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.