Simultaneous Refinements with Complex Compositional Constraints - Example of Singular Value Decomposition to Diagnose Poor Matrix Conditioning

A complex refinement of the structure of the battery cathode material, Li1.2Mn0.4Ni0.3Co0.1O2, using neutron and resonant diffraction data is used to demonstrate SVDdiagnostic as a useful tool to diagnose problems in the least squares normal matrix. SVDdiagnostic examines both the unprocessed and pre-conditioned matrices and outputs a 'condition' number as well as combinations of variables causing pseudo-singularity. Shrewd use of this diagnostic tool allowed structural details to be extracted that would otherwise be obliterated by large parameter correlations in such a complex simultaneous refinement. INTRODUCTION One common feature of all Rietveld refinement codes is the inversion of the least-squares normal matrix. This matrix becomes larger and more complex with the number of variables being refined simultaneously. Most Rietveld codes stabilize the least-squares matrix with some kind of preconditioning. This may take the form of an arbitrary diagonal matrix P to derive an alternative matrix B from C using C = P · B · P, where C = A · A is the matrix of normal equations used for solving an over-determined system of weighted observations A · x = b through the equation A ·A · x = A ·b. Solutions are obtained as x = C · A ·b. If the diagonal matrix P is set with element Pii = (Cii), it follows that Pii = (Cii). Accordingly, Bij = Cij(Cii Cjj) while Cij = Bij(Cii Cjj). Inverting B and obtaining C from B is usually more stable than inverting C directly. In the numerical inversion of a matrix, one of the most important values is the ratio of the largest to the smallest singular values in the matrix known as its condition number (CN). Numerical analysts regard log10(CN) to be the minimum number of decimal digits required to avoid catastrophic error propagation. Most Rietveld codes use double-precision, or 14 digits. Consequently a log10(CN) greater than 14 is a serious problem. The matrix that Rietveld codes actually invert is the preconditioned matrix, which usually has a much smaller CN. However, a large CN for the unprocessed matrix can still signal potential problems, even if the CN of the 133 Copyright ©JCPDS-International Centre for Diffraction Data 2007 ISSN 1097-0002