Approximating functions by Newton Polynomials. With an Implementation in Rust.

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In one of my recent stories I gave a little report about Taylor expansions and its applications.

In order to approximate a function by a Taylor polynomial, we need to know its derivatives. Though, in many data-driven applications we do not even know explicitly the function itself. That is, often we only know the function’s values at some discrete points coming either from data or simulation.

For such cases, the Taylor theorem is of no help and instead one often approximates the function by an interpolation polynomial.

There are several ways of arriving at an interpolation polynomial and one is to adjust the coefficients such that its values coincide at the known data points. A particular simply way of achieving this is by using so called Newton polynomials.

Newton polynomials:

Note, a proof of this theorem can be found in my story about the mean value theorem (see here).