Convergence requirements

To accommodate the step-by-step, recycling and checking for convergences requires input of vapor pressure relationships (such as Wilson s, Renon s, etc.) through the previously determined constants, latent heat of vaporization data (equations) for each component (or enthalpy of liquid and vapor), specific heat data per component, and possibly special solubility or Henry s Law deviations when the system indicates. 

Regarding accuracy, the finite difference approximations for the radial derivatives converge O(Ar ). The approximation for the axial derivative converges 0(Az), but the stability criterion forces Az to decrease at least as fast as Ar. Thus, the entire computation should converge O(Ar ). The proof of convergence requires that the computations be repeated for a series of successively smaller grid sizes. 

Record keeping is a fundamental requirement for any serious research activity, and of course drug discovery and development is no different. In the case of drug discovery, there are a number of convergent requirements for keeping good scientific records ... 

Later, Kuppermann and Belford (1962a, b) initiated computer-based numerical solution of (7.1), giving the space-time variation of the species concentrations from these, the survival probability at a given time may be obtained by numerical integration over space. Since then, this method has been vigorously followed by others. John (1952) has discussed the convergence requirement for the discretized form of (7.1), which must be used in computers this turns out to be AT/(Ap)2normalized forms of r and t. Often, Ar/(Ap)2 = 1/6 is used to ensure better convergence. Of course, any procedure requires a reaction scheme, values of diffusion and rate coefficients, and a statement about initial number of species and their distribution in space (vide infra). 

For low density (large Vm), the series (2.30) is expected to achieve useful accurary with only a few terms. Higher densities within the domain of convergence require additional terms to achieve a desired accuracy. For some densities, the virial series may not converge at all. 

Can display local quadratic convergence Requires construction and factorization of preconditioner Performance may be slow for highly nonlinear functions when directions of negative curvature are detected repeatedly... 

Here the gauge function for the first term must be independent of s because C = 1 at the boundary 7=0. Note again that asymptotic convergence requires that... 

In simple Cl, the expansion coefficients Dk are optimized in such a way that Tq has the minimum energy. The expansion in (22) can be slowly converging requiring millions of terms. The number of needed terms can be reduced in the multiconfiguration SCF procedure (MCSCF), where both Dk and the orbital expansion... 

Second-order convergence requires that the error in the (n + l)th iteration is the square of the error in the nth iteration. In the first iteration above the error is 0.001274 thus in the next iteration the error should be (0.001274) = 0,0000016 if we used a second-order procedure. Since the second iteration s error is 0.000204, the convergency of the above SCF procedure is linear rather than quadratic. 

The MDIIS or DR methods of convergence require definition of residuals. Their construction is illustrated below for the 3D-RISM/DRISM integral equations for aqueous solution discussed in Section 6. The residual of the Kohn-Sham DPT equations combined with the 3D-RISM approach in the SCF loop considered in Section 8 is considered as weU. 

In each iteration of the Newton-Raphson method, when the guesses are close to the true values, the length of the error vector, y, is the square of its length after the previous iteration that is, when the length of the initial error vector is 0.1, the subsequent error vectors are reduced to 0.01, 10 , 10", . However, this rapid rate of convergence requires that n" partial derivatives be evaluated at x. Since most recycle loops involve many process units, each involving many equations, the chain rule for partial differentiation cannot be implemented easily. Consequently, the partial derivatives are evaluated by numerical perturbation that is, each guess, Xj, i = 1,..., , is perturbed, one at a time. For each... 

Convergence requires checking the assumed and calculated values of the fractional recovery of the HNK in the bottoms. 

For the systems discussed earlier, the calculated specific optical properties were already converged to the limit values for a relatively small number of atoms (from several dozen up to a few hundreds). For nano-sized systems, the convergence requires from several hundred up to several thousand atoms. To demonstrate the ability of our method to describe nano-systems, we carried out calculations of a model macromolecular system with the structure presented in Fig. 3.20. The system represents a looped nanotube (nanotore). The looping of the nanotube allows for avoiding the end effects to appear in the calculations. The size of the nanotore is defined by the number of unit cells in the system. It can be expected that the C-C bonds... 

Binocular disparity and convergence require the involvement of both eyes, while motion parallax and accommodation could be observed eveu with a single eye. For natural 3-D objects, all the four major depth cues meutioned above should be present at the same lime. Various 3-D display technologies employ at least one of the four major depth cues to geuerate 3-D depth sensation. The more consistent the depth cues are, the more realistic and natural a 3-D image appears. 

Initial 3D simulations with patched boundary conditions prove to be extremely slow to converge, requiring very long simulation times, further refmement of methodologies and the solver are likely required to proceed. 

The convergence requirements restrict the applicability to times or reciprocal frequencies greater than 5ryv s OAa q kT. However, the values ofr fortunately depend only on fo and a, not on any properties of the artificial submole-culc. 

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