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 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). The normal equations are formed by multiplying both sides of the equation by . This produces an N by N square system, the solution to which is the desired least-squares solution for the parameter shifts. Similar equations are obtained by expanding (18.1.4.1) as a second-order Taylor series about the minimum and differentiating. 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 .