The normal equations

Cruickshank, D. W. J.

doi:10.1107/97809553602060000697

International
Tables for
Crystallography
Volume F
Crystallography of biological macromolecules
Edited by M. G. Rossmann and E. Arnold

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International Tables for Crystallography (2006). Vol. F. ch. 18.5, p. 404 | 1 | 2 |

Section 18.5.2.1. The normal equations

D. W. J. Cruickshank^a ^*^‡

^a Chemistry Department, UMIST, Manchester M60 1QD, England
Correspondence e-mail: dwj_cruickshank@email.msn.com

18.5.2.1. The normal equations

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In the unrestrained least-squares method, the residual $[R = \textstyle\sum\limits_{3}\displaystyle w(hkl)\Delta^{2} (hkl) \eqno(18.5.2.1)]$ is minimized, where Δ is either $[|F_{o}| - |F_{c}|]$ for $[R_{1}]$ or $[|F_{o}|^{2} - |F_{c}|^{2}]$ for $[R_{2}]$ , and w(hkl) is chosen appropriately. The summation is over crystallographically independent planes.

When R is a minimum with respect to the parameter $[u_{j}]$ , $[\partial R/\partial u_{j} = 0]$ , i.e., $[\textstyle\sum\limits_{3}\displaystyle w\Delta (\partial \Delta / \partial u_{j}) = 0. \eqno(18.5.2.2)]$ For $[R_{1}]$ , $[\partial \Delta / \partial u_{j} = -\partial |F_{c}|/\partial u_{j}]$ ; for $[R_{2}]$ , $[\partial \Delta / \partial u_{j} =]$ $[-2|F_{c}|\partial |F_{c}|/ \partial u_{j}]$ . The n parameters have to be varied until the n conditions (18.5.2.2) are satisfied. For a trial set of the $[u_{j}]$ close to the correct values, we may expand Δ as a function of the parameters by a Taylor series to the first order. Thus for $[R_{1}]$ , $[\Delta ({\bf u} + {\bf e}) = \Delta ({\bf u}) - \textstyle\sum\limits_{i}\displaystyle \varepsilon_{i} (\partial |F_{c}|/ \partial u_{i}), \eqno(18.5.2.3)]$ where $[\varepsilon_{i}]$ is a small change in the parameter $[u_{i}]$ , and u and e represent the whole sets of parameters and changes. The minus sign occurs before the summation, since $[\Delta = |F_{o}| - |F_{c}|]$ , and the changes in $[|F_{c}|]$ are being considered.

Substituting (18.5.2.3) in (18.5.2.2), we get the normal equations for $[R_{1}]$ , $[\openup6pt\displaylines{\textstyle\sum\limits_{i}\displaystyle \varepsilon_{i} \left[\textstyle\sum\limits_{3}\displaystyle w(\partial |F_{c}|/ \partial u_{i}) (\partial |F_{c}|/ \partial u_{j})\right] \cr\hfill= \textstyle\sum\limits_{3}\displaystyle w\Delta (\partial |F_{c}|/ \partial u_{j}).\hfill (18.5.2.4)}]$ There are n of these equations for $[j = 1,\ldots, n]$ to determine the n unknown $[\varepsilon_{j}]$ .

For $[R_{2}]$ the normal equations are $[\openup6pt\displaylines{\textstyle\sum\limits_{i}\displaystyle \varepsilon_{i} \left[\textstyle\sum\limits_{3}\displaystyle w(\partial |F_{c}|^{2} / \partial u_{i}) (\partial |F_{c}|^{2} / \partial u_{j})\right] \cr\hfill= \textstyle\sum\limits_{3}\displaystyle w\Delta (\partial |F_{c}|^{2} / \partial u_{j}). \hfill(18.5.2.5)}]$ Both forms of the normal equations can be abbreviated to $[\textstyle\sum\limits_{i}\displaystyle \varepsilon_{i} a_{ij} = b_{j}. \eqno(18.5.2.6)]$

For the values of $[\partial |F_{c}| / \partial u_{j}]$ for common parameters see, e.g., Cruickshank (1970).

Some important points in the derivation of the standard uncertainties of the refined parameters can be most easily understood if we suppose that the matrix $[a_{ij}]$ can be approximated by its diagonal elements. Each parameter is then determined by a single equation of the form $[\varepsilon_{i} \textstyle\sum\limits_{3}\displaystyle wg^{2} = \textstyle\sum\limits_{3}\displaystyle wg\Delta, \eqno(18.5.2.7)]$ where $[g = \partial |F_{c}| / \partial u_{i}]$ or $[\partial |F_{c}|^{2} / \partial u_{i}]$ . Hence $[\varepsilon_{i} = \left(\textstyle\sum\limits_{3}\displaystyle wg\Delta \right)\bigg/ \left(\textstyle\sum\limits_{3}\displaystyle wg^{2}\right). \eqno(18.5.2.8)]$ At the conclusion of the refinement, when R is a minimum, the variance (square of the s.u.) of the parameter $[u_{i}]$ due to uncertainties in the Δ's is $[\sigma_{i}^{2} = \left[\textstyle\sum\limits_{3}\displaystyle w^{2}g^{2}\sigma^{2}(F)\right] \bigg/ \left(\textstyle\sum\limits_{3}\displaystyle wg^{2}\right)^{2}. \eqno(18.5.2.9)]$ If the weights have been chosen as $[w(hkl) = 1 / \sigma^{2}(|F_{hkl}|)]$ or $[1 / \sigma^{2} (|F_{hkl}|^{2})]$ , this simplifies to $[\sigma_{i}^{2} = 1 \bigg/ \left(\textstyle\sum\limits_{3}\displaystyle wg^{2}\right) = 1 / a_{ii}, \eqno(18.5.2.10)]$ which is appropriate for absolute weights. Equation (18.5.2.10) provides an s.u. for a parameter relative to the s.u.'s $[\sigma (|F|)]$ or $[\sigma (|F|^{2})]$ of the observations.

In general, with the full matrix $[a_{ij}]$ in the normal equations, $[\sigma_{i}^{2} = (a^{-1})_{ii}, \eqno(18.5.2.11)]$ where $[(a^{-1})_{ii}]$ is an element of the matrix inverse to $[a_{ij}]$ . The covariance of the parameters $[u_{i}]$ and $[u_{j}]$ is $[\hbox{cov} (i, j) \equiv \sigma_{i}\sigma_{j}\hbox{correl} (i, j) = (a^{-1})_{ij}. ]$

References

Cruickshank, D. W. J. (1970). Least-squares refinement of atomic parameters. In Crystallographic computing, edited by F. R. Ahmed, S. R. Hall & C. P. Huber, pp. 187–196. Copenhagen: Munksgaard.Google Scholar

International Tables for Crystallography (2006). Vol. F. ch. 18.5, p. 404