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Show 55 Figure 11 - Iteration Increases Due to Overshooting Solution should have stopped. Thus, a solution to a diverging These two cases might be distinguished by the fact that the sign of f changes for the second and not the first. It is possible, however, to devise cases where this test will not work. In both cases, though, the increment in u was too large and brought the iteration past the point at which it iteration step is to shorten the amount by which u is incremented (e.g. halve it) and re-test the convergence criterion If(u+du) 1<lf(u) I. We can continue halving the increment until the criterion is met or until the increment becomes extremely small, indicating a local minimum or maximum. The algorithm is then while 1 f (u) 1 < £, d u = -f(u)/fl(u) while 1 f (u+du ) 1 > 1 f (u) 1 du = du/2 if l du l <E fail (local max/min) u = u + du succeed The addition of this halving step has been· called Damped Newton Iteration. It can also be invoked by other criteria which must be enforced to ensure not only convergence, but |
