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Show 62 each step we expect If' (u) I to be smeller than at the '_._. f(Lll+,) previous step. We are not, however, rally doing a Newton iteration at each step since we might be going uphill in a region of positive fIt. Thus a better iteration monitoring technique is to make sure that f(u) at each step is larger than it was at the previous step. Since we are guaranteed to always proceed in the "Jphill" direction by our choice of du, the only reason for f(u) to decrease is if we overshot the maximum and landed on the downhill slope of the function, below where we started. This is illustrated in figure 13. .....---;--, I Figure 13 - Overshooting Local Maximum Thus if f(ui+1)<f(ui) we can damp the iteration by halving the duo The algorithm now becomes while If' (u) I > E if f"(u)<O then du=-f'(u)/f"(u) else d u=f ' (u) while f(u+du)<f(u) du=du/2 if Idul<E return u = u + du Note the addition of the escape clause in the damping loop which exits if du gets exceedingly small. This has proven |
