Newton—Raphson method structure

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. 

This is the Newton-Raphson method. Starting with some structure, Rq, a new structure, R, can be obtained through Eq. (31) where h is given by Eq. (33), once the gradient vector g and the Hessian H have been calculated. 

Numerical Methods and Data Structure. Both EQ3NR and EQ6 make extensive use of a combined method, using a "continued fraction" based "optimizer" algorithm, followed by the Newton-Raphson method, to make equilibrium calculations. The method uses a set of master or "basis" species to reduce the number of iteration variables. Mass action equations for the non-basis species are substituted into mass balance equations, each of which corresponds to a basis species. 

In this study, conventional all-atom normal mode analysis was performed with the CHARMM package. Like other studies in the literature [29], the united atom CHARMM19 force field [33] with the EEFl solvation model [40] was used. Before normal mode calculation, the structures underwent many cycles of energy minimization by the adopted-basis Newton-Raphson method with decreasing harmonic constraints. 

Of course, the surface is not quadratic and the Hessian is not constant from step to step. However, near a critical point, the Newton-Raphson method (Eq. (2)) will converge rapidly. The main difficulty is that the convergence of the Newton-Raphson method is local. Thus the method will converge to the nearest critical point to the starting point. Consequently, one must start the optimization with a Hessian that contains no or one negative eigenvalue according to whether a minimum or a saddle point structure is required. For a minimum, the steepest descent direction... 

The choice of minimization method depends on two factors, the size of the systan and its conformational state. As a guiding principal, if the structure is far from the minimum then the steepest descent should be used for the first few steps, followed by a more precise conjugate gradient or Newton-Raphson methods. For detailed information, the reader is requested to refer to some of the standard textbooks (Allen and Tildesley 1987 Frenkel and Smit 1996X... 

Newton-Raphson method (NR.mh The structure of this function is the same as that of the two previous functions. The derivative of the function is taken numerically to reduce the inputs. It is also more applicable for complicated functions. The reader may simply introduce the derivative function in another MATLAB function and use it instead of numerical derivation. In the case of the Colebrook equation, the same MATLAB function Colebrookm, which represents Eq. (1,5), may be used with this function to calculate the value of the friction factor. 

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