Convergency problems

However, each of these forms possesses a spurious root and has other characteristics (maxima or minima) that often give rise to convergence problems with common iterative-solution techniques. 

Second card FORMAT(8F10.2), control variables for the regression. This program uses a Newton-Raphson type iteration which is susceptible to convergence problems with poor initial parameter estimates. Therefore, several features are implemented which help control oscillations, prevent divergence, and determine when convergence has been achieved. These features are controlled by the parameters on this card. The default values are the result of considerable experience and are adequate for the majority of situations. However, convergence may be enhanced in some cases with user supplied values. 

Z matrix generated using Cartesian —> Z-matrix conversion program. Severe converge problems with energy... 

The perfomiance of the penalty fiinction algoritlun is heavily influenced by the value chosen for a.. The larger the value of o. the better the constraints are satisfied but the slower the rate of convergence. Optimizations with very high values of a, encounter severe convergence problems. However, the method is very general and... 

The way in which the calculation is performed is also important. Unrestricted calculations will allow the system to shift from one spin state to another. It is also often necessary to run the calculation without using wave function symmetry. The calculation of geometries far from equilibrium tends to result in more SCF convergence problems, which are discussed in Chapter 22. 

Fixing Self-Consistent Field A A Convergence Problems... 

FIXING SELF-CONSISTENT FIELD CONVERGENCE PROBLEMS... 

Changing the constants in the SCF equations can be done by using a dilferent basis set. Since a particular basis set is often chosen for a desired accuracy and speed, this is not generally the most practical solution to a convergence problem. Plots of results vs. constant values are the bifurcation diagrams that are found in many explanations of chaos theory. 

Some convergence problems are due to numerical accuracy problems. Many programs use reduced accuracy integrals at the beginning of the calculation to save CPU time. However, this can cause some convergence problems for difficult systems. A course DFT integration grid can also lead to accuracy problems, as can an incremental Fock matrix formation procedure. 

Convergence problems are very common due to the number of orbitals available and low-energy excited states. The most difficult calculations are generally those with open-shell systems and an unfllled coordination sphere. All the techniques listed in Chapter 22 may be necessary to get such calculations to converge. 

The DIIS convergence accelerator is available for all the SCF semiempirical methods. This accelerator may be helpful in curing convergence problems. It often reduces the number of iteration cycles required to reach convergence. However, it may be slower because it requires time to form a linear combination of the Fock matrices during the SCF calculation. The performance of the DIIS accelerator depends, in part, on the power of your computer. 

H. B. Schlegel and J. J. W. McDouall, Do You Have SCF Stability and Convergence Problems in C. Ogretir and I. G. Csizmadia, eds.. Computational Advances in Organic Chemistry (Kluwer Academic Pubs., NATO-ASI Series C 330, The Netherlands, 1991), 167-85. 

Lowdin, P.-O., Phys. Rev. 97, 1474, 1490, 1509, Quantum theory of many-particle systems. I. Physical interpretations by means of density matrices, natural spin-orbitals and convergence problems in the method of configuration interaction. II. Study of the ordinary Hartree-Fock approximation. III. Extension of the Har-tree-Fock scheme to include degenerate systems and correlation effects. ... 

A common feature in the models reviewed above was to calculate pressure and temperature distributions in a sequential procedure so that the interactions between temperature and other variables were ignored. It is therefore desirable to develop a numerical model that couples the solutions of pressure and temperature. The absence of such a model is mainly due to the excessive work required by the coupling computations and the difficulties in handling the numerical convergence problem. Wang et al. [27] combined the isothermal model proposed by Hu and Zhu [16,17] with the method proposed by Lai et al. for thermal analysis and presented a transient thermal mixed lubrication model. Pressure and temperature distributions are solved iteratively in a iterative loop so that the interactions between pressure and temperature can be examined. 

Very often empirical equations can be developed from plant data using multiple regression techniques. The main advantage of this approach is that the correlations are often linear, can be easily coupled to optimization algorithms, do not cause convergence problems and are easily transferred from one computer to another. However, there are disadvantages, namely,... 

The above form of the Arrhenius equation takes into account the high degree of correlation that exists between the kinetic parameters. This pivoting method solves a convergence problem that can occur during parameter fitting if all six parameters (Fm, Em, Fdl, Edl, Fd2, and Ed2) are allowed to vary. 

Although the minimization of the objective function might run to convergence problems for different NN structures (such as backpropagation for multilayer perceptrons), here we will assume that step 3 of the NN algorithm unambiguously produces the best, unique model, g(x). The question we would like to address is what properties this model inherits from the NN algorithm and the specific choices that are forced. 

The aspect ratio of hexahedral cells should be not too high, typically below 20-100. If high-aspect ratio cells are used, the accuracy and possible convergence problems depend greatly on the flow direction. 

The only way to avoid this convergence problem is to terminate the infinite series (equation (G.49)) after a finite number of terms. If we let 2 take on the successive values / + 1, / + 2,..., then we obtain a series of acceptable solutions of the differential equation (G.43)... 

The calculation of y and P in Equation 14.16a is achieved by bubble point pressure-type calculations whereas that of x and y in Equation 14.16b is by isothermal-isobaric //cm-/(-type calculations. These calculations have to be performed during each iteration of the minimization procedure using the current estimates of the parameters. Given that both the bubble point and the flash calculations are iterative in nature the overall computational requirements are significant. Furthermore, convergence problems in the thermodynamic calculations could also be encountered when the parameter values are away from their optimal values. 

Because the current estimates of the interaction parameters are used when solving the above equations, convergence problems are often encountered when these estimates are far from their optimal values. It is therefore desirable to have, especially for multi-parameter equations of state, an efficient and robust estimation procedure. Such a procedure is presented next. 

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