Nonlinearity parameter

The method used here is based on a general application of the maximum-likelihood principle. A rigorous discussion is given by Bard (1974) on nonlinear-parameter estimation based on the maximum-likelihood principle. The most important feature of this method is that it attempts properly to account for all measurement errors. A discussion of the background of this method and details of its implementation are given by Anderson et al. (1978). 

Bard, Y., Nonlinear Parameter Estimation, Academic Press, New York (1974). ... 

The representation of trial fiinctions as linear combinations of fixed basis fiinctions is perhaps the most connnon approach used in variational calculations optimization of the coefficients is often said to be an application of tire linear variational principle. Altliough some very accurate work on small atoms (notably helium and lithium) has been based on complicated trial functions with several nonlinear parameters, attempts to extend tliese calculations to larger atoms and molecules quickly runs into fonnidable difficulties (not the least of which is how to choose the fomi of the trial fiinction). Basis set expansions like that given by equation (A1.1.113) are much simpler to design, and the procedures required to obtain the coefficients that minimize are all easily carried out by computers. 

Mishuk et a/.675,676 have applied the modified amplitude demodulation method to electrochemically polished pc-Bi in aqueous NaF solution. The curves of the real component of the nonlinear impedance Z" as a function of the electrode potential, unlike pc-Cd and pc-Pb, intersect for various cNaF at E - -0.62 V (SCE),674 i.e., at Ea=0 for pc-Bi, as obtained by impedance.666-672 The different behavior of pc-Bi from pc-Cd and pc-Pb at a > 0 has been explained by the semimetallic nature of pc-Bi electrodes. A comparison of inner-layer nonlinear parameter values for Hg, Cd, and Bi electrodes at a < 0 shows that the electrical double-layer structure at negative charges is independent of the metal.675,676... 

Brubaker, T. A., and O Keefe, K. R., Nonlinear Parameter Estimation, Anal. Chem. 51, 1979, 1385A-1388A. 

Even though excellent algorithms are available for estimating nonlinear parameters, the values of exponents a, b, and e for the system under consideration rendered the estimation techniques to yield poor results. Accordingly, Equation (7) was linearized. 

Figure 8. The fragility parameter m is plotted as a function of the NMT nonlinearity parameter Xnmt- The curve is predicted by the RFOT theory when the temperature variation of >o is neglected. The data are taken from Ref. [49]. The disagreement may reflect a breakdown of phenomenology for the history dependence of sample preparation. The more fragile substances consistently lie above the prediction, which has no adjustable parameters. This discrepancy may be due to softening effects.

On the other hand, the nonlinear optical properties of nanometer-sized materials are also known to be different from the bulk, and such properties are strongly dependent on size and shape [11]. In 1992, Wang and Herron reported that the third-order nonlinear susceptibility, of silicon nanocrystals increased with decreasing size [12]. In contrast to silicon nanocrystals, of CdS nanocrystals decreased with decreasing size [ 13 ]. These results stimulated the investigation of the nonlinear optical properties of other semiconductor QDs. For the CdTe QDs that we are concentrating on, there have been few studies of nonresonant third-order nonlinear parameters. 

The Z-scan is a sensitive technique for the measurement of nonlinear parameters developed by Sheik-Bahae at the end of thel980s [14, 15]. Figure 9.1 illustrates a... 

It should be noted that the high repetition rate ( 80MHz) laser may give a large deviation of nonlinear parameters from the expected values since the thermal effect may be superimposed on the Z-scan signals [16]. 

Table 9.1 Concentration-independent nonlinear parameters of water-soluble CdTe QDs.

The structure of such models can be exploited in reducing the dimensionality of the nonlinear parameter estimation problem since, the conditionally linear parameters, kl5 can be obtained by linear least squares in one step and without the need for initial estimates. Further details are provided in Chapter 8 where we exploit the structure of the model either to reduce the dimensionality of the nonlinear regression problem or to arrive at consistent initial guesses for any iterative parameter search algorithm. 

These problems refer to models that have more than one (w>l) response variables, (mx ) independent variables and p (= +l) unknown parameters. These problems cannot be solved with the readily available software that was used in the previous three examples. These problems can be solved by using Equation 3.18. We often use our nonlinear parameter estimation computer program. Obviously, since it is a linear estimation problem, convergence occurs in one iteration. 

In engineering we often encounter conditionally linear systems. These were defined in Chapter 2 and it was indicated that special algorithms can be used which exploit their conditional linearity (see Bates and Watts, 1988). In general, we need to provide initial guesses only for the nonlinear parameters since the conditionally linear parameters can be obtained through linear least squares estimation. 

Bard, Y "Comparison of Gradient Methods for the Solution of Nonlinear Parameter Estimation Problems", SIAM J. Sumer. Anal., 7,157-186 (1970). 

Marquardt, D.W., "An Algorithm for Least-Squares Estimation of Nonlinear Parameters", J. Soc. Indust. Appl. Math., 11(2), 431-441 (1963). 

Values of the rate constants kx and k2 can be obtained from experimental measurements of cA and cB at various times in a BR. The most sophisticated procedure is to use either equations 5.5-2 and -3 or equations 3.4-10 and 5.5-6 together in a nonlinear parameter-estimation treatment (as provided by the E-Z Solve software see Figure 3.11). A simpler procedure is first to obtain kx from equation 3.4-10, and second to obtain 2 fr°m and either of the coordinates of the maximum value of cB (tmax or cB max). These coordinates can be related to kx and k2, as shown in the following example. 

The application of optimisation techniques for parameter estimation requires a useful statistical criterion (e.g., least-squares). A very important criterion in nonlinear parameter estimation is the likelihood or probability density function. This can be combined with an error model which allows the errors to be a function of the measured value. 

Nash, J.C. and Walker-Smith, M. (1987) Nonlinear Parameter Estimation An Integrated System in Basic, Marcel Dekker. 

Until now, we have dealt with kinetic models and rate constants as the nonlinear parameters to be fitted to spectrophotometric absorbance data. However, measurements can be of a different kind and particularly titrations (e.g. pH-titrations) are often used for quantitative chemical analyses. In such instances concentrations can also be parameters. In fact, any variable used to calculate the residuals is a potential parameter to be fitted. 

M2. Marquardt, D. L., Least Squares Estimation of Nonlinear Parameters, IBM SHARE Library Program No. 3094, Exhibit B. 

A set of nonlinear parameters Aj, in general case, is unique for each function To satisfy the requirement of square integrability of the wave function, each matrix must be positively defined. It imposes certain restrictions on the values that the elements of matrix A may take. To ensure the positive definiteness and to simplify some calclations, it is very convenient to represent matrix A in a Cholesky factored form. 

Here, L is a lower triangular matrix (not to be confused with L, the Cholesky factor of the matrix of nonlinear parameters A ), and D is a diagonal matrix. The scheme of the solution of the generalized symmetric eigenvalue problem above has proven to be very efficient and accurate in numerous calculations. But the main advantage of this scheme is revealed when one has to routinely solve the secular equation with only one row and one column of matrices H and S changed. In this case, the update of factorization (117) requires only oc arithmetic operations while, in general, the solution from scratch needs oc operations. 

It is well known that the convergence of variational expansions in terms of correlated Gaussians, both the simple ones and those with premultipliers, strongly depends on how one selects the nonlinear parameters in the Gaussian... 

The non-BO wave functions of different excited states have to differ from each other by the number of nodes along the internuclear distance, which in the case of basis (49) is r. To accurately describe the nodal structure in aU 15 states considered in our calculations, a wide range of powers, m, had to be used. While in the calculations of the H2 ground state [119], the power range was 0 0, in the present calculations it was extended to 0-250 in order to allow pseudoparticle 1 density (i.e., nuclear density) peaks to be more localized and sharp if needed. We should notice that if one aims for highly accurate results for the energy, then the wave function of each of the excited states must be obtained in a separate calculation. Thus, the optimization of nonlinear parameters is done independently for each state considered. 

We tested a 76-term wave function for the system H3, including permutational and point group symmetry. The initial guess for the nonlinear parameters in the ECGs were generated randomly using Matlab. The Young... 

The inversion operator i acts on the electronic coordinates (fr = —r). It is employed to generate gerade and ungerade states. The pre-exponential factor, y is the Cartesian component of the i-th electron position vector (mf. — 1 or 2). Its presence enables obtaining U symmetry of the wave function. The nonlinear parameters, collected in positive definite symmetric 2X2 matrices and 2-element vectors s, were determined variationally. The unperturbed wave function was optimized with respect to the second eigenvalue of the Hamiltonian using Powell s conjugate directions method [26]. The parameters of were... 

Table 3. Energy (E) and other properties of He, computed from optimized one, two, and three-configuration wavefunction (nonlinear parameters given in Table 4)...

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