Difference higher order

One of the most commonly employed types of difference Fourier synthesis uses as coefficients 2/< / - Fcaic, and phases calculated from the current model. This shows the investigator positive density superimposed on the model where the model is correct, with no density appearing where the model contains incorrectly placed residues or atoms and positive density appears where the model should be. There are also other higher order difference syntheses, but they are less frequently used. 

In principal, the partial PDFs can be determined by taking second and higher order differences. For an -element compound there are n(n + l)/2 partials and therefore n(n + l)/2 independent measurements of the same sample are required to determine the full set of partials. For example, for a binary n = 2 and 3 therefore independent data-sets are required to recover the three partials. 

Higher-order differences are defined in a similar way. For data sets of ordinary accuracy, the values of y in Eqs. (11.33a) should be the smoothed values given by Eq. (11.32). This set of differences for point number i involves only points with subscripts greater than or equal to /. Other schemes can be defined that use points on both sides of the /th data point. 

Initial gradients are best estimated by higher-order difference formulas such as... 

The same procedure can be applied for higher order differences. 

J. A. Nohel and W. P. Timlake, Higher order differences in the numerical solution of two-dimensiomil neutron diffusion equations. Proceedings of the Second International Conference on the Peaceful Uses of Atomic Energy, Geneva, 1958, vol. 16, pp. 595-600. 

Neglecting fourth- and higher-order differences and expressing A/q, Arf, and A /o in terms of the function values, we get... 

Higher order differences derivatives Difference ratios Fourier coefficients Principal components scores Factor analysis scores Curve-fitting coefficients Partial least-squares scores... 

Next, consider a general higher-order difference equation given by... 

It should also be noted that the sum of the coefficients of the binomial expansion is always equal to zero. This can be used as a check to ensure that higher-order differences have been expanded correctiy. 

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