Damped iteration

The scalar t is in the range 0 to 1 and provides step-limiting, or damping, when required to obtain a convergent iteration. 

The RWP method also has features in common with several other accurate, iterative approaches to quantum dynamics, most notably Mandelshtam and Taylor s damped Chebyshev expansion of the time-independent Green s operator [4], Kouri and co-workers time-independent wave packet method [5], and Chen and Guo s Chebyshev propagator [6]. Kroes and Neuhauser also implemented damped Chebyshev iterations in the time-independent wave packet context for a challenging surface scattering calculation [7]. The main strength of the RWP method is that it is derived explicitly within the framework of time-dependent quantum mechanics and allows one to make connections or interpretations that might not be as evident with the other approaches. For example, as will be shown in Section IIB, it is possible to relate the basic iteration step to an actual physical time step. 

Equation (11) represents the first iteration of the RWP idea, but it is not the most efficient. It also represents the first appearance of the damping procedure as used in Mandelshtam and Taylor s Chebyshev iteration [4]. 

The magnitude of the discrete time step, 8, does not enter any final expressions for physical variables. Eor example, the physical time t in Eq. (27) involves k, the number of Chebyshev iterations taken, and not 8. Thus the core damped Chebyshev iteration, Eq. (18), may be taken to be... 

An iterative solution method for linear algebraic systems which damps the shortwave components of the iteration error very fast and, after a few iterations, leaves predominantly long-wave components. The Gauss-Seidel method [85] could be chosen as a suitable solver in this context. 

In one cycle of the multigrid method, first a few iterations are performed on the fine grid in order to obtain a comparatively smooth iteration error. After that the obtained residual is restricted to the coarse grid, where further iterations are performed in order to damp out the long-wave components of the solution error. Subsequently the coarse-grid solution is interpolated to the fine grid and the solution on the fine grid is updated. 

In order to regulate the convergence of the iterative process by damping or accelerating its rate according to convenience, the basic procedure has been further refined [88]. This is achieved by substituting the numerical reduced Hamiltonian matrix by a new one... 

For a/E <0 the convergence of the process is damped. This damping is necessary in the cases where large oscillations appear at the beginning of the iterative process. By introducing the damping, the modifications in the RDMs, from one iteration to the next one, are sufficiently small to allow the easy correction of the deviations of the N- and 5-representability, thus avoiding the appearance of oscillations. 

We first observe that the damping factor Q normally has a small value and after some iterations, two consecutive cp(i) will not differ too much. Then, one has approximately... 

Therefore the above expression provides an automatic selection of the damping or accelerating factor c in each iteration. The idea is easy to extend to a system of equations, if each element x of the vector x is regarded to be independent of the others when using the expressions (2.23) and (2.24). Thus the Wegstein method uses separate factors c for each variable ... 

The modified Newton iteration, and the reason that damping is effective, can be explained in physical terms. Chemical-kinetics problems often have an enormous range of characteristic scales—this is the source of stiffness, as discussed earlier. These problems are also highly nonlinear. 

Implement a damping strategy to maintain the solution within a feasible region 0 < x < 2. Test this algorithm beginning with an initial iterate of xo = 1.9. 

These methods for controlling convergence of an iterative solution to a complicated set of equations have wide applicability. The extrapolation and damping methods are based on well-known ideas for single variables while root-shifting may be a novel development by quantum chemists. 

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