Newton polynomials (named after their inventor Isaac Newton) are polynomials used for polynomial interpolation. Rather than solving the huge Vandermonde matrix equation obtained in the polynomial interpolation by Gauss-Jordan elimination, we notice that we can do clever tricks by writing the polynomial in a different way. Given a data set:
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2 Example 3 See Also |
We see that the polynomial for a certain may be defined recursively, thus:
Divided Differences
so that interpolates the function in the points . The coefficients are dependent only on the obtained function values of (our :s),
so it is natural to say that as only depends on , only on and only on , and we define a notation for the divided differences:
This definition gives us the formal definition of the divided differences:
So that for example:
These are not easily grasped when put like this, but once the functions are arranged in a tabular form, things look simpler. Here for example, for a data set of (and we know that for all ):
On the diagonal of this table you will find the coefficients, as . Insert these into the formula at the top and you have your unique interpolation polynomial:
Example
See Also