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International
Tables for Crystallography Volume F Crystallography of biological macromolecules Edited by M. G. Rossman and E. Arnold © International Union of Crystallography 2006 |
International Tables for Crystallography (2006). Vol. F. ch. 18.1, p. 372
Section 18.1.8.1. Solving the refinement equations
a
San Diego Supercomputer Center 0505, University of California at San Diego, 9500 Gilman Drive, La Jolla, CA 92093-0505, USA, and bStructural, Analytical and Medicinal Chemistry, Pharmacia & Upjohn, Inc., Kalamazoo, MI 49001-0119, USA |
Methods for solving the refinement equations are described in IT C Chapters 8.1![[link]](/graphics/greenarr.gif)
to 8.5
and in many texts. Prince (1994) provides an excellent starting point. There are two commonly used approaches to finding the set of parameters that minimizes equation (18.1.4.1). The first is to treat each observation separately and rewrite each term of (18.1.4.1) as ![[ w_{i} [y_{i} - f_{i} ({\bf x})] = w_{i} \sum\limits_{j = 1}^{N} \left({\partial f_{i} ({\bf x}) \over \partial x_{j}}\right)(x_{j}^{0} - x_{j}), \eqno(18.1.8.1)]](/teximages/fach18o1/fach18o1fd6.svg)
where the summation is over the N parameters of the model. This is simply the first-order expansion of ![[f_{i} ({\bf x})]](/teximages/fach18o1/fach18o1fi23.svg)
and expresses the hypothesis that the calculated values should match the observed values. The system of simultaneous observational equations can be solved for the parameter shifts provided that there are at least as many observations as there are parameters to be determined. When the number of observational equations exceeds the number of parameters, the least-squares solution is that which minimizes (18.1.4.1). This is the method generally used for refining small-molecule crystal structures, and increasingly for macromolecular structures at atomic resolution.
References
Prince, E. (1994). Mathematical techniques in crystallography and materials science. 2nd ed. Berlin: Springer-Verlag.Google Scholar
