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 Results for DC.creator="J." AND DC.creator="Zhang" in section 15.1.5 of volume F
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 ...

The full-matrix solution
Zhang, K. Y. J., Cowtan, K. D. and Main, P.  International Tables for Crystallography (2012). Vol. F, Section 15.1.5.2.2, p. 397 [ doi:10.1107/97809553602060000847 ]
The full-matrix solution 15.1.5.2.2. The full-matrix solution The equations to be solved for the electron-density shifts, , are from the Jacobian of equation (15.1.5.2), where is the residual to Sayre's equation, and is the residual to the linear density-modification equations, Starting from a trial solution of , the ...

The conjugate-gradient method
Zhang, K. Y. J., Cowtan, K. D. and Main, P.  International Tables for Crystallography (2012). Vol. F, Section 15.1.5.2.1, p. 397 [ doi:10.1107/97809553602060000847 ]
The conjugate-gradient method 15.1.5.2.1. The conjugate-gradient method The conjugate-gradient method does not require the inversion of the normal matrix, and therefore the solution to a large system of linear equations can be achieved very quickly. Starting from a trial solution to equations (15.1.5.4), such as a null vector ...

Least-squares solution to the system of nonlinear constraint equations
Zhang, K. Y. J., Cowtan, K. D. and Main, P.  International Tables for Crystallography (2012). Vol. F, Section 15.1.5.2, pp. 396-398 [ doi:10.1107/97809553602060000847 ]
... of shifts, to , through a system of linear equations, where J is a matrix of partial derivatives of F with respect ...

The system of nonlinear constraint equations
Zhang, K. Y. J., Cowtan, K. D. and Main, P.  International Tables for Crystallography (2012). Vol. F, Section 15.1.5.1, p. 396 [ doi:10.1107/97809553602060000847 ]
... can be obtained by solving the systems of simultaneous equations (Zhang & Main, 1990b) Equation (15.1.5.2) represents a system of nonlinear simultaneous ... the direct determination of phases. Acta Cryst. A46, 372-377. Zhang, K. Y. J. & Main, P. (1990b). The use of Sayre's ...

Combining constraints for phase improvement
Zhang, K. Y. J., Cowtan, K. D. and Main, P.  International Tables for Crystallography (2012). Vol. F, Section 15.1.5, pp. 396-398 [ doi:10.1107/97809553602060000847 ]
... constraints is obtained by a global minimization procedure (Main, 1990b; Zhang & Main, 1990b). 15.1.5.1. The system of nonlinear constraint equations ... can be obtained by solving the systems of simultaneous equations (Zhang & Main, 1990b) Equation (15.1.5.2) represents a system of nonlinear simultaneous ... of shifts, to , through a system of linear equations, where J is a matrix of partial derivatives of F with ...

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