Various algorithms are described, developed for the dm density modi®cation package, which have not been described elsewhere. Methods are described for the following problems: determination of the absolute scale and overall temperature factor of a data set, by a method which is less dependent on data resolution than Wilson statistics; an ef®cient interpolation algorithm for averaging and its application to re®nement of averaging operators; a method for the automatic determination of averaging masks.

A variety of density-modification techniques are now available for improving electron-density maps in accordance with known chemical information. This modification must, however, always be constrained by consistency with the experimental data. This is conventionally achieved by alternating cycles of map modification in real space with recombination with the experimental data in reciprocal space. The phase recombination is based upon the assumption that the density-modified map may be treated as a partial model of the structure which contains information independent of the experimentally derived phases. This assumption is shown to be incorrect, and an alternative procedure is investigated which as a side effect allows calculation of a free R factor.

An improvement is described in the automatic procedure for solving crystal structures incorporated in the computer program LSAM. The development of signs from an initial set containing symbols is carried only as far as is necessary to establish strong relationships between the symbols. The information so gained is used in a fresh beginning of the symbolic-addition process. Some failure of relationships between symbols is allowed to give a multisolution method. A phase-permutation computer program for non-centrosymmetric structures, MULTAN, incorporates a weighted tangent formula. This is of the form {\rm tan}\varphi_{\bf h} = {{\sum_{\bf h'}w_{\bf h'},w_{\bf h-h'}|E_{\bf h'}E_{\bf h-h'}| \sin (\varphi_{\bf h'} + \varphi_{\bf h-h'})}\over{\sum_{\bf h'}w_{\bf h'},w_{\bf h-h'}|E_{\bf h'}E_{\bf h-h'}| \cos (\varphi_{\bf h'} + \varphi_{\bf h-h'})}} = {{T_{\bf h}}\over{B_{\bf h}}} and w_{\bf h} = {\rm tanh}\{ \sigma_3\sigma_2^{-3/2}|E_{\bf h}|(T_{\bf h}^2 + B_{\bf h}^2)^{1/2}\}.All phases are accepted as soon as they are found with the associated weight. This gives a fourfold increase in speed in development of the complete phase set. An absolute figure of merit is described to indicate probably correct phase sets for multisolution methods.

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A general scheme for the improvement of electron-density maps is described which combines information from real and reciprocal space. The use of Sayre's equation, solvent flattening and histogram matching within this scheme has been described previously [Main (1990). Acta Cryst. A46, 372-377]. Non-crystallographic symmetry averaging, the use of a partial structure and constraints on individual structure factors have now been added. A computer program, SQUASH, is described which applies all these constraints simultaneously. Its application to the maps of several structures has been successful, particularly so when non-crystallographic symmetry is present. Uninterpretable maps have been improved to the point where a significant amount of the structure can be recognized. Applying the constraints simultaneously is more powerful than applying them all in series.