand-side except $x$ is a constant, it's clear that the $P(x)$ is a polynomial in $x$ of order 2. The generalization to higher-order polynomials is straightforward. | ||
| + | For example, let's fit a 2nd order polynomial to $(0,4), (1,3), (2,6)$. Plugging into the formula gives | ||
| + | \begin{equation*} | ||
| + | P(x) = 4 \frac{(x-1)(x-2)}{(-1)(-2)} + 3 \frac{(x)(x-2)}{(1)(-1)} + 6 \frac{(x)(x-1)}{(2)(1)} | ||
| + | \end{equation*} | ||
| + | |||
| + | This is mathematically clunky because it expresses a quadratic polynomial $P(x)$ as the sum of three quadratics. Simplifying takes a lot of multiplications and additions. If we do that, we get | ||
| + | |||
| + | \begin{equation*} | ||
| + | P(x) = 4 - 3x + 2x^2 | ||
| + | \end{equation*} | ||
| + | |||
| + | which can be easily verified as passing through the given data points. | ||
| + | |||
| + | Further reading | ||
| + | * [[http://mathworld.wolfram.com/LagrangeInterpolatingPolynomial.html | Lagrange Interpolating Polynomial]] (Wolfram Mathworld) | ||
| + | * [[https://en.wikipedia.org/wiki/Lagrange_polynomial| Lagrange Polynomial]] (Wikipedia) | ||
gibson/teaching/fall-2016/math753/lagrangepoly.txt ยท Last modified: 2016/11/11 12:39 by gibson
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