International
Tables for Crystallography Volume B Reciprocal space Edited by U. Shmueli © International Union of Crystallography 2010 
International Tables for Crystallography (2010). Vol. B, ch. 1.3, pp. 9596

Hughes (1941) was the first to use the already well established multivariate leastsquares method (Whittaker & Robinson, 1944) to refine initial estimates of the parameters describing a model structure. The method gained general acceptance through the programming efforts of Friedlander et al. (1955), Sparks et al. (1956), Busing & Levy (1961) and others.
The Fourier relations between and F (Section 1.3.4.2.2.6) are used to derive the `observational equations' connecting the structure parameters to the observations comprising the amplitudes and their experimental variances for a set of unique reflections.
The normal equations giving the corrections δu to the parameters are thenwhereTo calculate the elements of A, write:hence
In the simple case of atoms with realvalued form factors and isotropic thermal agitation in space group P1,where being a fractional occupancy.
Positional derivatives with respect to are given byso that the corresponding subvector of the righthand side of the normal equations reads:
The setting up and solution of the normal equations lends itself well to computer programming and has the advantage of providing a thorough analysis of the accuracy of its results (Cruickshank, 1965b, 1970; Rollett, 1970). It is, however, an expensive task, of complexity , which is unaffordable for macromolecules.
References
Busing, W. R. & Levy, H. A. (1961). Least squares refinement programs for the IBM 704. In Computing Methods and the Phase Problem in Xray Crystal Analysis, edited by R. Pepinsky, J. M. Robertson & J. C. Speakman, pp. 146–149. Oxford: Pergamon Press.Cruickshank, D. W. J. (1965b). Errors in leastsquares methods. In Computing Methods in Crystallography, edited by J. S. Rollett, pp. 112–116. Oxford: Pergamon Press.
Cruickshank, D. W. J. (1970). Leastsquares refinement of atomic parameters. In Crystallographic Computing, edited by F. R. Ahmed, pp. 187–197. Copenhagen: Munksgaard.
Friedlander, P. H., Love, W. & Sayre, D. (1955). Leastsquares refinement at high speed. Acta Cryst. 8, 732.
Hughes, E. W. (1941). The crystal structure of melamine. J. Am. Chem. Soc. 63, 1737–1752.
Rollett, J. S. (1970). Leastsquares procedures in crystal structure analysis. In Crystallographic Computing, edited by F. R. Ahmed, pp. 167–181. Copenhagen: Munksgaard.
Sparks, R. A., Prosen, R. J., Kruse, F. H. & Trueblood, K. N. (1956). Crystallographic calculations on the highspeed digital computer SWAC. Acta Cryst. 9, 350–358.
Whittaker, E. T. & Robinson, G. (1944). The Calculus of Observations. London: Blackie.