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0 and n 1 now zn de ned by 345 is identical to yn so

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Unformatted text preview: r and Step Size Control We would like to design software that automatically adjusts the step size so that some measure of the error, ideally the global error, is less than a prescribed tolerance. While automatic variation of the step size is easy with one-step methods, it is very di cult to compute global error measures. A priori bounds, such as (3.4.11), tend to be too conservative and, hence, use very small step sizes (cf. 16], Section II.3). Other more accurate procedures (cf. 15], pp. 13-14) tend to be computationally expensive. Controlling a 42 measure of the local (or local discretization) error, on the other hand, is fairly straight forward and this is the approach that we shall study in this section. A pseudo-code segment illustrating the structure of a one-step method yn = yn 1 + h (tn 1 yn 1 h) ; ; (3.5.1a) ; that performs a single integration step of the vector IVP y = f (t y) 0 y(0) = y0 (3.5.1b) is shown in Figure 3.5.1. On input, y contains an approximation of the solution at time t. On output, t is replaced by t + h and y contains the computed approximate solution at t + h. The step size must be de ned on input, but may be mod...
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