Iterative technique

N.P. Zhuk and D.O. Batrakov, Inverse scattering problem in the polarization parameters domain for isotropic layered media solution via Newfon-Kantorovich iterative technique, 1994, J. Electromagn. Waves AppL, vol. 8, No. 6, pp. 759-779. 

The question remains how to evaluate exp(—iTH(qo)/(2 )) i/ i while retaining the symmetric structure. In Sec. 4.2 we will introduce some iterative techniques for evaluating the matrix exponential but the approximative character of these techniques will in principle destroy the symmetry. 

I hese equations cannot be used directly, and numerical methods are needed to compute the velocity components. The velocity components can be found by implicit differentiation and using an iterative technique.-" ... 

The only generally applicable methods are CISD, MP2, MP3, MP4, CCSD and CCSD(T). CISD is variational, but not size extensive, while MP and CC methods are non-variational but size extensive. CISD and MP are in principle non-iterative methods, although the matrix diagonalization involved in CISD usually is so large that it has to be done iteratively. Solution of the coupled cluster equations must be done by an iterative technique since the parameters enter in a non-linear fashion. In terms of the most expensive step in each of the methods they may be classified according to how they formally scale in the large system limit, as shown in Table 4.5. 

Gauss-Siedel method is an iterative technique for the solution of sets of equations. Given, for example, a set of three linear equations... 

When the three coefficients oc12, oc13, and a23 are known, the coexistence curve can be found by simultaneous solution of Eqs. (119) and (120). A numerical iterative technique given by Hennico and Vermeulen (HI) was used by Balder for performing these calculations with a digital electronic computer. 

Lanteri, H., Roche, M., Cuevas, O., Aime, C., 2001, A general method to devise maximum-likelihood signal restoration multiplicative algorithms with nonnegativity constraints. Signal Processing 81, 945 Lucy, L.B., 1974, An iterative technique for the rectification of observed distributions, ApJ 79, 745... 

If the graph y vs. x suggests a certain functional relation, there are often several alternative mathematical formulations that might apply, e.g., y - /x, y = a - - exp(b (x + c))), and y = a-(l- l/(x + b)) choosing one over the others on sparse data may mean faulty interpretation of results later on. An interesting example is presented in Ref. 115 (cf. Section 2,3.1). An important aspect is whether a function lends itself to linearization (see Section 2.3.1), to direct least-squares estimation of the coefficients, or whether iterative techniques need to be used. 

S. HeikkilS and V. Lakshmikantham, Monotone iterative Techniques for Discontinuous Nonlinear Differential Equations (1994)... 

Monte Carlo simulation, an iterative technique which derives a range of risk estimates, was incorporated into a trichloroethylene risk assessment using the PBPK model developed by Fisher and Allen (1993). The results of this study (Cronin et al. 1995), which used the kinetics of TCA production and trichloroethylene elimination as the dose metrics relevant to carcinogenic risk, indicated that concentrations of 0.09-1.0 pg/L (men) and 0.29-5.3 pg/L (women) in drinking water correspond to a cancer risk in humans of 1 in 1 million. For inhalation exposure, a similar risk was obtained from intermittent exposure to 0.07-13.3 ppb (men) and 0.16-6.3 ppb (women), or continuous exposure to 0.01-2.6 ppb (men) and 0.03-6.3 ppb (women) (Cronin et al. 1995). 

Combining equations (7.4-24)-(7.4-26) gives a system of non-linear equations that can be solved using iterative techniques. Savings in equipment costs as compared to initial guesses are approximately 30 %. The real savings will be lower because the optimal choices for equipment units are usually not available on the chemical equipment market. The standard sizes greater but nearest to the optimal sizes will be selected. The total cost for the standard equipment is very close to the minimum found. Robinson and Loonkar (1972) extended their procedure for multiproduct batch plants. 

For the solution of Equation 10.25 the inverse of matrix A is computed by iterative techniques as opposed to direct methods often employed for matrices of low order. Since matrix A is normally very large, its inverse is more economically found by an iterative method. Many iterative methods have been published such as successive over-relaxation (SOR) and its variants, the strongly implicit procedure (SIP) and its variants, Orthomin and its variants (Stone, 1968), nested factorization (Appleyard and Chesire, 1983) and iterative D4 with minimization (Tan and Let-keman. 1982) to name a few. 

Sequential-modular programs in which the equations describing each process unit (module) are solved module-by-module in a stepwise manner and iterative techniques used to solve the problems arising from the recycle of information. 

The trial wave functions of a Schrodinger equation are expressed as determinant of the HF orbitals. This will give coupled nonlinear equations. The amplitudes were solved usually by some iteration techniques so the cc energy is computed as... 

Iterative Techniques. In estimating parameters in a model that is nonlinear in the parameters... 

NEWTON-RAPHSON ITERATIVE TECHNIQUE, THE FINAL ITERATIONS ON EACH ROOT ARE PER,FORMED USING THE ORIGINAL FOLYNOMIAL RATHER THAN THE RF-DUCEU POLYNOMIAL TO AVOID ACCUMULATED ERRORS IN THE REDUCED POLYNOMIAL. 

Accdg to Wilkins (Ref 20), it has been shown that hydrodynamic codes are capable of accurately solving the equations in hydrodynamics. They, therefore, offer the means to perform calculations in conjunction with experiments and use an iterative technique to obtain otherwise inaccessible data. In addition, they serve an important role in setting up and interpreting experiments in high explosive research... 

The unknowns in Eq. (38) are p and 2n. These are found by demanding hm to obey the image constraint equations (18) and normalization (3). Because the unknowns enter in a nonlinear way, the resulting M + 1 equations were solved by an iterative technique—Newton-Raphson relaxation (see, e.g., Hildebrand, 1956). Empirical cases are studied in Section XI. 

A program to solve these equations has been described by Orlov et al. (1998). O Keeffe (1989) has described an alternative method that is suitable for performing the calculation by hand. Rutherford (1990) has presented a way of inverting the matrix that retains the symmetry of the equations by including all Aa of the equations of type 3.3. Brown (1977) has described a robust iterative technique for solving the equations based on recognizing that eqn (3.4) is an expression of the principle of maximum symmetry (Rule 3.1). In this procedure... 

Our solution to this synthetic problem was the development of an iterative technique for preparing hydroxypropyl ethers from allyl ethers via oxymercuration-reduction. Figure 3 illustrates the process for the preparation of a series of three chain-extended hydroxypropyl derivatives of 2,6-dimethoxyphenol. Conversion of phenol 1 to the allyl ether 2 under phase-transfer conditions (6) was followed by oxymercuration (7) to give the intermediate organomercurial 3, which was reduced without isolation to give hydroxypropyl ether 4 in 64% overall yield. Ether 4. was then allylated to provide 5, which upon oxymercuration-reduction afforded hydroxypropyl derivative 6. One further iteration of the allylation-oxymercuration-reduction sequence yielded the hydroxypropyl compound 7. 

The computational problem, then, is determination of the cluster amplitudes t for aU of the operators included in tlie particular approximation. In the standard implementation, this task follows the usual procedure of left-multiplying the Schrodinger equation by trial wave functions expressed as dctcnninants of the HF orbitals. This generates a set of coupled, nonlinear equations in the amplitudes which must be solved, usually by some iterative technique. With the amplitudes in hand, the coupled-cluster energy is computed as... 

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