Tables for
Volume B
Reciprocal space
Edited by U. Shmueli

International Tables for Crystallography (2006). Vol. B, ch. 1.3, p. 89   | 1 | 2 |

Section Booth's method of steepest descents

G. Bricognea

aMRC Laboratory of Molecular Biology, Hills Road, Cambridge CB2 2QH, England, and LURE, Bâtiment 209D, Université Paris-Sud, 91405 Orsay, France Booth's method of steepest descents

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Having defined the now universally adopted R factors (Booth, 1945b[link]) as criteria of agreement between observed and calculated amplitudes or intensities, Booth proposed that R should be minimized with respect to the set of atomic coordinates [\{{\bf x}_{j}\}_{j\in J}] by descending along the gradient of R in parameter space (Booth, 1947c[link],d[link]). This `steepest descents' procedure was compared with Patterson methods by Cochran (1948d)[link].

When calculating the necessary derivatives, Booth (1948a[link], 1949[link]) used the formulae given above in connection with least squares. This method was implemented by Qurashi (1949)[link] and by Vand (1948[link], 1951[link]) with parameter-rescaling modifications which made it very close to the least-squares method (Cruickshank, 1950[link]; Qurashi & Vand, 1953[link]; Qurashi, 1953[link]).


Booth, A. D. (1945b). An expression for following the process of refinement in X-ray structure analysis using Fourier series. Philos. Mag. 36, 609–615.
Booth, A. D. (1947c). A new refinement technique for X-ray structure analysis. J. Chem. Phys. 15, 415–416.
Booth, A. D. (1947d). Application of the method of steepest descents to X-ray structure analysis. Nature (London), 160, 196.
Booth, A. D. (1948a). A new Fourier refinement technique. Nature (London), 161, 765–766.
Booth, A. D. (1949). The refinement of atomic parameters by the technique known in X-ray crystallography as `the method of steepest descents'. Proc. R. Soc. London Ser. A, 197, 336–355.
Cochran, W. (1948d). X-ray analysis and the method of steepest descents. Acta Cryst. 1, 273.
Cruickshank, D. W. J. (1950). The convergence of the least-squares and Fourier refinement methods. Acta Cryst. 3, 10–13.
Qurashi, M. M. (1949). Optimal conditions for convergence of steepest descents as applied to structure determination. Acta Cryst. 2, 404–409.
Qurashi, M. M. (1953). An analysis of the efficiency of convergence of different methods of structure determination. I. The methods of least squares and steepest descents: centrosymmetric case. Acta Cryst. 6, 577–588.
Qurashi, M. M. & Vand, V. (1953). Weighting of the least-squares and steepest-descents methods in the initial stages of the crystal-structure determination. Acta Cryst. 6, 341–349.
Vand, V. (1948). Method of steepest descents: improved formula for X-ray analysis. Nature (London), 161, 600–601.
Vand, V. (1951). A simplified method of steepest descents. Acta Cryst. 4, 285–286.

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