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--- gibson:teaching:fall-2016:math753:lagrangepoly [2016/11/11 12:24]
gibson created
+++ gibson:teaching:fall-2016:math753:lagrangepoly [2016/11/11 12:29]
gibson
@@ Line 9: / Line 9: @@
 \end{equation*}
-Obviously,
+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.

channelflow.org