Page 882 - Elementary_Linear_Algebra_with_Applications_Anton__9_edition

(2)

where , , …, are cubic polynomials. For convenience, we will write these in the form

(3)

The , , , and constitute a total of coefficients that we must determine to specify completely. If we

choose these coefficients so that interpolates the n specified points in the plane and , , and are continuous,

then the resulting interpolating curve is called a cubic spline.

Derivation of the Formula of a Cubic Spline

From Equations 2 and 3, we have

(4)
so

(5)

and

(6)

We will now use these equations and the four properties of cubic splines stated below to express the unknown coefficients , ,
, , , 2, …, , in terms of the known coordinates , , …, .

1. interpolates the points , , 2, …, n. Because interpolates the points , , 2, …, n, we have

(7)

From the first of these equations and 4, we obtain

(8)

From the last equation in 7, the last equation in 4, and the fact that , we obtain

(9)