Quadratic advantage

If the state and control variables in equations (9.4) and (9.5) are squared, then the performance index become quadratic. The advantage of a quadratic performance index is that for a linear system it has a mathematical solution that yields a linear control law of the form... 

The advantage of the NR method is that the convergence is second-order near a stationary point. If the function only contains tenns up to second-order, the NR step will go to the stationary point in only one iteration. In general the function contains higher-order terms, but the second-order approximation becomes better and better as the stationary point is approached. Sufficiently close to tire stationary point, the gradient is reduced quadratically. This means tlrat if the gradient norm is reduced by a factor of 10 between two iterations, it will go down by a factor of 100 in the next iteration, and a factor of 10 000 in the next ... 

Some experimenting nught be necessary if it turns out that quadratic terms such as improve the fit between the model and the data. However, the relevant point is that such a model is only a means to refine and speed up the process of finding optimal conditions. For this purpose it is counter-productive to try for a perfect fit, it might even be advantageous to keep the model simple and throw out all but the best five to 10 experiments, choose new conditions, and then return to the work bench. 

The Gauss-Newton method is directly related to Newton s method. The main difference between the two is that Newton s method requires the computation of second order derivatives as they arise from the direct differentiation of the objective function with respect to k. These second order terms are avoided when the Gauss-Newton method is used since the model equations are first linearized and then substituted into the objective function. The latter constitutes a key advantage of the Gauss-Newton method compared to Newton s method, which also exhibits quadratic convergence. 

Note that the eigenvalue pfx) does not depend on time, which is a consequence of Eq. (17). In particular, for a quadratic Hamiltonian, the operator satisfying Eq. (17) can be obtained explicitly. This canonical method has an advantage that quantum statistical information can... 

We applied the Liouville-von Neumann (LvN) method, a canonical method, to nonequilibrium quantum phase transitions. The essential idea of the LvN method is first to solve the LvN equation and then to find exact wave functionals of time-dependent quantum systems. The LvN method has several advantages that it can easily incorporate thermal theory in terms of density operators and that it can also be extended to thermofield dynamics (TFD) by using the time-dependent creation and annihilation operators, invariant operators. Combined with the oscillator representation, the LvN method provides the Fock space of a Hartree-Fock type quadratic part of the Hamiltonian, and further allows to improve wave functionals systematically either by the Green function or perturbation technique. In this sense the LvN method goes beyond the Hartree-Fock approximation. 

It is a quadratic form in the operators a, af and may therefore be diagonalized by means of a linear transformation among them. The special form of the interaction has the advantage that the transformation does not mix the a with the a1. It will later appear that this special form is less contrived than it looks. ... 

The zero product rule is useful when solving quadratic equations because you can rewrite a quadratic equation as equal to zero and take advantage of the fact that one of the factors of the quadratic equation is thus equal to 0. 

Finally, Equation 1 exhibits another important advantage of high fields in FTMS. Both the maximum trapping time and the maximum number of collisions (and gas-phase reactions) increase quadratically with B. Consequently increased magnetic field strength offers experimental access to larger ion clusters (Figure 2). 

Very recently, the study of linear birefringences has been extended to BE [32], where, as for ref. [31], furan and its homologues were investigated, in this case in solutions of cyclohexane. The latter were the subject of an experimental analysis by Dennis et al. [33], In ref. [32] advantage is taken of the recent development of frequency-dependent quadratic response in the nonequilibrium PCM solvation regime [34],... 

The coordinate map given by the variables (cD+, cD ) is a significant improvement as compared to eq. (3.25). Nevertheless, an explicit expression for an h matrix in its terms is still a clumsy combination of the trigonometric functions of two triples of reparametrizing angles w . It is known however that in the case of the SO(3) group [8] its quaternion [27] parameterization has the advantage that the matrix elements of SO(3) rotation matrices, when expressed in terms of the components of the normalized quaternion, are quadratic functions of these components. 

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