Quadratic form, minimization

Equation 2 is not a poor representation of the energy function because it can be shown that a Morse potential in real distances assumes a simple quadratic form if one uses a dimensionless bond order coordinate.1263 In any event, minimization of this function leads to the conclusion that the minimum energy is attained when all bonds have the same length. Furthermore, a bond alternating distortion that lengthens and shortens a pair of adjacent bonds by Ar can be shown to raise the cr-energy as in eq 3a. 

When one refers to the CG method, one often means the linear conjugate gradient, that is, the implementation for the convex quadratic form. In this case, minimizing IxTAx + bTx is equivalent to solving the linear system Ax = -b. Consequently, the conjugate directions pfe, as well as the lengths kh, can be computed in closed form. 

In a linear system Ax = b where the matrix A is symmetric and positive definite, the solution is obtained by minimizing the quadratic form (12.331). This implies that the gradient, / (x) = Ax — b, is zero. In the iteration procedure an approximate solution, x +i, can be expressed as a linear combination of the previous solution and a search direction, p, which is scaled by a scaling factor am-... 

Just as the lowest energy W of the differential equation can be obtained by minimizing the energy integral E = fdT with respect to the function 0, keeping /0 dr = 1, so the lowest value of W giving a solution of Equations 27-40 may be obtained by minimizing the quadratic form... 

The values of minimized quadratic form of (a) [Eq. (10)] follows a distribution with m-n degrees of freedom, i.e. the mean minimum is [27,30] ... 

Unlike LSM, Eqs. (20-21) were derived without direct consideration of noise statistics. Nevertheless, Eqs. (20-21) are based on the minimization of quadratic norms of deviations (f(a)- f ) which is formally equivalent to assuming normal noise with unit covariance matrix (i.e. C = I). Thus, Eqs. (20-21) minimize the quadratic forms that have the additional term... 

Thus, Eq. (38b) minimizes the quadratic form [Eq. (30a)] with two terms ( =1,2), where the second term 2(3) represents a priori constraints on the with derivatives. The inclusion of 2( ) ill th minimization can be considered as applying limitations on the quadratic norm of m-th derivatives of y(x) that are commonly used as a measure of smoothness (e. g. see [32]). Indeed, if one assumes the diagonal covariance matrix Cg. with diagonal elements... 

This equation minimizes three quadratic forms simultaneously ... 

It should be noted that the multi-term estimates [Eqs. (32), (34), (47)] retain the optimality of LSM estimates, i.e. they have smallest errors as determined by the Cramer-Rao inequality, Eq. (16). However, the Cramer-Rao inequality is valid only if all assumptions about noise in both the measurements and the a priori terms are correct. Therefore, the validation of the assumptions is important, while problematic in reality. A useful consistency check can be performed using the achieved value of the minimized quadratic form (a) [Eq. (30)]. For example, in case of zero biases, the minimum value of the three-term (a) [Eq (48)] has a distribution with mean... 

In case of over determined non-linear (a) in Eq. (1), statistical optimization of a solution can be included in the iterations by following MML as described in Section 2. Namely, the solution of Eq. (1) should be performed as minimization of quadratic form (a) given by Eq. (10) and the resulting non-... 

This equation is known as the Gauss-Newton method [49]. For square matrices K, Eq. (62) can be reduced to Newton iterations using matrix identity (K C K) = K C(K ) This is why solving Eq. (2) with square K, as well as, Newton method also can be considered as a minimization of quadratic form [45]. 

It should be noted that quadratic form (a) [Eq. (11a)] can be minimized by iterations different from Eq. (62). Many such methods also utihze gradient... 

Our calculations have shown that the single-scattering albedo can be obtained from Eq. (36) even if the error amplitude is d 0.25—0.3. The results for A in Fig. (3) with /tq = 0.5 are very nearly the same as for /tq = 01 reported earlier [24]. After having found A from Eq. (36) we use it in Eq. (37) first to determine the number of possible values of the depolarization factor c. If there are two or three roots we have to use each c-value in our model to determine which minimizes the quadratic form... 

Another way to interpret the procedure described is that the curvature along an arbitrary direction, in the surface a = r -f- wr, is a quadratic function of the values of i and m. Diagonalizing this quadratic form, subject to the constraint that a is a unit vector, is mathematically equivalent to the minimization/maximization of k. Thus, the extremal curvatures, Ka and Kb are determined by the extremal values of I and w, which we denote as I and m. For directions on the surface close to these extremal values, the expansion of the curvature as a function of ( — ) and (m — m ) has no linear terms since... 

To estimate Apa, we choose some values of T, py and therefore some values of rio, Apy, fay are fixed. For simplicity, we assume that quadratic form (4.180) is positive definite with elements fay of symmetrized matrix and denote by t y the elements of its inversion. Taking first derivative of fli (4.169) (in arbitrary real trDy at chosen T, py) as zero we obtain the extremal values trDy (in fact in minimum because second derivatives of (4.169) form positive definite matrix of (4.180), cf. [134, Sect. 11.3-3]). Inserting this values into (4.169) (for which this inequality is valid too) we obtain the following minimal values of n i... 

NPPC [22] is a binary classifier and it classifies a pattern by the proximity of a test pattern to one of the two planes as shown in Fig. 5. The two planes are obtained by solving two nonlinear programming problems (NPP) with a quadratic form of loss function. Each plane is clustered around a particular class of data by minimizing sum squared distances of patterns from it and considering the patterns of the others class at a distance of 1 with soft errors. Thus, the objective of NPPC is to find two hyperplanes ... 

The coefficients a,(x) are obtained by performing a weighted least square fit for the local approximation m (x). This is achieved by minimizing the following quadratic form... 

This (minimal) mechanism works as follows the (active reduced) enzyme exists in two different conformational states symbolized by the circular and the quadratic form. The quadratic form binds first ATP and then protons from the outsite are bound (pKout ) ... 

A similar Taylor s series expansion may be appHed to Gibbs potential of the system G (here for the vicinity of the minimal equilibrium value Gmin) as a quadratic form of the deviations of the state parameters from equilibrium. 

It is well known that the valne of CAfor hydrophilie snifaces is less than 90°. Fabrication of these snifaces has attracted considerable interest for both fundamental researeh and practical studies. So, the goal of the present stucfy is to minimize the CA of elec-trospun nanofibers. The optimal conditions of the electrospinning parameters were established from the quadratic form of the RSM. Independent variables namely, solution concentration, applied voltage, spinning distance, and volume flow rate were set in range and dependent variable (CA) was fixed at minimum. The optimal conditions in the tested range for minimum CA of electrospun fiber mat are shown in Table 5. 

A typical feed-forward GMDH-type network is shown in Figure 6.2. The coefficients a. in equation (5) are calculated using regression techniques (Farlow, 1984 Ivakh-nenko, 1971) so that the difference between actual output, y, and the calculated one, y, for each pair of x., Xj as input variables is minimized. Indeed, it can be seen that a tree of polynomials is constructed using the quadratic form given in equation (5) whose coefficients are obtained in a least-squares sense. In this way, the coefficients of each quadratic function are obtained to optimally fit the output in the whole set of input-output data pair that is... 

In the above expression, only the last term depends on which is a quadratic form with a constant coefficient matrix Ap. Minimizing it with respect to and subjected to the norm constraint = 1 gives the most probable mode shape as the eigenvector of Aq with the largest eigenvalue. 

Summary. In this chapter, we are concerned with the problem of multivariate data interpolation. The main focus hes on the concept of minimizing a quadratic form which, in practice, emerges from a physical model, subject to the interpolation constraints. The approach is a natural extension of the one-dimensional polynomial spline interpolation. Besides giving a basic outline of the mathematical framework, we design a fast numerical scheme and analyze the performance quality. We finally show that optimal interpolation is closely related to standard hnear stochastic estimation methods. 

By minimizing either of these quadratic forms, we can obtain a set of coefficients that provide the optimum vertical resolution in the sense of the particular form chosen. However, in the presence of measurement noise, the resulting retrievals would be unsatisfactory because of error propagation, and we must take the effects of noise explicitly into account. 

The present perturbative beatment is carried out in the framework of the minimal model we defined above. All effects that do not cincially influence the vibronic and fine (spin-orbit) stracture of spectra are neglected. The kinetic energy operator for infinitesimal vibrations [Eq. (49)] is employed and the bending potential curves are represented by the lowest order (quadratic) polynomial expansions in the bending coordinates. The spin-orbit operator is taken in the phenomenological form [Eq. (16)]. We employ as basis functions... 

Using Eq. (4.1.13), the estimation problem is cast in discrete form as the determination of the coefficients c = [c1 . .., cns] that minimize the quadratic performance index ... 

Another method for solving nonlinear programming problems is based on quadratic programming (QP)1. Quadratic programming is an optimization procedure that minimizes a quadratic objective function subject to linear inequality or equality (or both types) of constraints. For example, a quadratic function of two variables x and X2 would be of the general form ... 

Another simple optimization technique is to select n fixed search directions (usually the coordinate axes) for an objective function of n variables. Then fix) is minimized in each search direction sequentially using a one-dimensional search. This method is effective for a quadratic function of the form... 

The minimization of the quadratic performance index in Equation (16.2), subject to the constraints in Equations (16.5-16.7) and the step response model such as Equation (16.1), forms a standard quadratic programming (QP) problem, described in Chapter 8. If the quadratic terms in Equation (16.2) are replaced by linear terms, a linear programming program (LP) problem results that can also be solved using standard methods. The MPC formulation for SISO control problems described earlier can easily be extended to MIMO problems and to other types of models and objective functions (Lee et al., 1994). Tuning the controller is carried out by adjusting the following parameters ... 

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