Program construction for macromolecule atomic model refinement based on the fast Fourier transform and fast differentiation algorithms

Structure refinement may be considered as a minimization of a function R(_χ) of a large number of refineable parameters. A new type of function incorporating phase probability distribution is proposed. The minimization of the function utilizing gradient methods requires the computation of gradient ∇R, as well as the product of the gradient and the matrix of second derivatives with some direction. The algorithm of Kim, Nesterov & Cherkassky [Dokl. Akad. Nauk SSSR (1984), 275, 1306-1309] adapted to macromolecular structure refinement takes about four times longer for the computation of these values compared to the computation of the value of the minimized function. The matrix of second derivatives is used without any approximation.