Engineering Optimization: Theory and Practice, Fourth Edition

270 Nonlinear Programming I: One-Dimensional Minimization Methods

Eq. (5.21) can be expressed as

γ≃

1

γ

+ 1

that is,

γ^2 −γ− 1 = 0 (5.23)

This gives the rootγ= 1 .618, and hence Eq. (5.19) yields

Lk=

(

1

γ

)k− 1

L 0 = ( 0. 618 )k−^1 L 0 (5.24)

InEq. (5.18) the ratiosFN− 2 /FN− 1 andFN− 1 /FNhave been taken to be same

for large values ofN. The validity of this assumption can be seen from the following

table:

Value ofN 2 3 4 5 6 7 8 9 10 ∞

Ratio

FN− 1

FN

0.5 0.667 0.6 0.625 0.6156 0.619 0.6177 0.6181 0.6184 0.618

The ratioγ has a historical background. Ancient Greek architects believed that a

building having the sidesdandbsatisfying the relation

d+b

d

=

d

b

=γ (5.25)

would have the most pleasing properties (Fig. 5.11). The origin of the name,golden

section method, can also be traced to the Euclid’s geometry. In Euclid’s geometry,

when a line segment is divided into two unequal parts so that the ratio of the whole to

the larger part is equal to the ratio of the larger to the smaller, the division is called

the golden section and the ratio is called the golden mean.

Procedure. The procedure is same as the Fibonacci method except that the location

of the first two experiments is defined by

L∗ 2 =

FN− 2

FN

L 0 =

FN− 2

FN− 1

FN− 1

FN

L 0 =

L 0

γ^2

= 0. 382 L 0 (5.26)

The desired accuracy can be specified to stop the procedure.

Figure 5.11 Rectangular building of sidesbandd.