Newton-Raphson methods minima

An alternative, and closely related, approach is the augmented Hessian method [25]. The basic idea is to interpolate between the steepest descent method far from the minimum, and the Newton-Raphson method close to the minimum. This is done by adding to the Hessian a constant shift matrix which depends on the magnitude of the gradient. Far from the solution the gradient is large and, consequently, so is the shift d. One... 

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. 

It is instructive to note that both the steepest-descent and the Newton-Raphson methods lead in the direction of —VU however, the steepest-descent method is unable to tell us how far to go in each step and therefore we have to search for the minimum in a very ineffective way (see Fig. 4.3). 

FIGURE 4.3. Illustrating the effectiveness of different minimization schemes. The steepest-deicent method requires many steps to reach the minimum, while the Newton-Raphson method locates the minimum in a few steps (at the expense, however, of evaluating the second derivative matrix). 

The Newton-Raphson approach is another minimization method.f It is assumed that the energy surface near the minimum can be described by a quadratic function. In the Newton-Raphson procedure the second derivative or F matrix needs to be inverted and is then usedto determine the new atomic coordinates. F matrix inversion makes the Newton-Raphson method computationally demanding. Simplifying approximations for the F matrix inversion have been helpful. In the MM2 program, a modified block diagonal Newton-Raphson procedure is incorporated, whereas a full Newton-Raphson method is available in MM3 and MM4. The use of the full Newton-Raphson method is necessary for the calculation of vibrational spectra. Many commercially available packages offer a variety of methods for geometry optimization. 

The energies of most of the defects were minimised using the Newton-Raphson method with BFGS [14] updating of the Hessian. However, it was sometimes necessary to use the more demanding Rational Function Optimiser (RFO) [15] which enforces the required number of imaginary eigenvalues of the Hessian, to be zero at the minimum and one when used to locate a transition state as discussed later. 

When the initial estimate is far off, the Newton-Raphson method may not converge in fact, when the initial value is located at an x-value where F(x) goes through a minimum or maximum, the denominator in (8.1-1) will become zero, so that (8.1-1) will place the next iteration at either + °° or — . Furthermore, the Newton-Raphson algorithm will find only one root at a time, regardless of how many roots there are. On the other hand, when the method works, it is usually very efficient and fast. Exercise 8.1 illustrates how the Newton-Raphson algorithm works. 

The simple Newton-Raphson method attempts to find the minimum of this quadratic expression by solving the linear system... 

This formulation of the Newton-Raphson method for columns with infinitely many stages is analogous to the 2N Newton-Raphson method for a column with a finite number of stages. First the procedure is developed for a conventional distillation column with infinitely many stages for which the condenser duty Qc (or the reflux ratio Lx/D) and the reboiler duty QR (or the boilup ratio VN/B) are specified and it is required to find the product distribution. Then the procedure is modified as required to find the minimum reflux ratio required to effect the specified separation of two key components. 

Our earlier scheme is a modification of the Newton-Raphson method, where the atoms are moved one at a time rather than simultaneously (Allinger et al., 1971). It is therefore not necessary to calculate all of the elements to fill out a 3(N-6) x 3(N-6) matrix, nor is it necessary to diagonalize such a matrix. The principles of the method are outlined as follows. It is assumed that the energy surface in the vicinity of an energy minimum can be approximated by the function (10), where x, y, and z represent the Cartesian coordinates... 

Second-order methods use not only the first derivatives (i.e. the gradients) but also the second derivatives to locate a minimum. Second derivatives provide information about the curvature of the function. The Newton-Raphson method is the simplest second-order method. Recall our Taylor series expansion about the point r. . Equation (5.2) ... 

In practice, of course, the surface is only quadratic to a first approximation and so a number of steps will be required, at each of which the Hessian matrix must be calculated and inverted. The Hessian matrix of second derivatives must be positive definite in a Newton-Raphson minimisation. A positive definite matrix is one for which all the eigenvalues are positive. When the Hessian matrix is not positive definite then the Newton-Raphson method moves to points (e.g. saddle points) where the energy increases. In addition, far from a mimmum the harmonic approximation is not appropriate and the minimisation can become unstable. One solution to this problem is to use a more robust method to get near to the minimum (i.e. where the Hessian is positive definite) before applying the Newton-Raphson method. 

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