Matrix Factorisations for the Estimation of NMR Relaxation Distributions

The two most successful methods of estimating the distribution of NMR relaxation times from two dimensional data are firstly a data compression stage followed by application of the Butler-Reeds-Dawson (BRD) algorithm, and secondly a primal dual interior point method using a preconditioned conjugate gradient (PCG). Both of these methods have been presented in the literature as requiring a truncated singular value decomposition of matrices representing the exponential kernels. Other matrix factorisations are applicable to each of these algorithms, and which demonstrate the different fundamental principles behind the operation of the algorithms. In the case of the data compression approach the most appropriate matrix decomposition specifically designed for this task is the rank-revealing QR (RRQR) factorisation. In the case of the interior point method, the most appropriate method is the LDL factorisation with diagonal pivoting, also known as the Bunch-Kaufman-Parlett factorisation. The details of these differences are discussed, and the performances of the algorithms are compared numerically.