A bound constrained conjugate gradient solution method as applied to crystallographic refinement problems

Small modifications to the conjugate gradient method for solving symmetric positive definite systems have resulted in an increase in performance over LU decomposition by a factor of around 84 for solving a dense system of 1325 unknowns. Performance is further increased in the case of applying upper- and lower-bound parameter constraints. For structure solution employing simulated annealing and the Newton–Raphson method of non-linear least squares, the overall performance gain can be a factor of four, depending on the applied constraints. In addition, the new algorithm with bounding constraints often seeks out lower minima than would otherwise be attainable without constraints. The behaviour of the new algorithm has been tested against the crystallographic problems of Pawley refinement, rigid-body and general crystal structure refinement.