The corrector

Since ym+1 is known approximately, the approximate slope at the point (tm+i, ym+1) can be calculated. This is shown as the line L2 in Fig. 1.14. The line L1 is the same as that in Fig. 1.13 and its slope is given by f[tm, ym]. Then the slopes of L1 and L2are averaged to give the line L. Finally, the line L3 is drawn parallel to L through the point (tm, ym). Its intersection with the ordinate erected at t tm+1 yields the new approximation to ym+1. This value is called the corrected value y^1. In algebraic terms it is given by y£+1 = ym + 2 {f[tm, ym] + f[tm+1, yL0+1 ]} . [1.61]

Another, and hopefully even better, estimate of f[tm+1, ym+1 ] is obtained by using ym+1 and recorrecting its value. Thus, in general, the i-th approximation to ym+1 is given by ym+1 = ym + 2 f[tm, ym] + f[tm+1, y£-+?]} . [1.62]

Finally, the iteration of this step is stopped when

where s is a value that we specify and which is called the tolerance.

It can be shown mathematically that if the partial derivative -p- is bounded and some number M, then if the step-size h is less than 2/M, the solution will converge to a finite value. But it is a curious result that this convergence limit is not necessarily, but almost always, the 'correct' value.

Figure 1.14. Graphical analysis of the corrector part of the predictor-corrector method of numerical integration.

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