Page 884 - Elementary_Linear_Algebra_with_Applications_Anton__9_edition

Moreover, we already know from 8 that and . The final

We leave it as an exercise for the reader to derive the expressions for the and in terms of the
result is as follows:
THEOREM 11.5.1

Cubic Spline Interpolation with , , 2, …, , the cubic spline
Given n points , , …,

that interpolates these points has coefficients given by

(14)

for , 2, …, , where , , 2, …, n.

From this result, we see that the quantities , , …, uniquely determine the cubic spline. To find these quantities, we
substitute the expressions for , , and given in 14 into 12. After some algebraic simplification, we obtain

(15)

or, in matrix form,

This is a linear system of equations for the n unknowns , , …, . Thus, we still need two additional equations to
determine , , …, uniquely. The reason for this is that there are infinitely many cubic splines that interpolate the given
points, so we simply do not have enough conditions to determine a unique cubic spline passing through the points. We discuss
below three possible ways of specifying the two additional conditions required to obtain a unique cubic spline through the points.
(The exercises present two more.) They are summarized in Table 1.

Table 1