International
Tables for
Crystallography
Volume B
Reciprocal space
Edited by U. Shmueli

International Tables for Crystallography (2010). Vol. B, ch. 1.3, pp. 95-96   | 1 | 2 |

## Section 1.3.4.4.7.1. The method of least squares

G. Bricognea

aGlobal Phasing Ltd, Sheraton House, Suites 14–16, Castle Park, Cambridge CB3 0AX, England, and LURE, Bâtiment 209D, Université Paris-Sud, 91405 Orsay, France

#### 1.3.4.4.7.1. The method of least squares

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Hughes (1941) was the first to use the already well established multivariate least-squares 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 real-valued 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 right-hand 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 X-ray 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 least-squares methods. In Computing Methods in Crystallography, edited by J. S. Rollett, pp. 112–116. Oxford: Pergamon Press.
Cruickshank, D. W. J. (1970). Least-squares 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). Least-squares 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). Least-squares 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 high-speed digital computer SWAC. Acta Cryst. 9, 350–358.
Whittaker, E. T. & Robinson, G. (1944). The Calculus of Observations. London: Blackie.