Multidimensional Generalisations
Two variants are commonly encountered. One is the sum of uncoupled 2D Rosenbrock problems,
![f(\mathbf{x}) = f(x_1, x_2, \dots, x_N) = \sum_{i=1}^{N/2} \left[100(x_{2i-1}^2 - x_{2i})^2
+ (x_{2i-1} - 1)^2 \right].](http://upload.wikimedia.org/math/1/9/2/192457688ec048767a93e8c04d519d90.png)
This variant is only defined for even and has predictably simple solutions.
A more involved variant is
This variant has been shown to have exactly one minimum for (at ) and exactly two minim for -- the global minimum of all ones and a local minimum near . This result is obtained by setting the gradient of the function equal to zero, noticing that the resulting equation is a rational function of . For small the polynomials can be determined exactly and Sturm's theorem can be used to determine the number of real roots, while the roots can be bounded in the region of . For larger this method breaks down due to the size of the coefficients involved.
Read more about this topic: Rosenbrock Function