Stirling central difference

Stirling central differences represent a horizontal line using averages of the differences i.e.,... 

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. 

C THIS PROGRAM PERFORMS THE STIRLING S CENTRAL DIFFERENCE... 

The central difference formula of Stirling thus furnishes the same result as the ordinary difference formula of Newton. We get different results when the higher orders of differences are neglected. For instance, if we neglect differences of the second order in formulae (7) and (20), Stirling s formula would furnish more accurate results, because, in virtue of the substitution A1 = A1 - JA2 v we have really retained a portion of the second order of differences. If, therefore, we take the difference formula as far as the first, third, or some odd order of differences, we get the same results with the central and the ordinary difference formulae. One more term is required to get an odd order of differences when central differences are employed. Thus, five terms are required to get... 

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 ... 

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. 

In principle, the nucleophile can attack the allene at two different positions, but the products show exclusive attack at the central carbon atom, similarly to other nucleophilic additions to allenes (Eglinton et al., 1954 Stirling, 1964b Taylor, 1967). This may result from the stabilization of the carbanion (194), formed by attack at this position, by the two phenyl groups. The ion (194) may be protonated at either one of the terminal positions of the allenic system, and low amounts of... 

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