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The use of magic integers in phase determination is examined in the light of the theory of Main [Acta Cryst. (I 977). A33, 750-757]. The integers may be used in the economical search of an n-dimensional function of the phases. An interpolation procedure in the n-dimensional phase space allows the use of integer sequences of quite high error, with a consequent reduction in the magnitudes of the integers used. The number of variables to be associated with the magic-integer sequences is also examined. It is found that this number has virtually no effect on either accuracy of phase representation or on computing time. The range of one of the variables can be restricted, where necessary, in order to define the enantiomorph, thus using several phases simultaneously to give a strong enantiomorph definition. A convergence procedure is described for choosing the phases to be represented by magic integers. Magic integers may also be used to choose sets of phase values for the reflexions used in MULTAN to start phase determination. This replaces the more usual quadrant permutation method and results in large savings in the number of starting sets to be explored. MULTA N is thus made more powerful for the same computing time as before.

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