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
Famous quotes containing the words addition of, addition and/or points:
“Depend upon it there comes a time when for every addition of knowledge you forget something that you knew before. It is of the highest importance, therefore, not to have useless facts elbowing out the useful ones.”
—Sir Arthur Conan Doyle (1859–1930)
“The force of truth that a statement imparts, then, its prominence among the hordes of recorded observations that I may optionally apply to my own life, depends, in addition to the sense that it is argumentatively defensible, on the sense that someone like me, and someone I like, whose voice is audible and who is at least notionally in the same room with me, does or can possibly hold it to be compellingly true.”
—Nicholson Baker (b. 1957)
“A few ideas seem to be agreed upon. Help none but those who help themselves. Educate only at schools which provide in some form for industrial education. These two points should be insisted upon. Let the normal instruction be that men must earn their own living, and that by the labor of their hands as far as may be. This is the gospel of salvation for the colored man. Let the labor not be servile, but in manly occupations like that of the carpenter, the farmer, and the blacksmith.”
—Rutherford Birchard Hayes (1822–1893)