What We Mean by LSQ

The term LSQ is used here to mean Least Squares Refinement , which uses observations of a function to improve the values of some set of parameters on which that function depends.

Refinement involving a crystal structure could be either standard single crystal LSQ refinement, in which each observation depends on a single structure factor, or profile refinement (PR) for which several structure factors contribute to one observation.

The PR routines are written to use many of the same CCSL routines as do standard LSQ main programs. The PR programs and Libraries are available separately, with a separate Manual.

CCSL also deals with simpler LSQ problems. For example, the main program FWLSQ fits the 5,7 or 9 parameters of the exponential approximation to a scattering factor curve. The simplicity in this case is that no crystal structure is involved. Another simple case is the refinement of (up to) 6 cell parameters, given an observed list of d-spacings (actually d* squared values, in the main program DSLSQ). This must deal with the constraints which are necessarily imposed on the cell parameters by the space group symmetry.

The essentials of a LSQ problem are:

a)

a set of observations of something, (the observed function ) usually with their estimated standard deviations, $\sigma$'s, measured at different values of some argument ARG ; ARG may be $h,k,l$ (for standard LSQ), $\sin\theta/\lambda$, $\theta$ (for Rietveld PR) or $\lambda$, or energy, or time of flight; it is a quantity which takes different values, and will identify the observation.
For crystallographic applications the observations are often all of the same physical thing, but this is not necessary. For example, geometric constraints may be introduced by giving bond lengths and/or angles as additional types of observation, with $\sigma$'s.

b)

some calculated function involving ARG , which defines a mathematical model to be compared with an observation. This calculated function depends on parameters , things which may possibly be varied in order to improve the fit of the function to its related observations.
By fit we mean minimisation of the weighted sum of squares of differences between observed and calculated function values (where the summing is over the different values of ARG ). It is desirable to use statistical weights ($1/\sigma^2$) for the differences.
The theory may be viewed as using the beginning of a Taylor series expansion, and therefore requires that the parameters be close to their correct values, making the required shifts so small that their squares may be neglected.