and F3, F3 is greater and Xs replaces Xu as the upper bound. This process continues
iteratively until the bounds are narrowed to a desired tolerance.
For this particular method the golden section is used as a sequence for dividing the
interval of the interior points to find the minimum value of the function, F, with as few
function evaluations as possible. The golden section ratio is .38197 or 38%. This
iterative procedure can be applied once a convergence criterion has been determined; this
criterion will indicate when the process has converged to an acceptable solution.
An initial interval of uncertainty has been defined, as XU-XL and it may be desired to
reduce the interval by a fraction, E, of the initial interval or by a magnitude Ax. Eis
referred to as a local tolerance and Ax as an absolute tolerance because it is independent
of the initial interval. The relative tolerance can be defined as follows:
E = (3.1)
X XL
This tolerance, E, can be converted to a maximum number of function evaluations in
addition to the three required to evaluate FL, F1, and Fu. This can be done because the
interval is reduced by the golden ratio, r(38%), for each function evaluation. Therefore,
for a specified tolerance, E:
S= (1- )(N3) (3.2)
where N is the total number of evaluations. Solving the equation for N yields:
In c
N= n +3= -2.078*lne+3 (3.3)
In(1 r)
Choosing the value of E is left to the programmer. For example, if it is desired to reduce
the interval to 2% of the initial interval (E=0.02) then N= 8.13+3 = 11.13->12 function