Lecture 19.
Polynomial interpolation with the Newton
basis
Today we looked at Newton divided differences. I
wrote out the divided difference for a degree two polynomial, but I did not
derive it on the board. Please note the derivation in Recktenwald leads to a
formulation that does not match the usual divided difference form. There is also
an error in the way f[x1,x2,x3] is defined on page 540. If you follow the
derivation the first term in the numerator should be f[x1,x3]. This error is
reported in the errata updates on the NMM web site.
I also 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'r 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.
I closed with a brief
description of piece-wise polynomial interpolation. On Thursday I hope to
present some Matlab demonstrations of various interpolation methods and on
Friday I will continue with piece-wise interpolation.
Posted: Tue - October 25, 2005 at 11:14 AM