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1.4 Numerical Example and Programming Notes     continued...

The contour plot of Rosenbrock's function for n=2 is shown in Figure 3. In general, contour maps show surfaces in the -dimensional space defined by where is a constant. For n=2, plane curves correspond to various values of . We see from the figure that the minimum point (dark circle) is at , where . The gradient components of this function are given by

and the Hessian is the block diagonal matrix with entries

(These formulas are given in a form most efficient for programming.) For n=2, the two eigenvalues of the Hessian at the minimum are and , and thus the condition number . The function contours, whose axes lengths are proportional to the inverse of the eigenvalues, are thus quite elongated near the minimum (see Figure 3).