5.8 Golden Section Method 267
SOLUTION Heren=6 andL 0 = 3. 0 , which yield
L∗ 2 =
Fn− 2
Fn
L 0 =
5
13
( 3. 0 )= 1. 153846
Thus the positions of the first two experiments are given byx 1 = 1. 1 53846 and
x 2 = 3. 0 − 1. 153846 = 1 .846154 withf 1 = f(x 1 ) =− 0 .207270 and f 2 = f(x 2 )=
− 0. 1 15843. Sincef 1 is less thanf 2 , we can delete the interval [x 2 , 3.0] by using
the unimodality assumption (Fig. 5.10a). The third experiment is placed atx 3 = 0 +
(x 2 −x 1 )= 1. 846154 − 1. 153846 = 0 .692308, with the corresponding function value
off 3 = − 0. 2 91364.
Sincef 1 >f 3 , we delete the interval [x 1 , x 2 ] (Fig. 5.10b). The next experiment
is located at x 4 = 0 +(x 1 −x 3 ) = 1. 153846 − 0. 692308 = 0 .461538 with f 4 =
− 0. 3 09811. Nothing thatf 4 < f 3 , we delete the interval [x 3 , x 1 ] (Fig. 5.10c). The
location of the next experiment can be obtained asx 5 = 0 +(x 3 −x 4 ) = 0. 692308 −
0. 461538 = 0 .230770 with the corresponding objective function value of f 5 =
− 0. 2 63678. Sincef 5 >f 4 , we delete the interval [0,x 5 ] (Fig. 5.10d). The final exper-
iment is positioned at x 6 =x 5 + (x 3 −x 4 ) = 0. 230770 +( 0. 692308 − 0. 461538 )=
0 .461540 withf 6 = − 0. 3 09810. (Note that, theoretically, the value ofx 6 should be
same as that ofx 4 ; however, it is slightly different fromx 4 , due to round-off error).
Sincef 6 >f 4 , we delete the interval [x 6 , x 3 ] and obtain the final interval of uncer-
taintyasL 6 =[x 5 , x 6 ] =[0.230770, 0.461540] (Fig. 5.10e). The ratio of the final to
the initial interval of uncertainty is
L 6
L 0
=
0. 461540 − 0. 230770
3. 0
= 0. 076923
This value can be compared with Eq. (5.15), which states that ifnexperiments (n=6)
are planned, a resolution no finer than 1/Fn= 1 /F 6 = 131 = 0. 0 76923 can be expected
from the method.
5.8 Golden Section Method
Thegolden section methodis same as the Fibonacci method except that in the Fibonacci
method the total number of experiments to be conducted has to be specified before
beginning the calculation, whereas this is not required in the golden section method.
In the Fibonacci method, the location of the first two experiments is determined by
the total number of experiments,N. In the golden section method we start with the
assumption that we are going to conduct a large number of experiments. Of course,
the total number of experiments can be decided during the computation.
The intervals of uncertainty remaining at the end of different number of experiments
can be computed as follows:
L 2 = iml
N →∞
FN− 1
FN
L 0 (5.17)
L 3 = iml
N →∞
FN− 2
FN
L 0 = iml
N →∞
FN− 2
FN− 1
FN− 1
FN
L 0
≃ iml
N →∞
(
FN− 1
FN
) 2
L 0 (5.18)
