Introduction to Numerical Analysis I MATH/CMPSC 455 Splines

S PLINE Suppose 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

C UBIC S PLINE 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!

D ERIVE THE C UBIC S PLINE  Step 1: 2 nd order derivative is piecewise linear; (use the continuity of 2 nd 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 1 st order derivative)

Where: Nature Cubic Spline:

Clamped Cubic Spline: