Lecture 21 


Piece wise cubic interpolation. 

Hermite interpolation and Cubic Spline interpolation are piece-wise cubic polynomial interpolation methods. They both use cubic polynomials to interpolate each of the pieces. Hermite interpolation is named after the French mathematician Charles Hermite. Let me emphasize that for material on piecewise polynomial interpolation the notation Pi(x) the i in the subscript refers to the polynomial interpolating the ith piece, and not the degree of the polynomial. For each piece a separate cubic polynomial is found.

We then turned our attention to cubic splines. No explicit knowledge of an underlying function is needed to determine the cubic spline polynomial, rather we constrain the curve by insisting that the polynomial on both sides of each know agrees with the function at the knot, and that the first and second derivative of the cubic polynomial agree at the knot. We then used the coefficients of the Hermite polynomial as a base to derive the coefficients for the spline polynomials. The derivation was rather lengthy and technical. The point of this exercise was to give you an idea, even if you didn't get all of the details, of how splines work.

We will continue to look at splines the next time we meet which is next Tuesday.  

Posted: Fri - October 28, 2005 at 12:41 PM          

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