Definition

Given a function defined on and a set of nodes with inside the interval . Choose and . Then a cubic spline interpolant , , for is a function which satisfies the following conditions:

(i) consists of , a cubic polynomial on the subinterval for each ;

(ii) for all ;

(iii) for all ;

(iv) for all ;

(v) for all ;

and a set of boundary conditions, either

(a) = 0 (free/natural boundary);

or

(b) and (clamped boundary)

>

Clamped conditions normally lead to a more accurate approximation but it requires knowledge about the derivative of the function at the two end points.