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Phase improvement by iterative density modification
Zhang, K. Y. J., Cowtan, K. D. and Main, P.  International Tables for Crystallography (2012). Vol. F, ch. 15.1, pp. 385-400 [ doi:10.1107/97809553602060000847 ]
... formula can be expressed as The parameters a, b, c, d in the cubic polynomial are calculated by matching function values ... scaling non-atomic resolution data is through the density histogram (Cowtan & Main, 1993; Zhang, 1993). The ideal density histogram defines ... single Fourier transform by the use of spectral B-splines (Cowtan & Main, 1998). is derived analytically based on the ...

Averaging
Cowtan, K. D., Zhang, K. Y. J. and Main, P.  International Tables for Crystallography (2012). Vol. F, Section 15.3.5.6, pp. 411-412 [ doi:10.1107/97809553602060000849 ]
... map to obtain the density value at an arbitrary coordinate (Cowtan & Main, 1998). This approach gives very accurate interpolation from ... provides gradient information for the refinement of averaging operators. References Cowtan, K. D. & Main, P. (1998). Miscellaneous algorithms for density ...
     [more results from section 15.3.5 in volume F]

Phase-extension schemes
Cowtan, K. D., Zhang, K. Y. J. and Main, P.  International Tables for Crystallography (2012). Vol. F, Section 15.3.4.3, pp. 409-410 [ doi:10.1107/97809553602060000849 ]
Phase-extension schemes 15.3.4.3. Phase-extension schemes When performing phase extension, the order in which the structure factors are included will affect the final accuracy of the extended phases. The phases obtained from previous cycles of phase extension will be included in the calculation of new phases for the unphased structure ...
     [more results from section 15.3.4 in volume F]

Preparation of input data
Cowtan, K. D., Zhang, K. Y. J. and Main, P.  International Tables for Crystallography (2012). Vol. F, Section 15.3.3, pp. 407-408 [ doi:10.1107/97809553602060000849 ]
... the mask will be calculated by a local-correlation approach (Cowtan & Main, 1998; Vellieux et al., 1995). If multiple domains ... X-ray crystallography and NMR. Yale University Press, New Haven. Cowtan, K. D. & Main, P. (1998). Miscellaneous algorithms for density ...

Program operation
Cowtan, K. D., Zhang, K. Y. J. and Main, P.  International Tables for Crystallography (2012). Vol. F, Section 15.3.2, p. 407 [ doi:10.1107/97809553602060000849 ]
Program operation 15.3.2. Program operation DM and DMMULTI are largely automatic; in order to perform a phase-improvement calculation only two tasks are required of the user: (1) Provide the input data. These must include the reflection data and solvent content, and may also include averaging operators, solvent mask and averaging ...

Introduction
Cowtan, K. D., Zhang, K. Y. J. and Main, P.  International Tables for Crystallography (2012). Vol. F, Section 15.3.1, p. 407 [ doi:10.1107/97809553602060000849 ]
Introduction 15.3.1. Introduction DM is an automated procedure for phase improvement by iterated density modification. It is used to obtain a set of improved phases and figures of merit, using as a starting point the observed diffraction amplitudes and some initial poor estimates for the phases and figures of merit. DM ...

DM/DMMULTI software for phase improvement by density modification
Cowtan, K. D., Zhang, K. Y. J. and Main, P.  International Tables for Crystallography (2012). Vol. F, ch. 15.3, pp. 407-412 [ doi:10.1107/97809553602060000849 ]
... the mask will be calculated by a local-correlation approach (Cowtan & Main, 1998; Vellieux et al., 1995). If multiple domains ... are somewhat prone to producing grossly overestimated figures of merit (Cowtan & Main, 1996). Users should be aware of this. In ... by fitting the data to a semi-empirical scattering curve (Cowtan & Main, 1998). This curve is prepared using Parseval' ...

Example
Zhang, K. Y. J., Cowtan, K. D. and Main, P.  International Tables for Crystallography (2012). Vol. F, Section 15.1.7, pp. 398-399 [ doi:10.1107/97809553602060000847 ]
... initio when high-order noncrystallographic symmetry is present. References Wigley, D. B., Roper, D. I. & Cooper, R. A. (1989). Preliminary crystallographic analysis of ... from Escherichia coli. J. Mol. Biol. 210, 881-882. Zhang, K. Y. J., Cowtan, K. D. & Main, P. (1997). ...

Statistical density-modification methods
Zhang, K. Y. J., Cowtan, K. D. and Main, P.  International Tables for Crystallography (2012). Vol. F, Section 15.1.6, p. 398 [ doi:10.1107/97809553602060000847 ]
Statistical density-modification methods 15.1.6. Statistical density-modification methods Statistical density-modification methods arise from a reinterpretation of the problem of phase improvement in statistical terms, and as a result reduce the problems of bias associated with the classical density-modification methods described above. This is achieved by expressing all information ...

The diagonal approximation
Zhang, K. Y. J., Cowtan, K. D. and Main, P.  International Tables for Crystallography (2012). Vol. F, Section 15.1.5.2.3, pp. 397-398 [ doi:10.1107/97809553602060000847 ]
The diagonal approximation 15.1.5.2.3. The diagonal approximation 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 ...
     [more results from section 15.1.5 in volume F]

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