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73b 0 newton iteration converges when the initial

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Unformatted text preview: evaluations and, thus, important to select an accurate initial guess yn . Initial guess are usually chosen by a separate explicit method called a predictor. The implicit method is then called a corrector. For example, the Adams-Bashforth methods furnish predictors for use with Adams-Moulton correctors. Two possible termination criteria for the corrector iteration are: 46 1. Iterate the corrector to convergence, e.g., terminate when ( ( jyn ) ; yn ;1) j where is on the order of the unit round o error of the computer. The local discretization and stability of this predictor-corrector combination are determined by the properties of the corrector alone however, more function evaluations than necessary may be required. 2. Iterate the corrector a xed number of times. This procedure reduces the number of function evaluations, but now the discretization error and stability characteristics of the result contain a mixture of the properties of both the predictor and corrector. A compromise between these two strategies is to iterate the corrector a xed number of times and to repeat the step with a smaller step size if adequate convergence was not obtained. In describing predictor-corrector methods, we will use the notation P to denote an application of the predictor step, C to denote an application of the corrector step, and E to denote a function evaluation, i.e., an evaluation of f . (0) Example 5.7.1. A method that uses a predictor to evaluate yn , does a function (0) (0) (1) evaluation fn = f (tn yn ), and one corrector iteration to obtain yn = yn is a PEC method. If corrector iterations are performed and each iteration requires one evaluation of f , then the method is a P (EC ) method. In this case, the last function evaluation ( ( f (tn yn ;1)) is saved as fn for the next time step. Since we save yn = yn ), it may be ( preferable to make an additional function evaluation and save fn = f (tn yn )). This method then becomes a P (EC ) E method. The orders of the predictor and corrector formulas need not be the same. Assume that the predictor has an order p and the corrector has an order...
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