The use of first and second derivatives in optical model parameter searches

This paper compares the efficiencies of standard minimization procedures in carrying out nuclear optical-model fits. It is shown that first-order perturbation theory permits computation of the gradient of x² more than five times as fast as is possible by difference methods. A linear approximation to the second-derivative matrix in terms of first derivatives of residuals is found to be very accurate in the neighborhood of minima; it provides a way of introducing second-derivative information that is significantly superior to the use of variable-metric algorithms. The resulting restricted-step Gauss-Newton procedures are shown to be about five times as fast as direct-search methods. The use of the methods of pseudo-inverses to “freeze” linear combinations of parameters poorly determined by the data is discussed.