Accelerated convergence method

The Accelerated Convergence Method and the Electrostatic Potential in a Point-Charge Crystal... 

With K in the convergence function of Eq. (9.18) equal to 0.4 A-1, the accelerated convergence method reaches an accuracy of nine significant figures with inclusion of only 1000 unit cells. By comparison, about 50,000 unit cells would be required for a summation performed exclusively in direct space. 

Roussel, M. R. Fraser, S. J. Geometry of the steady-state approximation. Perturbation and Accelerated Convergence Methods 1990,93 1072-1081. 

M.R. Roussel and S.J. Fraser, Geometry of the Steady-State Approximation - Perturbation and Accelerated Convergence Methods, J. Chem. Phys. 93 (1990) 1072-1081. 

Thus, HyperChem occasionally uses a three-point interpolation of the density matrix to accelerate the convergence of quantum mechanics calculations when the number of iterations is exactly divisible by three and certain criteria are met by the density matrices. The interpolated density matrix is then used to form the Fock matrix used by the next iteration. This method usually accelerates convergent calculations. However, interpolation with the MINDO/3, MNDO, AMI, and PM3 methods can fail on systems that have a significant charge buildup. 

Method ofWegstein This is a variant of the method of successive substitutions which forces and/or accelerates convergence. The iterative procedure Xi + = F xC) is revised by setting x + i = F xi) and then taking Xi + = qxi -i- (1 — q)xi + i, where is a suitably chosen number which may be taken as constant throughout or may be adjusted at each step. Wegstein found that suitable q s are ... 

Enhanced sampling in conformational space is not only relevant to sampling classical degrees of freedom. An additional reason to illustrate this particular method is that the delocalization feature of the underlying distribution in Tsallis statistics is useful to accelerate convergence of calculations in quantum thermodynamics [34], We focus on a related method that enhances sampling for quantum free energies in Sect. 8.4.2. 

Performance of the CG method is generally very sensitive to roundoff in the computations that may destroy the mutual conjugacy property. The method was actually neglected for many years until it was realized that a preconditioning technique can be used to accelerate convergence significantly.14 80 81... 

Quasi-Newton methods can be viewed as extensions of nonlinear CG methods, in which additional curvature information is used to accelerate convergence. Thus, the required analytic Hessian information, memory, and computational requirements are kept as low as possible, and the main strength of Newton methods—employing curvature information to detect and move away from saddle points efficiently—is retained. 

This chapter listed many of the possible units in the Model Library of Aspen Plus. The ammonia process illustrated the procedures (and computer windows) you used to set the process conditions and examine the results. The thermodynamics choices can be verified by comparison with data reported in the literature. Sometimes the calculations do not converge, and then the Wegstein method, or Broyden s method, are useful for accelerating convergence. 

Wegstein s method can be employed to accelerate convergence when the method of successive substitutions requires a large number of iterations. As shown in Figure 4.11b, the previous two iterates of/ x and x are extrapolated linearly to obtain the next value of x as... 

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