Fast differentiation algorithm and efficient calculation of the exact matrix of second derivatives.

Many crystallographic problems are reduced to the optimization of some functional. In most cases, this functional is expressed in terms of structure factors and depends on a large number of variables; an example is the refinement of atomic models. Calculation of the functional derivatives, necessary for different optimization methods, is a time-consuming procedure. Previously, a technique to calculate the exact gradient of any crystallographic functional for the time equal to that for a single-function-value calculation has been proposed [Lunin & Urzhumtsev (1985). Acta Cryst. A41, 327-333]. Currently, a similar scheme is proposed to calculate the exact matrix of the second derivatives of these functionals. The accuracy of this matrix is crucial for the calculation of the inverted matrix, which can be used in optimization methods of the second order.