Z-Rietveld is a program suite for Rietveld analysis and the Pawley method; it was developed for analyses of powder diffraction data in the Materials and Life Science Facility of the Japan Proton Accelerator Research Complex. Improvements have been made to the nonlinear least-squares algorithms of Z-Rietveld so that it can deal with singular matrices and intensity non-negativity constraints. Owing to these improvements, Z-Rietveld successfully executes the Pawley method without requiring any constraints on the integrated intensities, even in the case of severely or exactly overlapping peaks. In this paper, details of these improvements are presented and their advantages discussed. A new approach to estimate the number of independent reflections contained in a powder pattern is introduced, and the concept of good reflections proposed by Sivia [J. Appl. Cryst. (2000), 33, 1295-1301] is shown to be explained by the presence of intensity non-negativity constraints, not the intensity linear constraints. research papers J. Appl. Cryst. (2012). 45, 299-308 R. Oishi-Tomiyasu et al. Application of matrix decomposition in Z-Rietveld 305 Figure 3 Complementary cumulative distribution function of the chi-squared distribution f ðx; NÞ.

A new powder auto-indexing method for the CONOGRAPH software [Oishi-Tomiyasu (2013). Acta Cryst. A69, 603-610] can carry out exhaustive powder auto-indexing in a short time, even if the q values of many peaks are used, with sufficient consideration given to their observational errors. This article explains that the use of many q values is essential to make powder auto-indexing robust against dominant zones and missing or false peaks in the input. Results from CONOGRAPH for 25 real diffraction patterns, including difficult cases, are presented. Owing to a sorting criterion for zones defined in the previous article, the computation times were reduced by a factor of between 18 and 250, and exhaustive powder auto-indexing was completed in 5 min at most.

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A new Bravais-lattice determination algorithm is introduced herein. For error-stable Bravais-lattice determination, Andrews & Bernstein [Acta Cryst. (1988), A44, 1009-1018] proposed the use of operations to search for nearly Buerger-reduced cells. Although these operations play an essential role in their method, they increase the computation time, in particular when lattice parameters obtained in (powder) auto-indexing are supposed to contain large errors. The new algorithm requires only several permutation matrices in addition to the operations that are necessary when the lattice parameters have exact values. As a result, the computational efficiency of error-stable Bravais-lattice determination is improved considerably. Furthermore, the new method is proved to be error stable under a very general assumption. The detailed algorithms and the set of matrices sufficient for error-stable determination are presented.