Stirlings Interpolation

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  

We replace the derivatives off x) with the differential operators to obtain 

The odd-order differential operators in Eq. (3.127) are replaced by averaged central differences and the even-order differential operators by central differences, all taken from Table 3.3. Substituting these into Eq. (3.127) and regrouping of terms yield the formula 

By applying Eq. (3.121) into Eq. (3.128), we obtain the final form of Sliding s interpolation 

The general formula for determining the higher-order terms containing odd differences in the above series is 

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These have to be calculated only once before starting the propagation. Eqn. (67) can be interpreted as Stirling interpolation to the first order. For a detailed derivation of expression (68-69) see Ref. [71]. 

Therefore, one has recourse to other interpolation polynomials associated with the names of Lagrange, Newton, Stirling, Hermite, etc. Let us give the following formulae, for equally spaced points [136]. 

Recently, an alternative approach has been developed by Zou [71], where within a given basis set size the kinetic matrix elements can be evaluated to a desired order of accuracy using Stirling s interpolation formula. The kinetic energy matrix elements can then be written in terms of the discretized position space as... 

P.F. Zou, Accurate solution to the time-independent Schrodinger equation using Stirling s interpolation formula, Chem. Phys. Lett., 222 (1994) 287. 

PROG 17 uses the interpolation methods of determine the values of F, Fj, and F3 at t = 575°F. The values from the Stirling s central difference formula are ... 

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

H - wi) = (xo xi) - H - wi) ( 2 i)-When the intervals between the two terms are large, or the differences between the various members of the series decrease rapidly, simple proportion cannot be used with confidence. To take away any arbitrary choice in the determination of the intermediate values, it is commonly assumed that the function can be expressed by a limited series of powers of one of the variables. Thus we have the interpolation formulas of Newton, Bessel, Stirling, Lagrange, and Gauss. 

Let us now return to Stirling s interpolation formula. Differentiate (19), page 317, with respect to x, and if we take the difference between y0 and yx to be infinitely small, we must put x = 0 in the result. In this way, we find that... 

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. 

Other forms of Stirling s interpolation formula exist, which make use of base points spaced at half intervals (i.e., at h/2). Our choice of using averaged central differences to replace the odd differential operators eliminated the need for having base points located at the midpoints. The central differences for Eq. (3.129) are tabulated in Table 3.6. 

Write a MATLAB function which uses the Stirling s interpolation formula to evaluate the function /(.x) from a set of (n -i-1) 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... 

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