Engineering Optimization: Theory and Practice, Fourth Edition

5.12 Direct Root Methods 291

f′(l)

l

A = li

li+ 2

li+ 1

l*

Figure 5.19 Iterative process of the secant method.

the secant method can also be considered as a quasi-Newton method. It can also be
considered as a form of elimination technique since part of the interval, (A, λi+ 1 ) ni
Fig. 5.19, is eliminated in every iteration. The iterative process can be implemented by
using the following step-by-step procedure.

1.Setλ 1 = A= 0 and evaluatef′(A) .The value off′(A) will be negative.

Assume an initial trial step lengtht 0. Set i= 1.

2.Evaluatef′(t 0 ).

3 .Iff′(t 0 ) < 0,setA=λi=t 0 ,f′(A)=f′(t 0 ) new, t 0 = 2 t 0 , and go to step 2.

4 .Iff′(t 0 )≥ 0 , setB=t 0 ,f′(B)=f′(t 0 ) and go to step 5.,

5.Find the new approximate solution of the problem as

λi+ 1 =A−

f′(A)(B −A)

f′(B)−f′(A)

(5.74)

6.Test for convergence:

|f′(λi+ 1 )|≤ε (5.75)

whereεis a small quantity. If Eq. (5.75) is satisfied, takeλ∗≈λi+ 1 and stop

the procedure. Otherwise, go to step 7.

7.Iff′(λi+ 1 )≥ 0 , set newB=λi+ 1 ,f′(B)=f′(λi+ 1 ) ,i=i+ 1 , and go to

step 5.

8.Iff′(λi+ 1 ) < 0 , set newA=λi+ 1 ,f′(A)=f′(λi+ 1 ) ,i=i+ 1 , and go to

step 5.