SVDdiagnostic, a program to diagnose numerical conditioning of Rietveld refinements

Singular value decomposition (SVD) of the matrix of normal equations is used here both passively to assess numerical stability, and actively to troubleshoot problem refinements, singular or not. Such systems can then either be cured by rank reduction or solved with arbitrary-precision arithmetic carrying a number of digits known to be sufficient. SVD analysis provides objective information about such required rank reduction or number of digits. Pre-conditioning of the normal matrix is seen to decrease its condition number by many orders of magnitude in actual cases, illustrating its great practical usefulness. The methods and tools developed here have general applicability to diagnose problems with least squares, in particular ill-conditioned Rietveld refinements. Crystal-chemical and standard refinements described in the work by Mercier et al. [J. Appl. Cryst. (2006), 39, 369-375] are shown to have similar numerical stability. The program SVDdiagnostic is freely available at http://www.tothcanada.com .