International
Tables for
Crystallography
Volume B
Reciprocal space
Edited by U. Shmueli

International Tables for Crystallography (2010). Vol. B, ch. 1.3, p. 88   | 1 | 2 |

## Section 1.3.4.3.5.4. Real symmetric transforms

G. Bricognea

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#### 1.3.4.3.5.4. Real symmetric transforms

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Conjugate symmetric (Section 1.3.2.4.2.3) implies that if the data X are real and symmetric [i.e. and ], then so are the results . Thus if contains a centre of symmetry, F is real symmetric. There is no distinction (other than notation) between structure-factor and electron-density calculation; the algorithms will be described in terms of the former. It will be shown that if , a real symmetric transform can be computed with only partial transforms instead of .

 (i) Decimation in time Since we have and . The decimated vectors are not only real, but have an internal symmetry expressed byThis symmetry, however, is different for each so that we may multiplex two such vectors and into a general real vectorfor each of the pairs . The Hermitian-symmetric transform vectorscan then be evaluated by the methods of Section 1.3.4.3.5.1(b) at the cost of only general complex . The demultiplexing relations by which the separate vectors and may be recovered are most simply obtained by observing that the vectors Z after the twiddle-factor stage are real-valued since F(2I) has a real matrix. Thus, as in Section 1.3.4.3.5.1(c)(i),where and are real vectors and where the multipliers and are the inverse twiddle factors. Therefore,and hence the demultiplexing relation for each :The values of and at those points where can be evaluated directly while forming Y. This demultiplexing and the final stage of the calculation, namelyneed only be carried out for the unique half of the range of . (ii) Decimation in frequency Similarly, the vectors of decimated and scrambled results are real and obey internal symmetrieswhich are different for each . For each of the pairs the multiplexed vectoris a Hermitian-symmetric vector without internal symmetry, and the real vectorsmay be evaluated at the cost of only general complex by the methods of Section 1.3.4.3.5.1(c). The individual transforms and may then be retrieved via the demultiplexing relationswhich can be solved recursively as described in Section 1.3.4.3.5.1(b)(ii). This yields the unique half of the real symmetric results F.