35.2 Caveats

Note that minimization functions can only search for one minimum at a time. When there are several minima in the search area, the first minimum to be found will be returned; however it is difficult to predict which of the minima this will be. In most cases, no error will be reported if you try to find a minimum in an area where there is more than one.

With all minimization algorithms it can be difficult to determine the location of the minimum to full numerical precision. The behavior of the function in the region of the minimum x^* can be approximated by a Taylor expansion,

y = f(x^*) + (1/2) f''(x^*) (x - x^*)^2

and the second term of this expansion can be lost when added to the first term at finite precision. This magnifies the error in locating x^*, making it proportional to \sqrt \epsilon (where \epsilon is the relative accuracy of the floating point numbers). For functions with higher order minima, such as x^4, the magnification of the error is correspondingly worse. The best that can be achieved is to converge to the limit of numerical accuracy in the function values, rather than the location of the minimum itself.