Addition of New Points
As with other difference formulas, the degree of a Newton's interpolating polynomial can be increased by adding more terms and points without discarding existing ones. Newton's form has the simplicity that the new points are always added at one end: Newton's forward formula can add new points to the right, and Newton's backwards formula can add new points to the left. Unfortunately, the accuracy of polynomial interpolation depends on how close the interpolated point is to the middle of the x values of the set of points used; as Newton's form always adds new points at the same end, an increase in degree cannot be used to increase the accuracy anywhere but at that end. Gauss, Stirling, and Bessel all developed formulae to remedy that problem.
Gauss's formula alternately adds new points at the left and right ends, thereby keeping the set of points centered near the same place (near the evaluated point). When so doing, it uses terms from Newton's formula, with data points and x values renamed in keeping with one's choice of what data point is designated as the data point.
Stirling's formula remains centered about a particular data point, for use when the evaluated point is nearer to a data point than to a middle of two data points. Bessel's formula remains centered about a particular middle between two data points, for use when the evaluated point is nearer to a middle than to a data point. They achieve that by sometimes using the average of two differences where Newton's or Gauss's would use just one difference. Stirling's does that in odd-degree terms; Bessels does that in even-degree terms. Calculating and averaging two differences need not involve extra work, since it can be done by formula, in advance—the expression for the averaged difference is not more complicated than that of the simple difference.
Read more about this topic: Newton Polynomial
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