Lagrange Polynomial: Even/Odd Sums
- Author:
- Ryan Hirst
Lagrange Interpolating Polynomial through n equally spaced points.
The meat of this worksheet is in the Algebra view.
The polynomial coefficients are , where M (n x n) was made constant by using equally spaced point. Further breaking up the polynomial into even and odd terms, this multiplication can be written
Where the matrices Mev, Mod are order , and the y-vectors are first and second differences of the given y-values (see the Algebra view).
For large n, this simplifies both the matrix inverse and the coefficient calculation.
However, I think you will find this is an ill-conditioned problem, and that the polynomial above will start to lose integrity at roughly the same order as the previous worksheet, even though the matrix order is halved. For a very high order polynomial, with sufficient care and numerical precision (e.g. through programming), we can always generate an accurate matrix inverse in advance, since M^(-1) depends only on n.
TOOL: Lagrange Polynomial through n equally spaced points: {link}