CS3414 Afterclass Notes --- 3 June, 2002

3. Piecewise Polynomials

   1. Intro (last week)

   2. Splines

      • To complete our discussion of splines, we looked at how to construct a cubic spline interpolant. The straightforward approach is to set up and solve a 4n x 4n linear system for the 4n coefficients defining a piecewise cubic written in the standard form, i.e., n different cubic polynomials defined in the power or shifted-power form, one polynomial for each piece. By enforcing the interpolation and continuity conditions, we have 4n-2 equations. Need two more. In the case of ``natural'' splines, the two additional equations are p''(t0) = 0 and p''(tn) = 0. (Note: with shifted power basis and a little cleverness, cost to solve for the coefficients can be reduced to O(n)).

      • By choosing basis functions cleverly, the cost can be reduced further, e.g., using B-spline basis functions (see Section 7.3), or Hermite cubics.

   3. Hermite cubics

      • Definition: a piecewise cubic with only 1 continuous derivative at the knots.

      • The Hermite cubic basis functions are defined to make it easy to define an interpolating piecewise cubic. This approach is especially convenient if you know both function values and derivatives at the knots.