Next: Lagrange Interpolation

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

tabulated below for small .
 0.1 0.2 0.3 0.4 0.5
This function diverges as at . To evaluate it requires doing the integral, or summing a series, so it is somewhat expensive. Suppose we want the value of . A very crude approximation chooses either of the two nearby values or . 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 .