Newton-Raphson minimizer

For nonquadratic but monotonic surfaces, the Newton-Raphson minimization method can be applied near a minimum in an iterative way [24]. 

There are several reasons that Newton-Raphson minimization is rarely used in mac-romolecular studies. First, the highly nonquadratic macromolecular energy surface, which is characterized by a multitude of local minima, is unsuitable for the Newton-Raphson method. In such cases it is inefficient, at times even pathological, in behavior. It is, however, sometimes used to complete the minimization of a structure that was already minimized by another method. In such cases it is assumed that the starting point is close enough to the real minimum to justify the quadratic approximation. Second, the need to recalculate the Hessian matrix at every iteration makes this algorithm computationally expensive. Third, it is necessary to invert the second derivative matrix at every step, a difficult task for large systems. 

Schaumann T, Braun W, Wilthrich K. The program FANTOM for energy refinement of polypeptides and proteins using a Newton-Raphson minimizer in torsion angle space. Biopolymers 1990 29 679-694. 

The final step in the MM analysis is based on the assumption that, with all force constants and potential functions correctly specified in terms of the electronic configuration of the molecule, the nuclear arrangement that minimizes the steric strain corresponds to the observable gas-phase molecular structure. The objective therefore is to minimize the intramolecular potential energy, or steric energy, as a function of the nuclear coordinates. The most popular procedure is by computerized Newton-Raphson minimization. It works on the basis that the vector V/ with elements dVt/dxn the first partial derivatives with respect to cartesian coordinates, vanishes at a minimum point, i.e. = 0. This condition implies zero net force on each atom... 

A new feature in MM3 is the full Newton-Raphson minimization algorithm. This allows for the location and verification of transition states and for the calculation of vibrational spectra. Indeed, many of the new potential functions in MM3 were included to provide a better description of the potential energy surface which is required for an accurate calculation of vibrational spectra. 

Several modifications of the Engler-Schleyer force field have appeared. White (27) added olefin parameters and used an efficient two-stage Newton-Raphson minimization modification. The accuracy of the White force field regarding heat of formation calculations is... 

In the block-diagonal Newton-Raphson minimization, the generally small size of the off-diagonal terms is exploited and the matrix describing the curvature is reduced to N 3 3 matrices, i.e., to 9N elements (Fig. 3.8). Due to the approximations... 

Fig. 3.8. Arrangement of the blocks in the block-diagonal Newton-Raphson minimization procedure.

FANTOM for Energy Refinement of Polypeptides and Proteins Using a Newton—Raphson Minimizer in Torsion Angle Space. 

In these calculations, the isolated defect or defect cluster is embedded in the crystal, which extends to infinity, and the contrast between this approach and that used in the supercell methods is illustrated diagrammatically in Figure 1. The normal procedure in a Mott-Littleton calculation is to relax all the atoms in a region of crystal surrounding the defect, containing typically 100-300 atoms, until all are at zero force. Newton-Raphson minimization methods are generally used. The relaxation of the remainder of the crystal is then described by more approximate methods in which the polarization, P at a point r, is calculated for crystals that have dielectric isotropy, from the expression ... 

The multidimensional version of the Newton-Raphson minimization is employed for functions of many variables. The matrix of second derivatives is inverted and multiplied by the first derivatives to obtain the optimum step for all parameters (Fig. 8). Again, convergence can be problematic if the curvature is negative or close to zero. Note that the matrix is positive definite if the approximate form of the second derivative is used (last equation. Fig. 5). Thus, only eigenvalues close to zero can give problems. However, parameters are frequently... 

The Banded-Matrix Approximation. The linear nature of the polymer chains allows an approximation which greatly speeds up the individual Newton-Raphson minimizations. As discussed cibove, the Newton-Raphson method results in a set of linear equations (represented by the coefficient or "C matrix in Eq. (4)). The operations in the Gauss reduction of the system increase as N (where N is the number of variables). Atoms that are spatially... 

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