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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 18.1.8.2. Normal 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 |
In matrix form, the observational equations are written as ![[ {\bf A}\Delta = {\bf r},]](/teximages/fach18o1/fach18o1fd7.svg)
where A is the M by N matrix of derivatives, Δ is the parameter shifts and r is the vector of residuals given on the left-hand sides of equation (18.1.8.1![[link]](/graphics/greenarr.gif)
). The normal equations are formed by multiplying both sides of the equation by ![[{\bf A}^{T}]](/teximages/fach18o1/fach18o1fi24.svg)
. This produces an N by N square system, the solution to which is the desired least-squares solution for the parameter shifts. ![[ \displaylines{{\bf A}^{T} {\bf A}\Delta = {\bf A}^{T} {\bf r} \hbox{ or } {\bf M}\Delta = {\bf b},\quad \cr m_{ij} = \sum\limits_{k = 1}^{M} w_{k} \left({\partial f_{k} ({\bf x}) \over \partial x_{i}}\right) \left({\partial f_{k} ({\bf x}) \over \partial x_{j}}\right),\cr b_{i} = \sum\limits_{k = 1}^{M} w_{k} [y_{k} - f_{k} ({\bf x})] \left({\partial f_{k} ({\bf x}) \over \partial x_{i}}\right). } ]](/teximages/fach18o1/fach18o1fd8.svg)
Similar equations are obtained by expanding (18.1.4.1) as a second-order Taylor series about the minimum ![[{\bf x}_{0}]](/teximages/fach18o1/fach18o1fi25.svg)
and differentiating. ![[ \eqalignno{\Phi ({\bf x} - {\bf x}_{0}) &\approx \Phi ({\bf x}_{0}) + \Bigg \langle \left({\partial \Phi \over \partial x_{i}}\right)_{{\bf x}_{0}} \Bigg|({\bf x} - {\bf x}_{0})\Bigg \rangle &\cr&{\phantom\approx}+ {1 \over 2} \Bigg \langle ({\bf x} - {\bf x}_{0})\Bigg| \left({\partial^{2} \Phi \over \partial x_{i} \partial x_{j}}\right)_{{\bf x}_{0}} \Bigg|({\bf x} - {\bf x}_{0}) \Bigg \rangle , &\cr\bigg| \left({\partial \Phi \over \partial {\bf x}} \right) \Bigg \rangle &\approx \Bigg| \left({\partial^{2} \Phi \over \partial x_{i} \partial x_{j}}\right)_{{\bf x}_{0}} \Bigg|({\bf x} - {\bf x}_{0}) \Bigg \rangle . }]](/teximages/fach18o1/fach18o1fd9.svg)
The second-order approximation is equivalent to assuming that the matrix of second derivatives does not change and hence can be computed at x instead of at .
