Newton backward interpolation formula

Having defined the necessary backward difference relations, we present the Newton backward interpolation formula, written generally as... 

In a similar derivation, using backward differences, the Gregory-Newton backward interpolation formula is derived as... 

Derive the Gregory-Newton backward interpolation formula. 

Write a MATLAB function that uses the Gregory-Newton backward interpolation formula to evaluate the function f x) from a set of (n + I) equally spaced input values. Write the function in a general fashion so that n can be any positive integer. Also write a MATLAB script that reads the data and shows how this MATLAB function fits the data. Use the experimental data of Table 3.3 to verify the program, and evaluate the function atx= 10, 50,90,130, 170, and 190. 

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

The forward and the backward difference approximations give the derivatives of the Newton interpolation polynomial at the edges of the interpolation range. However, the central difference is derived from the Newton interpolation at the center of the range of interpolation. Accuracy of an interpolation formula based on equispaced points is highest at the center of the interpolation range. Therefore, the central difference interpolation formula is always more accurate than the forward or backward difference approximations. 

To derive implicit integration methods, we start with the Newton backward difference interpolation formula starting from the point backward (Eq. 7.85) rather than from the point backward as used in the generation of the explicit methods. With a single equation of the type shown by Eq. 7.87, we have... 

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. 

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