Newton Polynomial - Strengths and Weaknesses of Various Formulae

Strengths and Weaknesses of Various Formulae

The suitability of Stirling's, Bessel's and Gauss's formulae depends on 1) the importance of the small accuracy gain given by average differences; and 2) if greater accuracy is necessary, whether the interpolated point is closer to a data point or to a middle between two data points.

In general, the difference methods can be a good choice when one does not know how many points, what degree of interpolating polynomial, will be needed for the desired accuracy, and when one wants to look first at linear and other low-degree interpolation, successively judging accuracy by the difference in the results of two successive polynomial degrees. Lagrange's formula (not a difference formula) allows that also, but going to the next higher degree without re-doing work requires that each term's value be recorded—not a problem with a computer, but maybe awkward with a calculator.

Other than that, Lagrange is easier to calculate than the difference methods, and is (probably rightly) regarded by many as the best choice when one already knows what polynomial degree will be needed. And when all the interpolation will be done at one x value, with only the data points' y values varying from one problem to another, Lagrange's formula becomes so much more convenient that it begins to be the only choice to consider.

Lagrange's formula's ease of calculation is best achieved by its "barycentric forms". Its 2nd barycentric form might be the most efficient of all when using a computer, but its 1st barycentric form might be more convenient when using a calculator.