Quadratic approximants

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). 

On the basis of tbe above analyses, it follows that them is no need to compute multidimensional potential surfaces if one wishes to handle the R-T effect in the framework of the model proposed. In spite of that, such computations were carried out in [152] in order to demonstrate the reliability of the model for handling the R-T effect and to estimate the range in which it can safely be applied in its lowest order (quadratic) approximation. The 3D potential surfaces involving the variation of the bending coordinates pi, p2 and the relative azimuth angle 7 = 4 2 1 were computed for both component of the state. 

In the case of ethylene, because of 2-fold symmetry, odd terms drop out of the series, V3, V5,... = 0. In the case of ethane, because of 3-fold symmeti-y, even temis drop out, V2, V4,... = 0. Terms higher than three, even though permitted by symmetry, are usually quite small and force fields can often be limited to three torsional terms. Like cubic and quaitic terms modifying the basic quadratic approximation for stretching and bending, terms in the Fourier expansion of Ftors (to) beyond n = 3 have limited use in special cases, for example, in problems involving octahedrally bound complexes. In most cases we are left with the simple expression... 

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 superscript (2) marks the quadratic approximation of the electrostatic energy. 

The above formula is obtained by differentiating the quadratic approximation of S(k) with respect to each of the components of k and equating the resulting expression to zero (Edgar and Himmelblau, 1988 Gill et al. 1981 Scales, 1985). It should be noted that in practice there is no need to obtain the inverse of the Hessian matrix because it is better to solve the following linear system of equations (Peressini et al. 1988)... 

In the lowest order (quadratic) approximation for n electronic states of asymmetrical (ABCD) tetraatomics, the electronic matrix elements (60) have the forms [18,152,153] ... 

INTERPRETATION OF THE OBJECTIVE FUNCTION IN TERMS OF ITS QUADRATIC APPROXIMATION... 

In doing the line search we can minimize a quadratic approximation in a given search direction. This means that to compute the value for a for the relation x ""1 = x + ask we must minimize... 

From one viewpoint the search direction of steepest descent can be interpreted as being orthogonal to a linear approximation (tangent to) of the objective function at point x examine Figure 6.9a. Now suppose we make a quadratic approximation of/(x) at x ... 

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. 

The minimum of the quadratic approximation of fix) in Equation (6.10) is obtained by differentiating (6.10) with respect to each of the components of Ax and equating the resulting expressions to zero to give... 

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... 

Representation of the trust region to select the step length. Solid lines are contours of fix). Dashed lines are contours of the convex quadratic approximation of fix) at x. The dotted circle is the trust region boundary in which 8 is the step length. x0 is the minimum of the quadratic model for which H(x) is positive-definite. 

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. 

The concept of dipole hardness permit to explore the relation between polarizability and reactivity from first principles. The physical idea is that an atom is more reactive if it is less stable relative to a perturbation (here the external electric field). The atomic stability is measured by the amount of energy we need to induce a dipole. For very small dipoles, this energy is quadratic (first term in Equation 24.19). There is no linear term in Equation 24.19 because the energy is minimum relative to the dipole in the ground state (variational principle). The curvature hi of E(p) is a first measure of the stability and is equal exactly to the inverse of the polarizability. Within the quadratic approximation of E(p), one deduces that a low polarizable atom is expected to be more stable or less reactive as it does in practice. But if the dipole is larger, it might be useful to consider the next perturbation order ... 

The Hessian matrix of the quadratic approximation (3.42) of tine objective... 

Their conclusion tended to the negative on this. They also concluded that quadratic approximations for energy variations were not applicable. 

Find the exact mean and variance of X. Now, suppose Y = MX. Find the exact mean and variance of Y. Find the mean and variance of the linear and quadratic approximations to Y=j(X). Are the mean and variance of the quadratic approximation closer to the true mean than those of the linear approximation ... 

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