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Next: Lagrange Interpolation

\fbox{\Large \bf Physics 6720 - Neville Interpolation -
\normalsize October 25, 2002}

When it is expensive or difficult to evaluate a function $f(x)$ at an arbitrary value of $x$, we might consider, instead, interpolating from a table of values. Consider the exponential integral,

{\rm Ei}(x) = P\int_{-\infty}^x e^{-t}dt/t

tabulated below for small $x$.
$x$ ${\rm Ei}(x)$
0.1 $-1.6228$
0.2 $-0.8218$
0.3 $-0.3027$
0.4 $ 0.1048$
0.5 $ 0.4542$
This function diverges as $\log(x)$ at $x = 0$. To evaluate it requires doing the integral, or summing a series, so it is somewhat expensive. Suppose we want the value of ${\rm Ei}(0.15)$. A very crude approximation chooses either of the two nearby values ${\rm Ei}(0.1) =
-1.6228$ or ${\rm Ei}(0.2) = -0.8218$. A common and somewhat better approach makes a linear interpolation. Since 0.15 is midway between 0.1 and 0.2 the linear interpolation just averages the two values, giving $-1.2223$.