Newton’s interpolation formula

This is Newton s interpolation formula (Newton s Principia, 3, lem. 5, 1687) employed in finding or interpolating one or more terms when n particular values of the function are known. Let us write y0 in place of yx for the first term, then... 

There are two types of interpolating polynomials that can be used. These are open formulas which are used to predict the Xj+i value based on known information up to Xj, and closed formulas which are used to correct the Xi+i. In both cases backward difference interpolating polynomials are used since we are using previous time information to determine current or future time behavior. In order to develop the backward difference formulas, we use Newton s fundamental formula for interpolating polynomials,... 

In this section, we will develop two interpolation methods for equally spaced data (I) the Gregory-Newton formulas, which are based on forward and backward differences, and (2) Stirling s interpolation formula, based on central differences. 

Stirling s interpolation formula is based on central differences. Its derivation is similar to that of the Gregory-Newton formulas and can be arrived at by using either the symbolic operator relations or the Taylor series expansion of the function. We will use the latter and expand the function fix + nh) in a Taylor series around jc ... 

Use of Interpolation Formula If the data are given over equidistant values of the independent variable x, an interpolation formula such as the Newton formula (see Refs. 143 and 18.5) may be used and the resulting formula differentiated analytically. If the independent variable is not at equidistant values, then Lagrange s formulas must be used. By differentiating three- and five-point Lagrange interpolation formulas the following differentiation formulas result for equally spaced tabular points ... 

If the accuracy afforded by a linear approximation is inadequate, a generally more accurate result may be based upon the assumption that fix) may be approximated by a polynomial of degree 2 or higher over certain ranges. This assumption leads to Newton s fundamental interpolation formula with divided differences... 

These values give the best interpolation at t = 575°F rather than the Newton-Gregory s forward or backward interpolation formulae. 

A comparison of the difference table, page 309, with Newton s formula will show that the interpolated term yx is built up by taking the algebraic sum of certain proportions of each of the terms employed. The greatest proportions are taken from those terms nearest the interpolated term. Consequently we should expect more accurate results when the interpolated term occupies a central position among the terms employed rather than if it were nearer the beginning or end of the given series of terms. 

Then, to find 74 = y( o + 4//), we substitute Newton s forward interpolation formula... 

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