Gradient method iterative procedure

The Sentinel method is the outstanding exponent of the group of interpretive methods, as it has already been applied successfully for selectivity optimization in programmed solvent LC. However, other interpretive methods, based either on fixed experimental designs or on iterative procedures, can be applied along the same lines. It was seen in section 6.3.2.3 that the extension of the Sentinel method to incorporate gradient optimization was fairly straightforward. 

As far as the Fletcher-Reeves method is concerned, it must clearly be the method of choice in linear coefficient optimization as it involves only the storage of gradient and direction vectors between iterations. It has been used by a number of authors (Sleeman,29 Fletcher,5 Kari and Sutcliffe,32 Claxton and Smith,51 and Weinstein and Pauncz52). It [is unfortunately possible, however, to sum up the experience so far gained of the method in quantum chemistry as disappointing, in the sense that in SCF caclulations the authors have found that the calculations proceed significantly more slowly than the conventional iterative procedure, when the conventional procedure converges at all. 

The iterative procedure for the solution of the inverse problem by the regular conjugate gradient method was performed assuming the functions T(x,y), AT(x,y), X(x,y), and J x) are available at the -th iteration. The wall heat flux, q(x), at step n+1 is obtained from... 

All matrix elements in the Newton-Raphson methods may be constructed from the one- and two-particle density matrices and transition density matrices. The linear equation solutions may be found using either direct methods or iterative methods. For large CSF expansions, such micro-iterative procedures may be used to advantage. If a micro-iterative procedure is chosen that requires only matrix-vector products to be formed, expansion-vector-dependent effective Hamiltonian operators and transition density matrices may be constructed for the efficient computation of these products. Sufficient information is included in the Newton-Raphson optimization procedures, through the gradient and Hessian elements, to ensure second-order convergence in some neighborhood of the final solution. 

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