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2.4.2 Steepest Descent     continued...

We show progress by superimposing the resulting iterates from each minimization application on the contour plot. Recall that the minimum point for Rosenbrock's function lies at , where . We clearly see in Figure 9 the characteristic behavior of the SD method: relatively good progress at first, but very slow convergence in the vicinity of the solution. The method was stopped after 1200 iterations, where a gradient norm of only was obtained. For the remaining methods, gradient norms of -- were realized.

Note from the contour plots of Beale's function, Figure 13, that the function has a narrow curving valley in the vicinity of the minimum, which occurs at . The function value at the minimum is zero and increases sharply, particularly as increases.

From Figure 13, we clearly note how the SD search vectors (the negative gradient) are perpendicular to the contour lines; progress is initially rapid but then becomes very slow.