Quadratic approximation method

This equation can be solved in terms of the eigenvectors of F(rQ) to advance one step forward from 5( o) on the MEP. Much larger steps can be taken using this local quadratic approximation method [41,44]. Even more substantial increases in the step size have been achieved by accounting approximately for third derivatives of the energy along the path (although only ab initio second derivatives are actually computed [45]). 

Reaction Coordinate Comparison of Gradient and Local Quadratic Approximation Methods. 

A great number of studies indicated that quadratic approximation methods, which are characterized by solving a sequence of quadratic subproblems recursively belong to the most efficient and reliable nonlinear programming algorithms presently available. This method combines the most efficient characteristics of different optimization techniques (see e.g, [19]). For equality constrained problems, the general nonlinear constrained optimization problem can be formulated by an... 

The simultaneous measurement of Ip and Is is performed by the quadratic approximation method depicted in Figure 13.13. When yr = 90°(7t/2 radians),... 

The simplest smooth fiuictioii which has a local miiiimum is a quadratic. Such a function has only one, easily detemiinable stationary point. It is thus not surprising that most optimization methods try to model the unknown fiuictioii with a local quadratic approximation, in the fomi of equation (B3.5.1). 

Two quadratic equations in two variables can in general be solved only by numerical methods (see Numerical Analysis and Approximate Methods ). If one equation is of the first degree, the other of the second degree, a solution may be obtained by solving the first for one unknown. This result is substituted in the second equation and the resulting quadratic equation solved. 

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. 

Anotlrer way of choosing A is to require that the step length be equal to the trust radius R, this is in essence the best step on a hypersphere with radius R. This is known as the Quadratic Approximation (QA) method. ... 

The techniques for calculating the electronic states of an impurity in a metal from first principles are well understood and have already been implemented. An approximate method that leads to much simpler calculations has been proposed recently. We investigate this method within the framework of the quadratic Korringa-Kohn-Rostoker formalism, and show that it produces surprisingly good predictions for the charge on the impurity. 

In this case, the approximate solution gives almost the same answer as the exact solution. In general, you should use the approximate method and check your answer to sec that it is reasonable only if it is not should you use the quadratic equation. 

Newton s method makes use of the second-order (quadratic) approximation of fix) at x and thus employs second-order information about fix), that is, information obtained from the second partial derivatives of fix) with respect to the independent variables. Thus, it is possible to take into account the curvature of fix) at x and identify better search directions than can be obtained via the gradient method. Examine Figure 6.9b. 

Difficulty 3 can be ameliorated by using (properly) finite difference approximation as substitutes for derivatives. To overcome difficulty 4, two classes of methods exist to modify the pure Newton s method so that it is guaranteed to converge to a local minimum from an arbitrary starting point. The first of these, called trust region methods, minimize the quadratic approximation, Equation (6.10), within an elliptical region, whose size is adjusted so that the objective improves at each iteration see Section 6.3.2. The second class, line search methods, modifies the pure Newton s method in two ways (1) instead of taking a step size of one, a line search is used and (2) if the Hessian matrix H(x ) is not positive-definite, it is replaced by a positive-definite matrix that is close to H(x ). This is motivated by the easily verified fact that, if H(x ) is positive-definite, the Newton direction... 

Successive quadratic programming (SQP) methods solve a sequence of quadratic programming approximations to a nonlinear programming problem. Quadratic programs (QPs) have a quadratic objective function and linear constraints, and there exist efficient procedures for solving them see Section 8.3. As in SLP, the linear constraints are linearizations of the actual constraints about the selected point. The objective is a quadratic approximation to the Lagrangian function, and the algorithm is simply Newton s method applied to the KTC of the problem. 

In some cases when estimates of the pure-error mean square are unavailable owing to lack of replicated data, more approximate methods of testing lack of fit may be used. Here, quadratic terms would be added to the models of Eqs. (32) and (33), the complete model would be fitted to the data, and a residual mean square calculated. Assuming this quadratic model will adequately fit the data (lack of fit unimportant), this quadratic residual mean square may be used in Eq. (68) in place of the pure-error mean square. The lack-of-fit mean square in this equation would be the difference between the linear residual mean square [i.e., using Eqs. (32) and (33)] and the quadratic residual mean square. A model should be rejected only if the ratio is very much greater than the F statistic, however, since these two mean squares are no longer independent. 

Fig. 6.7 Comparison of the maximum of the neural network approximation of the ODHE ethylene yield obtained in 10 runs of the genetic algorithm with a population size 60, and the global maximum obtained with a sequential quadratic programming method run for 15 different starting points.

Instead of the very demanding CCSDT calculations one often performs CCSD (T) (note the parentheses), in which the contribution of triple excitations is represented in an approximate way (not refined iteratively) this could be called coupled cluster approximate (or perturbative) triples. The quadratic configuration method (QCI) is very similar to the CC method. The most accurate implementation of this in common use is QCISD(T) (quadratic Cl singles, doubles, triples, with triple excitations treated in an approximate, non-iterative way). The CC method, which is usually only moderately slower than QCI (Table 5.6), is apparently better [102]. CCSD(T) calculations are, generally speaking, the current benchmark for practical molecular calculations on molecules of up to moderate size. 

This method, first introduced by Isaac Newton and better formulated in the actual form by Joseph Raphson, is the simplest second-order algorithm. The basic idea is to use a quadratic approximation to the objective function around the initial parameter estimate and, then, to adjust the parameters in order to minimize the quadratic approximation until their values converge. 

The separation of the PES into a part determined by the reaction coordinate and a part described by a quadratic approximation in a subspace of the remaining coordinates has recently often been used, typically with the WKB approximation (236,237) Yamashita and Miller (238) utilized the reaction-path Hamiltonian method combined with the path-integral method to calculate the rate constant of the reaction of H + H2. 

Truncated Newton methods were introduced in the early 1980s111-114 and have been gaining popularity ever since.82-109 110 115-123 Their basis is the following simple observation. An exact solution of the Newton equation at every step is unnecessary and computationally wasteful in the framework of a basic descent method. That is, an exact Newton search direction is unwarranted when the objective function is not well approximated by a convex quadratic and/or the initial point is distant from a solution. Any descent direction will suffice in that case. As a solution to the minimization problem is approached, the quadratic approximation may become more accurate, and more effort in solution of the Newton equation may be warranted. 

If the approximation had caused an error of 10% or more, you would not be able to use it. You would have to solve by a more rigorous method, such as the quadratic equation or using an electronic calculator to solve for the unknown value. (Ask your instructor if you are responsible for the quadratic equation method.)... 

There have been a number of correction algorithms formulated over the years to help improve this size consistency problem but they do not entirely resolve it °. Another means of introducing size consistency is by a quadratic approximation, QCISD. The approach achieves this size consistency by sacrificing its variational character. It can be considered as a simplified approximate form of CCSD (see below) the method may cease to remain size consistent on going to higher levels of substitution. ... 

Quadratic Approximation (QA) method. 14.3.3 Storing and Diagonalizing the Hessian... 

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