Tables for
Volume B
Reciprocal space
Edited by U. Shmueli

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

Section 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 The method of least squares

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Hughes (1941)[link] was the first to use the already well established multivariate least-squares method (Whittaker & Robinson, 1944[link]) 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)[link], Sparks et al. (1956)[link], Busing & Levy (1961)[link] and others.

The Fourier relations between [\rho\llap{$-\!$}] and F (Section[link]) are used to derive the `observational equations' connecting the structure parameters [\{u_{p}\}_{p = 1, \ldots, n}] to the observations [\{|F_{{\bf h}}|^{\rm obs}, (\sigma_{{\bf h}}^{2})^{\rm obs}\}_{{\bf h} \in {\scr H}}] comprising the amplitudes and their experimental variances for a set [{\scr H}] of unique reflections.

The normal equations giving the corrections δu to the parameters are then[({\bf A}^{T}{\bf WA})\delta {\bf u} = - {\bf A}^{T}{\bf W}\Delta,]where[\eqalign{A_{{\bf h}p} &= {\partial | F_{{\bf h}}^{\,\rm calc}| \over \partial u_{p}}\cr \Delta_{{\bf h}} &= |F_{{\bf h}}^{\,\rm calc}| - |F_{{\bf h}}|^{\rm obs}\cr {\bf W} &= \hbox{diag } (W_{{\bf h}}) \quad \hbox{with} \quad W_{{\bf h}} = {1 \over (\sigma_{{\bf h}}^{2})^{\rm obs}}.}]To calculate the elements of A, write:[F = |F| \exp (i\varphi) = \alpha + i\beta\hbox{\semi}]hence[\eqalign{ {\partial |F| \over \partial u} &= {\partial \alpha \over \partial u} \cos \varphi + {\partial \beta \over \partial u} \sin \varphi\cr &= {\scr Re} \left[{\partial F \over \partial u} \overline{\exp (i\varphi)}\right] = {\scr Re} \left[{\overline{\partial F} \over \partial u} \exp (i\varphi)\right].}]

In the simple case of atoms with real-valued form factors and isotropic thermal agitation in space group P1,[F_{{\bf h}}^{\,\rm calc} = {\textstyle\sum\limits_{j \in J}}\, g_{j} ({\bf h}) \exp (2\pi i{\bf h}\cdot {\bf x}_{j}),]where[g_{j} ({\bf h}) = Z_{j}\,f_{j} ({\bf h}) \exp [-{\textstyle{1 \over 4}} B_{j} (d_{{\bf h}}^{*})^{2}],][Z_{j}] being a fractional occupancy.

Positional derivatives with respect to [{\bf x}_{j}] are given by[\eqalign{ {\partial F_{{\bf h}}^{\,\rm calc} \over \partial {\bf x}_{j}} &= (2\pi i{\bf h}) g_{j} ({\bf h}) \exp (2\pi i{\bf h}\cdot {\bf x}_{j})\cr {\partial |F_{{\bf h}}^{\,\rm calc}| \over \partial {\bf x}_{j}} &= {\scr Re} [(- 2\pi i{\bf h}) g_{j} ({\bf h}) \exp (-2 \pi i{\bf h}\cdot {\bf x}_{j}) \exp (i\varphi_{{\bf h}}^{\rm calc})]}]so that the corresponding [3 \times 1] subvector of the right-hand side of the normal equations reads:[\eqalign{&- \sum\limits_{{\bf h}\in {\scr H}} W_{{\bf h}} {\partial |F_{{\bf h}}^{\,\rm calc}| \over \partial {\bf x}_{j}} (|F_{{\bf h}}^{\,\rm calc}| - |F_{{\bf h}}|^{\rm obs})\cr &\quad= - {\scr Re} \left [\sum\limits_{{\bf h}\in {\scr H}} g_{j} ({\bf h}) (-2\pi i{\bf h}) W_{{\bf h}} (|F_{{\bf h}}^{\rm calc}| - |F_{{\bf h}}|^{\rm obs})\right.\cr &\qquad \times \left.\exp (i\varphi_{{\bf h}}^{\rm calc}) \exp (-2\pi i{\bf h}\cdot {\bf x}_{j})\right ].}]

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[link], 1970[link]; Rollett, 1970[link]). It is, however, an expensive task, of complexity [\propto n \times |{\scr H}|^{2}], which is unaffordable for macromolecules.


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.

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