Interpolating Cubic Polynomial Splines

Cubic Polynomial Splines

Assume that values of a function are known at the nodes , . We desire to construct a smooth curve passing through the given points with polynomial splines of third degree, also called  cubic splines. If we assume that the nodes are ordered , then we can represent the desired curve by a spline function with , which is composed of cubic polynomials for and . These are differently for each subinterval and must satisfy certain connecting conditions at the nodes.

Definition of Interpolating Cubic Spline Functions

Depending on the conditions imposed at and , different types of spline functions will result. For a set of ordered nodes with , the spline functions mentioned previously can be defined by the following conditions:

(1)

.

(2)	is a cubic polynomial in each subinterval , .

(3)	interpolates , that is, for .

Types of Interpolating Cubic Spline Functions

(a)	Natural Cubic Splines: For or , is represented by the tangent of at or , thus .

(b)	Complete Cubic Splines: Cubic spline with specified first end point derivatives — For the end point conditions , .

(c)	Generalized Natural Cubic Spline: Cubic spline with specified second end point derivatives — For the end point conditions , .

(d)	Cubic Spline with Specified Third End-Point Derivatives: For the end point conditions , .

(e)	Cubic Spline with Not-a-Knot Condition: If satisfies the endpoint conditions and . This condition means that the third derivative of the spline function is continuous at the nodes and . Thus and are not “genuine” nodes for the spline function.

To construct a function that satisfies (1) and (2) above, we set

, for ,

A cubic polynomial spline is twice continuously differentiable, depending on 4 parameters. Hence it is a spline function of odd degree for [-times continuously differentiable, -parametric].

Properties (1) and (3) for lead to conditions for the :

(A)

,

(B)

,

(C)

,

(D)

,

Here we have set and . Two additional end point conditions can be specified at will. Then we have to satisfy conditions for the  coefficients for and . Property (1) is the single most rigorous requirement of a spline function . It gives rise to conditions (C) and (D) and effects a smooth transition between the graphs of the polynomials and at the node for each : The graphs of adjacent polynomials and  have the same curvature at . This property of spline functions is especially useful for approximating a function which is known empirically and which one can draw using a flexible ruler, commonly known as a shipbuilder’s spline.