Lecture 18. 



Interpolating with the Newton basis.
Today I showed how Horner's rule can be used to evaluated the interpolating polynomial using the coefficients of the Newton basis. The advantage of using Horner's rule is misstated in the Ellis notes. Ellis remarks that Horner's rule uses half the number of multiplications when in fact it uses roughly the square root of the number of multiplications when compared to evaluating a polynomial naively.

Demonstrations with Matlab
Today we saw some illuminating demonstrations, using Matlab, pertaining to polynomial interpolation.
We started with an exploration of full degree polynomial interpolation and the demonstration programs from Recktenwald:
demoGasNewt, demoGasLag, and demoGasVand. The main observation is that the poor scaling of the Vandermonde matrix yields visual artifacts in the polynomial plot. Then we looked at Moler's interpgui. This emphasized the difference between local interpolation methods, such as straight line and Hermite interpolation, and interpolation where data points effect the interpolation curve globally, such as full degree polynomial interpolation and cubic splines.

Piecewise Polynomial Interpolation
Hermite interpolation and straight line interpolation are piece-wise polynomial interpolation methods. They both use polynomial functions to locally 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.  

Posted: Fri - October 20, 2006 at 03:24 PM          

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