Fletcher-Powell method

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. 

For an iV-dimensional surface, 2iV + 4 -i- m(N + 3) steps are required, where m is the number of passes through step 7. (typically two or three). A test of the goodness of an individual structure requires iV + 1 energy computations to evaluate the first derivative numerically. As N increases, the modified Fletcher-Powell method is significantly more efficient than the axial iteration... 

Within some programs, the ROMPn methods do not support analytic gradients. Thus, the fastest way to run the calculation is as a single point energy calculation with a geometry from another method. If a geometry optimization must be done at this level of theory, a non-gradient-based method such as the Fletcher-Powell optimization should be used. 

Davidsou-Fletcher-Powell (DFP) a geometry optimization algorithm De Novo algorithms algorithms that apply artificial intelligence or rational techniques to solving chemical problems density functional theory (DFT) a computational method based on the total electron density... 

A more sophisticated version of the sequential univariate search, the Fletcher-Powell, is actually a derivative method where elements of the gradient vector g and the Hessian matrix H are estimated numerically. 

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