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

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