The normal equations can be represented in matrix notation as:


where

and the Δx column matrix contains the parameter shifts that will improve the model. In matrix shorthand,


This matrix equation can be solved for Δx by inverting the [A] matrix:



The A matrix is an (m x m) square matrix where m is the number of the adjustable parameters; each element of A, aij, includes a summation of the n Fcalc's where n is the number of measured intensities; each of the n Fcalc's has been expanded in a Taylor series in Δxi.
For an illustrative example of a three parameter problem (x, y and z-coordinates) involving four hkl reflections (yes, a rather trivial example, but perhaps a useful one for those of us who don't routinely converse in the language of summations and matrices), download this PDF file.

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