International
Tables for Crystallography Volume B Reciprocal space Edited by U. Shmueli © International Union of Crystallography 2010 
International Tables for Crystallography (2010). Vol. B, ch. 1.3, pp. 9495

Some methods of phase determination rely on maximizing a certain global criterion involving the electron density, of the form , under constraint of agreement with the observed structurefactor amplitudes, typically measured by a residual C. Several recently proposed methods use for various measures of entropy defined by taking or (Bricogne, 1982; Britten & Collins, 1982; Narayan & Nityananda, 1982; Bryan et al., 1983; Wilkins et al., 1983; Bricogne, 1984; Navaza, 1985; Livesey & Skilling, 1985). Sayre's use of the squaring method to improve protein phases (Sayre, 1974) also belongs to this category, and is amenable to the same computational strategies (Sayre, 1980).
These methods differ from the densitymodification procedures of Section 1.3.4.4.3.2 in that they seek an optimal solution by moving electron densities (or structure factors) jointly rather than pointwise, i.e. by moving along suitably chosen search directions [or ].
For computational purposes, these search directions may be handled either as column vectors of sample values on a grid in real space, or as column vectors of Fourier coefficients in reciprocal space. These column vectors are the coordinates of the same vector in an abstract vector space of dimension over , but referred to two different bases which are related by the DFT and its inverse (Section 1.3.2.7.3).
The problem of finding the optimum of S for a given value of C amounts to achieving collinearity between the gradients and of S and of C in , the scalar ratio between them being a Lagrange multiplier. In order to move towards such a solution from a trial position, the dependence of and on position in must be represented. This involves the Hessian matrices H(S) and H(C), whose size precludes their use in the whole of . Restricting the search to a smaller search subspace of dimension n spanned by we may build local quadratic models of S and C (Bryan & Skilling, 1980; Burch et al., 1983) with respect to n coordinates X in that subspace:The coefficients of these linear models are given by scalar products:which, by virtue of Parseval's theorem, may be evaluated either in real space or in reciprocal space (Bricogne, 1984). In doing so, special positions and reflections must be taken into account, as in Section 1.3.4.2.2.8. Scalar products involving S are best evaluated by realspace grid summation, because H(S) is diagonal in this representation; those involving C are best calculated by reciprocalspace summation, because H(C) is at worst blockdiagonal in this representation. Using these Hessian matrices in the wrong space would lead to prohibitively expensive convolutions instead of scalar (or at worst matrix) multiplications.
References
Bricogne, G. (1982). Generalised density modification methods. In Computational Crystallography, edited by D. Sayre, pp. 258–264. New York: Oxford University Press.Bricogne, G. (1984). Maximum entropy and the foundations of direct methods. Acta Cryst. A40, 410–445.
Britten, P. L. & Collins, D. M. (1982). Information theory as a basis for the maximum determinant. Acta Cryst. A38, 129–132.
Bryan, R. K., Bansal, M., Folkhard, W., Nave, C. & Marvin, D. A. (1983). Maximumentropy calculation of the electron density at 4 Å resolution of Pf1 filamentous bacteriophage. Proc. Natl Acad. Sci. USA, 80, 4728–4731.
Bryan, R. K. & Skilling, J. (1980). Deconvolution by maximum entropy, as illustrated by application to the jet of M87. Mon. Not. R. Astron. Soc. 191, 69–79.
Burch, S. F., Gull, S. F. & Skilling, J. (1983). Image restoration by a powerful maximum entropy method. Comput. Vision Graphics Image Process. 23, 113–128.
Livesey, A. K. & Skilling, J. (1985). Maximum entropy theory. Acta Cryst. A41, 113–122.
Narayan, R. & Nityananda, R. (1982). The maximum determinant method and the maximum entropy method. Acta Cryst. A38, 122–128.
Navaza, J. (1985). On the maximumentropy estimate of the electron density function. Acta Cryst. A41, 232–244.
Sayre, D. (1974). Leastsquares phase refinement. II. Highresolution phasing of a small protein. Acta Cryst. A30, 180–184.
Sayre, D. (1980). Phase extension and refinement using convolutional and related equation systems. In Theory and Practice of Direct Methods in Crystallography, edited by M. F. C. Ladd & R. A. Palmer, pp. 271–286. New York, London: Plenum.
Wilkins, S. W., Varghese, J. N. & Lehmann, M. S. (1983). Statistical geometry. I. A selfconsistent approach to the crystallographic inversion problem based on information theory. Acta Cryst. A39, 47–60.