5.9 Comparison of Elimination Methods 271
Example 5.8 Minimize the function
f (x)= 0. 65 −[0. 75 /( 1 +x^2 ) ]− 0. 65 xtan−^1 ( 1 /x)
using the golden section method withn=6.
SOLUTION The locations of the first two experiments are defined by L∗ 2 =
0. 382 L 0 = ( 0. 382 )( 3. 0 )= 1 .1460. Thusx 1 = 1. 1 460 andx 2 = 3. 0 − 1. 1460 = 1. 8540
with f 1 = f(x 1 ) =− 0 .208654 and f 2 = f(x 2 ) =− 0 .115124. Since f 1 < f 2 , we
delete the interval [x 2 , 3.0] based on the assumption of unimodality and obtain the ne w
interval of uncertainty asL 2 = 0,[ x 2 ] =[0.0, 1.8540]. The third experiment is placed
atx 3 = 0 +(x 2 −x 1 ) = 1. 8540 − 1. 1460 = 0 .7080. Sincef 3 = − 0. 2 88943 is smaller
thanf 1 = − 0. 2 08654, we delete the interval [x 1 , x 2 ] and obtain the new interval of
uncertainty as [0.0,x 1 ] =[0.0, 1.1460]. The position of the next experiment is given
byx 4 = 0 +(x 1 −x 3 ) = 1. 1460 − 0. 7080 = 0 .4380 withf 4 = − 0. 3 08951.
Sincef 4 < f 3 , we delete [x 3 , x 1 ] and obtain the new interval of uncertainty as [0,
x 3 ]=[0.0, 0.7080]. The next experiment is placed atx 5 = 0 +(x 3 −x 4 ) = 0. 7080 −
0. 4380 = 0 .2700. Sincef 5 = − 0. 2 78434 is larger thanf 4 = − 0. 3 08951, we delete the
interval [0,x 5 ] and obtain the new interval of uncertainty as [x 5 , x 3 ] 0.2700, 0.7080].=[
The final experiment is placed atx 6 =x 5 + (x 3 −x 4 ) = 0. 2700 +( 0. 7080 − 0. 4380 )=
0 .5400 withf 6 = − 0. 3 08234. Sincef 6 >f 4 , we delete the interval [x 6 , x 3 ] and obtain
the final interval of uncertainty as [x 5 , x 6 ] =[0.2700, 0.5400]. Note that this final
interval of uncertainty is slightly larger than the one found in the Fibonacci method,
[0.461540, 0.230770]. The ratio of the final to the initial interval of uncertainty in the
present case is
L 6
L 0
=
0. 5400 − 0. 2700
3. 0
=
0. 27
3. 0
= 0. 09
5.9 Comparison of Elimination Methods
The efficiency of an elimination method can be measured in terms of the ratio of the
final and the initial intervals of uncertainty,Ln/L 0. The values of this ratio achieved
in various methods for a specified number of experiments (n=5 andn=10) are
compared in Table 5.3. It can be seen that the Fibonacci method is the most effi-
cient method, followed by the golden section method, in reducing the interval of
uncertainty.
A similar observation can be made by considering the number of experiments (or
function evaluations) needed to achieve a specified accuracy in various methods. The
results are compared in Table 5.4 for maximum permissible errors of 0.1 and 0.01. It
can be seen that to achieve any specified accuracy, the Fibonacci method requires the
least number of experiments, followed by the golden section method.
INTERPOLATION METHODS
The interpolation methods were originally developed as one-dimensional searches
within multivariable optimization techniques, and are generally more efficient than
Fibonacci-type approaches. The aim of all the one-dimensional minimization methods
