Gregory-Newton Interpolation

we consider a set of known values of the function f x) at equally spaced values of x  

These points are represented graphically in Fig. 3.4 and are tabulated in Tables 3.4 and 3.5, The first, second, and third forward differences of these base points are also tabulated in Table 3.4 and the corresponding backward differences in Table 3.5. 

The Gregory-Newton forward interpolation formula can be derived using the forward finite difference relations derived in Secs. 3.2 and 3.4. Eq. (3.17), written for the function/ 

When n is a positive integer, the binomial series has in + 1) terms therefore, Eq. (3.119) is a polynomial of degree n. If ( + 1) base-point values of the function / are known, this polynomial fits all (n + 1) points exactly. Assume that these (n-r 1) base-points are (X( /(X(,)), (xi,/(j ,)).(x ,/(x )), where (Xo,/(xo)) is the pivot point and jc, is defined as 

We can now designate the distance of the point of interest from the pivot point as ( r - jc ). The value of Tt is no longer an integer and is replaced by 

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Equation (11.34) is based on the Gregory-Newton interpolation formula. The second derivative is given by... 

It was stated earlier that the binomial series [Eq. (3.118)] has a finite number of terms, (n +1), when n is a positive integer. However, in the Gregory-Newton interpolation formulas,... 

Extrapolation is required if f(x) is known on the interval [a,b], but values of f(x) are needed for x values not in the interval. In addition to the uncertainties of interpolation, extrapolation is further complicated since the function is fixed only on one side. Gregory-Newton and Lagrange formulas may be used for extrapolation (depending on the spacing of the data points), but all results should be viewed with extreme skepticism. 

THIS PROGRAM PERFORMS THE NEWTON-GREGORY FORWARD INTERPOLATION OF 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. 

This is the Gregory-Newton forward interpolation formula. The general formula of the above... 

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

Example 3.1 Gregory-Newton Method for Interpolation of Equally Spaced Data. 

An exothermic, relatively slow reaction takes place in a reactor under your supervision. Yesterday, after you left the plant, the temperature of the reactor went out of control, for a yet unknown reason, until the operator put it under control by changing the cooling water flow rate. Your supervisor has asked you to prepare a report regarding this incident. As the first step, you must know when the reactor reached its maximum temperature and what was the value of this maximum temperature. A computer was recording the temperature of the reactor at one-hour intervals. These time-temperature data are given in Table E3.1. Write a general MATLAB function for -order one-dimensional interpolation by Gregory-Newton forward interpolation formula to solve this problem. 

Method of Solution The function uses the general formula of the Gregory-Newton forward interpolation [Eq. (3.123)] to perform the n-order interpolation. The input to the function specifying the number of base points must be at least (n + 1). 

Solution to Example 3.1. It interpolates the time-temperature data % given in Table E3.1 by Gregory-Newton forward interpolation % formula and finds the maximum temperature and the time this % maximum happened. 

YI = GregoryNewton(X,Y,XI,N) applies the Nth-order % Gregory-Newton forward interpolation to find YI, the % values of the underlying function Y at the points in the vector XI. The vector X specifies the points at which the data Y is given. 

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 ... 

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. 

This method is accomplished by first replacing the function y = f x) with a polynomial approximation, such as the Gregory-Newton forward interpolation formula [Eq. (3.122)], In practice, the interval [x, x ] is being divided into several segments, each of width h, and the Gregory-Newton forward interpolation formula becomes note that =x + h) ... 

Gregory-Newton forward interpolation method. Lagrange polynomial interpolation method. Cubic splines interpolation method. 

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

THE PROGRAM USES THE NEWTON-GREGORY FORWARD AND BACKWARD INTERPOLATIONS AND STIRLING S CENTRAL DIFFERENCE METHOD. 

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