A polynomial 
is defined by its degree, representation, and coefficients. The polynomial representation shown in the preceding equation is called a monomial representation. In this representation, the polynomial is written as a linear combination of monomials 
.
Alternatively, the polynomial can be written in:
- Newton representation with the coefficients
and 
points, 
:
- Lagrange representation with the coefficients
and 
points, 
:
with the cardinal functions:
There are infinitely many representations, but we restrict ourselves here to these three typical ones.
A polynomial can be determined from interpolation conditions:
with the given distinct values 
and arbitrary values 
as input. In the Lagrange formulation, the interpolation polynomial is directly available, as its coefficients are the interpolation data. The coefficients for the interpolation polynomial in Newton representation can be obtained...