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