The diagonal approximation

Zhang, K. Y. J.; Cowtan, K. D.; Main, P.

doi:10.1107/97809553602060000687

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. 15.1, p. 323 | 1 | 2 |

Section 15.1.5.2.3. The diagonal approximation

K. Y. J. Zhang,^a K. D. Cowtan^b ^* and P. Main^c

^a Division of Basic Sciences, Fred Hutchinson Cancer Research Center, 1100 Fairview Ave N., Seattle, WA 90109, USA,^bDepartment of Chemistry, University of York, York YO1 5DD, England, and ^cDepartment of Physics, University of York, York YO1 5DD, England
Correspondence e-mail: cowtan+email@ysbl.york.ac.uk

15.1.5.2.3. The diagonal approximation

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The full-matrix solution to equation (15.1.5.4) requires a significant amount of computing, although it can be achieved using FFTs. The diagonal approximation to the normal matrix has been used as an alternative method of solution to the electron-density shift in equation (15.1.5.4) (Main, 1990b). As with the full-matrix calculation, it can be done entirely by FFTs and a linear combination of vectors.

The diagonal element of the normal matrix, $[{\bf J}^{T}{\bf J}]$ , in equation (15.1.5.7) is $[d_{0} \left({\bf x}\right) = (4/N)\rho\left({\bf x}\right)\left[\rho\left({\bf x}\right){\textstyle\sum\limits_{\bf h}} \left| \theta \left({\bf h}\right) \right|^{2} - {\textstyle\sum\limits_{\bf h}} \theta \left({\bf h}\right)\right] + 2. \eqno(15.1.5.30)]$ The right-hand side of equation (15.1.5.7), $[-{\bf J}^{T} \varepsilon \left({\bf x}\right)]$ , is identical to the residual vector, $[r_{0}\left({\bf x}\right)]$ , which can be calculated from equation (15.1.5.22). Therefore, the solution to the electron-density shift, $[\delta\rho\left({\bf x}\right)]$ , can be calculated from $[\delta\rho\left({\bf x}\right) = r_{0} \left({\bf x}\right)/d_{0} \left({\bf x}\right). \eqno(15.1.5.31)]$

Compared with the full-matrix solution, all the calculations involved in between equations (15.1.5.12) and (15.1.5.18) and the subsequent iterations are spared in the diagonal approximation. This makes calculation by the diagonal approximation much faster than by the full-matrix method.

References

Main, P. (1990b). The use of Sayre's equation with constraints for the direct determination of phases. Acta Cryst. A46, 372–377.Google Scholar

International Tables for Crystallography (2006). Vol. F. ch. 15.1, p. 323