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Stereochemical constraints as observational equations
International Tables for Crystallography (2006). Vol. C, Section 8.3.2.1, pp. 698-701 [ doi:10.1107/97809553602060000611 ]
... 3.65937 O1 -0.48899 -2.49684 3.46331 N2 -0.00450 -0.81846 4.87646 Glu E C[beta] -0.06551 -0.87677 1.25157 C[gamma] 1.15947 -1.71468 1.59818 ... 13.01-13.26. Bangalore: Indian Academy of Sciences. Hestenes, M. & Stiefel, E. (1952). Methods of conjugate gradients for solving linear systems. ... Cryst. A34, 578-582. Schomaker, V., Waser, J., Marsh, R. E. & Bergman, G. (1959). To fit a plane or ...
[more results from section 8.3.2 in volume C]
Direct application of constraints
International Tables for Crystallography (2006). Vol. C, Section 8.3.1.2, pp. 693-698 [ doi:10.1107/97809553602060000611 ]
... factors, and possibly higher cumulants of an atomic density function (Prince, 1994). The constrained calculation is usually performed by evaluating ... beta]13 = -[beta]23; one principal axis parallel to [110] , , , , , , , , , , , , , , , , , , , , , , , , , , , (e) [beta]11 = [beta]22, [beta]13 = [beta]23; one principal ... beta]13 = [beta]23; one principal axis parallel to [100] , , , , , , , , , , , , , , , , , (e) [beta]22 = 2[beta]12, [beta]23 = 0; one ...
[more results from section 8.3.1 in volume C]
Constraints and restraints in refinement
International Tables for Crystallography (2006). Vol. C, ch. 8.3, pp. 694-701 [ doi:10.1107/97809553602060000611 ]
... factors, and possibly higher cumulants of an atomic density function (Prince, 1994). The constrained calculation is usually performed by evaluating ... beta]13 = -[beta]23; one principal axis parallel to [110] , , , , , , , , , , , , , , , , , , , , , , , , , , , (e) [beta]11 = [beta]22, [beta]13 = [beta]23; one principal ... beta]13 = [beta]23; one principal axis parallel to [100] , , , , , , , , , , , , , , , , , (e) [beta]22 = 2[beta]12, [beta]23 = 0; one ...
Some examples
International Tables for Crystallography (2006). Vol. C, Section 8.2.3.2, pp. 691-692 [ doi:10.1107/97809553602060000610 ]
... Scaling by entropy maximization. Acta Cryst. A40, 705-708. Jaynes, E. T. (1979). Where do we stand on maximum entropy ...
[more results from section 8.2.3 in volume C]
Robust/resistant methods
International Tables for Crystallography (2006). Vol. C, Section 8.2.2, pp. 689-691 [ doi:10.1107/97809553602060000610 ]
... by Tukey was applied to crystal structure refinement by Nicholson, Prince, Buchanan & Tucker (1982). It corresponds to a fitting function ... be introduced by this effect. References Belsley, D. A., Kuh, E. & Welsch, R. E. (1980). Regression diagnostics. New York: John Wiley. Box, ...
Maximum-likelihood methods
International Tables for Crystallography (2006). Vol. C, Section 8.2.1, p. 689 [ doi:10.1107/97809553602060000610 ]
Maximum-likelihood methods 8.2.1. Maximum-likelihood methods In Chapter 8.1 , structure refinement is presented as finding the answer to the question, `given a set of observations drawn randomly from populations whose means are given by a model, M(x), for some set of unknown parameters, x, how can we best determine ...
Other refinement methods
International Tables for Crystallography (2006). Vol. C, ch. 8.2, pp. 689-692 [ doi:10.1107/97809553602060000610 ]
... by Tukey was applied to crystal structure refinement by Nicholson, Prince, Buchanan & Tucker (1982). It corresponds to a fitting function ... to the model's correctness. References Belsley, D. A., Kuh, E. & Welsch, R. E. (1980). Regression diagnostics. New York: John Wiley. Box, ...
Software for least-squares calculations
International Tables for Crystallography (2006). Vol. C, Section 8.1.7, p. 688 [ doi:10.1107/97809553602060000609 ]
Software for least-squares calculations 8.1.7. Software for least-squares calculations Giving even general recommendations on software is a difficult task for several reasons. Clearly, the selection of methods discussed in earlier sections contains implicitly some recommendations for approaches. Among the reasons for avoiding specifics are the following: (1) Assessing differences ...
Orthogonal distance regression
International Tables for Crystallography (2006). Vol. C, Section 8.1.6, pp. 687-688 [ doi:10.1107/97809553602060000609 ]
... Stat. Comput. 8, 1052-1078. Boggs, P. T. & Rogers, J. E. (1990). Orthogonal distance regression. Contemporary mathematics: statistical analysis of ...
Conjugate-gradient methods
International Tables for Crystallography (2006). Vol. C, Section 8.1.5.2, pp. 686-687 [ doi:10.1107/97809553602060000609 ]
Conjugate-gradient methods 8.1.5.2. Conjugate-gradient methods A numerical procedure that is applicable to large-scale problems that may not be sparse is called the conjugate-gradient method. Conjugate-gradient methods were originally designed to solve the quadratic minimization problem, find the minimum of where H is a symmetric, positive-definite ...
[more results from section 8.1.5 in volume C]
International Tables for Crystallography (2006). Vol. C, Section 8.3.2.1, pp. 698-701 [ doi:10.1107/97809553602060000611 ]
... 3.65937 O1 -0.48899 -2.49684 3.46331 N2 -0.00450 -0.81846 4.87646 Glu E C[beta] -0.06551 -0.87677 1.25157 C[gamma] 1.15947 -1.71468 1.59818 ... 13.01-13.26. Bangalore: Indian Academy of Sciences. Hestenes, M. & Stiefel, E. (1952). Methods of conjugate gradients for solving linear systems. ... Cryst. A34, 578-582. Schomaker, V., Waser, J., Marsh, R. E. & Bergman, G. (1959). To fit a plane or ...
[more results from section 8.3.2 in volume C]
Direct application of constraints
International Tables for Crystallography (2006). Vol. C, Section 8.3.1.2, pp. 693-698 [ doi:10.1107/97809553602060000611 ]
... factors, and possibly higher cumulants of an atomic density function (Prince, 1994). The constrained calculation is usually performed by evaluating ... beta]13 = -[beta]23; one principal axis parallel to [110] , , , , , , , , , , , , , , , , , , , , , , , , , , , (e) [beta]11 = [beta]22, [beta]13 = [beta]23; one principal ... beta]13 = [beta]23; one principal axis parallel to [100] , , , , , , , , , , , , , , , , , (e) [beta]22 = 2[beta]12, [beta]23 = 0; one ...
[more results from section 8.3.1 in volume C]
Constraints and restraints in refinement
International Tables for Crystallography (2006). Vol. C, ch. 8.3, pp. 694-701 [ doi:10.1107/97809553602060000611 ]
... factors, and possibly higher cumulants of an atomic density function (Prince, 1994). The constrained calculation is usually performed by evaluating ... beta]13 = -[beta]23; one principal axis parallel to [110] , , , , , , , , , , , , , , , , , , , , , , , , , , , (e) [beta]11 = [beta]22, [beta]13 = [beta]23; one principal ... beta]13 = [beta]23; one principal axis parallel to [100] , , , , , , , , , , , , , , , , , (e) [beta]22 = 2[beta]12, [beta]23 = 0; one ...
Some examples
International Tables for Crystallography (2006). Vol. C, Section 8.2.3.2, pp. 691-692 [ doi:10.1107/97809553602060000610 ]
... Scaling by entropy maximization. Acta Cryst. A40, 705-708. Jaynes, E. T. (1979). Where do we stand on maximum entropy ...
[more results from section 8.2.3 in volume C]
Robust/resistant methods
International Tables for Crystallography (2006). Vol. C, Section 8.2.2, pp. 689-691 [ doi:10.1107/97809553602060000610 ]
... by Tukey was applied to crystal structure refinement by Nicholson, Prince, Buchanan & Tucker (1982). It corresponds to a fitting function ... be introduced by this effect. References Belsley, D. A., Kuh, E. & Welsch, R. E. (1980). Regression diagnostics. New York: John Wiley. Box, ...
Maximum-likelihood methods
International Tables for Crystallography (2006). Vol. C, Section 8.2.1, p. 689 [ doi:10.1107/97809553602060000610 ]
Maximum-likelihood methods 8.2.1. Maximum-likelihood methods In Chapter 8.1 , structure refinement is presented as finding the answer to the question, `given a set of observations drawn randomly from populations whose means are given by a model, M(x), for some set of unknown parameters, x, how can we best determine ...
Other refinement methods
International Tables for Crystallography (2006). Vol. C, ch. 8.2, pp. 689-692 [ doi:10.1107/97809553602060000610 ]
... by Tukey was applied to crystal structure refinement by Nicholson, Prince, Buchanan & Tucker (1982). It corresponds to a fitting function ... to the model's correctness. References Belsley, D. A., Kuh, E. & Welsch, R. E. (1980). Regression diagnostics. New York: John Wiley. Box, ...
Software for least-squares calculations
International Tables for Crystallography (2006). Vol. C, Section 8.1.7, p. 688 [ doi:10.1107/97809553602060000609 ]
Software for least-squares calculations 8.1.7. Software for least-squares calculations Giving even general recommendations on software is a difficult task for several reasons. Clearly, the selection of methods discussed in earlier sections contains implicitly some recommendations for approaches. Among the reasons for avoiding specifics are the following: (1) Assessing differences ...
Orthogonal distance regression
International Tables for Crystallography (2006). Vol. C, Section 8.1.6, pp. 687-688 [ doi:10.1107/97809553602060000609 ]
... Stat. Comput. 8, 1052-1078. Boggs, P. T. & Rogers, J. E. (1990). Orthogonal distance regression. Contemporary mathematics: statistical analysis of ...
Conjugate-gradient methods
International Tables for Crystallography (2006). Vol. C, Section 8.1.5.2, pp. 686-687 [ doi:10.1107/97809553602060000609 ]
Conjugate-gradient methods 8.1.5.2. Conjugate-gradient methods A numerical procedure that is applicable to large-scale problems that may not be sparse is called the conjugate-gradient method. Conjugate-gradient methods were originally designed to solve the quadratic minimization problem, find the minimum of where H is a symmetric, positive-definite ...
[more results from section 8.1.5 in volume C]
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