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linear system in 1 is said to be a Vandermonde system.

EXAMPLE 5 Interpolating a Cubic
To interpolate a polynomial to the data (−2, 11), (−1, 2), (1, 2), (2, −1), we form the Vandermonde system 1:

For this data, we have

The solution, found by Gaussian elimination, is

and so the interpolant is . This is plotted in Figure 4.4.3, together with the data points, and we see that

does indeed interpolate the data, as required.

Figure 4.4.3
The interpolant of Example 4

Newton Form

The interpolating polynomial is said to be written in its natural, or standard, form. But

there is convenience in using other forms. For example, suppose we seek a cubic interpolant to the data , ,

, . If we write

(2)

in the equivalent form