First-order predictor algorithm

Predictive methods that calculate u for the next time step of a MD simulation based on information from previous timesteps have been developed to minimize the computational cost. Ahlstrom et al. [13] used a first-order predictor algorithm, in which values of u from the two previous times steps are used to determine u at the next time step. A very serious drawback of this method is that it is not stable for long simulation times. However, it has been combined with iterative solutions, either by providing the initial iteration of the electric field values [163, 164], or by performing an iterative SCF step less frequently than every step [13,165], Higher-order predictor algorithms have also been described in the literature [13,163, 166],... 

The predictive methods determine p for the next time step based on information from previous time steps. Ahlstrom et al. used a first-order predictor algorithm, which uses the p values from the two previous times steps to predict p at the next time step. 

SERIES FIRST ORDER REACTION PULSE CHASE EXPERIMENTS Predictor-corrector algorithm,... 

A related approach which has been used successfully in industrial applications occurs in discrete-time control. Both Dahlin (43) and Higham (44) have developed a digital control algorithm which in essence specifies the closed loop response to be first order plus dead time. The effective time constant of the closed loop response is a tuning parameter. If z-transforms are used in place of s-transforms in equation (11), we arrive at a digital feedback controller which includes dead time compensation. This dead time predictor, however, is sensitive to errors in the assumed dead time. Note that in the digital approach the closed loop response is explicitly specified, which removes some of the uncertainties occurring in the traditional root locus technique. 

Schrodinger s equation leads then to Nx + second-order coupled differential equations with first derivatives. These are solved transforming the set into 2(N + Ne) first-order coupled equations, and generating N2 + N linearly independent solutions by choosing suitable boundary conditions at small rAC. A Bashforth-Moulton fourth-order predictor-corrector algorithm was used in the integration. 

If j30 = 0, the method is explicit and the computation of is straightforward. If 30 + 0, the method is implicit because an implicit algebraic equation is to be solved. Usually, two algorithms, a first one explicit and called the predictor, and a second one implicit and called the corrector, are used simultaneously. The global method is called a predictor-corrector method as, for example, the classical fourth-order Adams method, viz. 

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