Davidon-Fletcher-Powell

EX242 2.4.2 Rosenbrock problem by Davidon-Fletcher-Powell method M36... 

It is widely believed that, generally speaking, methods such as the Davidon-Fletcher-Powell method are superior to the Fletcher-Reeves method and, indeed, Fletcher suggests (see p. 82 of ref. 8) that typically the Fletcher-Reeves method will take about twice as many iterations as the Davidon-Fletcher-Powell method. 

This rank two correction has no numerical problems with small denominators, and it can be shown that is always positive definite if H is. This guarantees that d will always be a descent direction, thus overcoming one of the serious difficulties of the pure Newton method. The Davidon-Fletcher-Powell method works quite well, but it turns out that the slight modification below gives experimentally better results, even though it is theoretically equivalent. 

I is the identity matrix. The six first derivatives of the energy with respect to the strain components e, measure the forces acting on the unit cell. When combined with the atomic coordinates we get a matrix with 3N - - 6 dimensions. At a minimum not only should there be no force on any of the atoms but the forces on the unit cell should also be zero. Application of a standard iterative minimisation procedure such as the Davidon-Fletcher-Powell method will optimise all these degrees of freedom to give a strain-free final structure. In such procedures a reasonably accurate estimate of the initial inverse Hessian matrix is usually required to ensure that the changes in the atomic positions and in the cell dimensions are matched. 

Davidon-Fletcher-Powell. Choose an initial point Xj = (0, 0) , an initial deflection matrix Dj =... 

The first formulation of the rank-2 updating was the Davidon-Fletcher-Powell (DFP) formula that updates the inverse Hessian matrix ... 

Blanco M, Coello J, Gene J, et al. (1996) Application of the Davidon-Fletcher-Powell algorithm to the resolution of multicomponent mixtures using UV-Vis spectrophotometry. Analytica Chimica Acta 327(2) 145-152. 

Quasi-Newton, variable metric, conjugate gradient, Fletcher-Powell, Davidon-Fletcher-Powell, Murtagh-Sargent, Broyden-Fletcher-... 

The program uses several minimizing routines that can easily be chosen, such as the pit-mapping method developed by Sillen, but also Simplex, Davidon-Fletcher-Powell, and Monte Carlo methods of MINUIT package [53]. As for AR, up to third-degree equations can be used ... 

J. Cioslowski and M. Kertesz, QCPE Bull., 7, 159 (1987). Davidon-Fletcher-Powell vs. Broyden-Fletcher-Goldfarb-Shanno Optimization Procedures in Semiempirical and Solid-State MNDO Calculations. 

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