Introduction to Numerical Analysis I

MATH/CMPSC 455

Splines

SPLINESuppose that n+1 points has been specified and satisfy . A spline of degree k is a function such that:

On each subinterval , is a polynomial of degree

has a continuous (k-1)-th derivate on

Spline is a piecewise polynomial of degree at most k, and has continuous derivatives of all order up to k-1.

Example: Spline of degree 0

Example: Spline of degree 1

CUBIC SPLINE

A cubic spline is a piecewise cubic polynomial

x …y …

is cubic polynomial (piecewise polynomial)

(Interpolation)

, (Continuity)

Question: Can we uniquely determine the cubic spline?

Unknowns (coefficients):

Conditions:

• Interpolation:• Continuity of 1st order derivative:• Continuity of 2nd order derivative:

Total:

We have two degrees of freedom!

DERIVE THE CUBIC SPLINE Step 1: 2nd order derivative is piecewise

linear; (use the continuity of

2nd order derivative)

Step 2: Take integration twice, get the cubic spline with undetermined coefficient;

Step 3: Determine the coefficient of the low order terms; (use the interpolation property)

Step 4: Determine the remaining coefficient

by solving a symmetric, tri-diagonal system;(use the continuity of 1st order derivative)

Where:

Nature Cubic Spline:

Clamped Cubic Spline: